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# how many equivalence classes are there

Use MathJax to format equations. Must a creature with less than 30 feet of movement dash when affected by Symbol's Fear effect? So the set of elements equal to (related to) a, namely the equivalence class of a is {a,b,c,d,e}. The minimum sum is 2 (from (1,1)), ant the max is 12 (from (6,6)). It's easy to see, the we will also have all the numbers in a range [2,12]. So look for all the pairs that satisfy this condition. Thanks for contributing an answer to Mathematics Stack Exchange! For the equivalence relation on Z, (mod 2), there are two equiv-alence classes, [0], which is the set of even integers, and [1], which is the set of odd integers. There are many Sudoku variants, partially characterized by size (N), ... Equivalence class identification and linkage uses the lowest ID within the class. But as we have seen, there are really only three distinct equivalence classes. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The answer is provided in the solution. Not really a programming question. You're told $R$ contains those 4 pairs; you're not meant to conclude that $R$ contains only those 4 pairs. Let $S \subseteq \mathbb{Z}$, and define a relation $R$ on $S \times S$ by, $$(m, n)R(s, t) \quad \text{ if and only if } \quad m + n = s + t$$. So in particular, since it contains $(a, d)$, it must also contain $(d, a)$, since it is symmetric. More generally, given a positive integer n, the equivalence classes for … Since you're told that $R$ is an equivalence relation, two elements must be related if there is any path between them. The bothering part is how to write an efficient and non-naive "equal" operator. When there is a strong need to avoid redundancy. Note that $(m,n)R(s,t)\iff f(m,n)=f(s,t)$ where function $f:S\times S\to\mathbb Z$ is prescribed by $(m,n)\to m+n$. So the set of elements equal to (related to) $a$, namely the equivalence class of $a$ is $\{a,b,c,d,e\}$. So the set of elements equal to (related to) a, namely the equivalence class of a is {a,b,c,d,e}. So the reduced echelon form is a canonical form [2] for row equivalence: the reduced echelon form matrices are representatives of the classes. $a=a$, since anything is equal to itself (i.e., by reflexivity). If I choose one of the equivalence classes and give a DFA for the class, then the DFA is a "subDFA" of M, with states from the class. How many equivalence classes are there Describe each of them the best you can from MATH 440 at University of Southern California Your list of the elements of $R$ is incomplete; $R$ is, in fact. There are as many equivalence classes as there are which of the following? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The fact that if a language L is regular if it has a finite set of equivalence classes is proven in chapter 7. Use MathJax to format equations. (Did I miss any?). We have proved there is one and only one reduced echelon form matrix in each row equivalence class. University Math Help. You're told that, The equivalence class of, say, $a$ will be all the elements equal to $a$ so we can argue. If we use a mapping x->1, y->1, z->1, w->2, h->2 for the equivalence class of S, one has to consider the mapping x->10, y->10, z->10, w->20, h->20 as the same equivalence class. So, the partition has to be done … Here the ''connection'' is made through certain walks along the directions of the arrows. share | improve this question | follow | edited Aug 20 '14 at 0:11. templatetypedef. For that, we want $c+d=6$. Can playing an opening that violates many opening principles be bad for positional understanding? 5. In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = g –1 ag.This is an equivalence relation whose equivalence classes are called conjugacy classes.. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. Why was Warnock's election called while Ossof's wasn't? Last updated at Jan. 28, 2020 by Teachoo. What are the equivalence classes? Thanks for contributing an answer to Mathematics Stack Exchange! 1. c.) 6. d.) Infinite. Example 5 Example 6 Important . How many equivalence classes does this relation have? There are currently four math class options, which are scheduled to have their last assessments in November 2020: Mathematical Studies SL (known as Math Studies) Mathematics SL (available online) Mathematics HL (available online) Further Mathematics HL (final exam in May 2020 canceled due to coronavirus) The IB does not do a good job of distinguishing between the four courses, but from what … This is equivalent to (a/b) and (c/d) being equal if ad-bc=0. Will a divorce affect my co-signed vehicle? You are TOLD that it is an equivalence relation, so you can assume the transitive, symmetric, and reflexive property all hold, and therefore any consequence of these properties and the given ordered pairs being in the relation. Similarly, it must also be transitive…. The problmm does not state that this is the entire relationship. The second is quite difficult and I … Aren't they both on the same ballot? For the set $S$ there are $2^6$ subsets. • The resulting automaton is minimal for this language. He goes on to say that in this case, the equivalence class of A consists of all the integers of the form a+2m, where m=0,+-1,+-2,+3...; and in this example there are only two distinct equivalence classes,namely cl(0) and cl(1). Your question doesn't seem to quite make sense. Corresponding to every equivalent relation there is an equivalent partitioning of the set.. Can I hang this heavy and deep cabinet on this wall safely? Indeed, you're told $R$ is an equivalence relation, so it must be reflexive, so it must have, for example, $(a,a)$; it must be symmetric, so, for example, since it has $(a,d)$, it must have $(d,a)$; it must be transitive, so, for example, since it has $(b,c)$ and $(c,e)$, it must have $(b,e)$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Making statements based on opinion; back them up with references or personal experience. Gwen, atb+ c = 100 a , b, c EN ( a , b, came negative integers) ( a) No. There are ve distinct equivalence classes, modulo 5: [0];[1];[2];[3], and [4]. For simplicity I'm taking mothers to represent each of the kids' parents. (Select all that apply.) How many equivalence classes does this set have? multiplicative identity, there is no n such that 2 n = 1. Monopoly) for some time, knows that every roll from $2$ to $12$ can be made with a pair of dice... $$[(1,5)]=\{(1,5), (5,1), (3,3), (2,4), (4,2)\}.$$, $$[(1,1)]=\{(1,1)\} \qquad \text{and} \qquad [(5,6)]=\{(5,6), (6,5)\}.$$. Hope this helps! It's easy to see, the we will also have all the numbers in a range [2,12]. How many equivalent classes are there? For example eRa and cRe, you can conclude aRc. So the number of equivalence classes equals the cardinality of the image of $f$. We have proved there is one and only one reduced echelon form matrix in each row equivalence class. 2. symmetric (∀x,y if xRy then yRx): every e… Your question doesn’t make sense. @HHH You remembered to include $(1,1)$ but you forgot about $(6,6)$. (There are infinitely many matrices in the pictured class, but we've only got room to show two.) Median response time is 34 minutes and may be longer for new subjects. Problem 3. ... element of A belongs to exactly one equivalence class. The equivalence classes of this relation are the $$A_i$$ sets. It is beneficial for two cases: When exhaustive testing is required. B. distinct integers. Example 48 - Chapter 1 Class 12 Relation and Functions. Origin of “Good books are the warehouses of ideas”, attributed to H. G. Wells on commemorative £2 coin? Misc 17 - Chapter 1 Class 12 Relation and Functions. The… Subscribe to our Youtube Channel - https://you.tube/teachoo. Beethoven Piano Concerto No. The algebra of continuous functions on Cantor set. check out the Wikipedia page on # of connected components in a graph, Equivalence relation and its equivalence classes. Each equivalence class of this relation will consist of a collection of subsets of X that all have the same cardinality as one another. How can I identify how many equivalence classes are there? GeoPandas: How to convert DataFrame to GeoDataFrame with Polygon? 4.4 Threshold Values and Equivalence Classes. How many equivalent classes there are? Equivalence class testing is a black box software testing technique that divides function variable ranges into classes/subsets that are disjoint. Every class has a feature that every pair in this class has the same sum. C. distinct real numbers. The equivalence class for [6], if there were such a thing, would be the same as the equivalence class for [0]. We verify that R is re±exive, symmetric, and transitive. bcmwl-kernel-source broken on kernel: 5.8.0-34-generic. Thus the equivalence classes are such as {1/2, 2/4, 3/6, … } {2/3, 4/6, 6/9, … } A rational number is then an equivalence class. Hope that helps! The important point is that if √2 and √3 are thought of as particular infinite sequences of digits there is no way to define (√2 + √3). (b,b),(e,a),(a,e),(e,e),(c,e),(e,c)\}$$MathJax reference. In the above example, for instance, the class of 0, [0], may also be called the class of 5, [5], or the class of 10, [ 10]. It is only representated by its lowest or reduced form. The maximum number of equivalence relations on the set A = {1, 2, 3} are. Add comment More. How many equivalence We never learned to make diagrams of equivalence classes, sadly. Proof: We will show that every a ∈ A belongs to at least one equivalence class and to at most one equivalence class. That means, that you have 11 classes. How many equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) are there in all ? So for example [(1,5)] and [(5,1)] will have the same elements, so overall I have the sum of 2,3,4,...,11 , hence 10 equivalence classes ? … We can draw a binary relation A on R as a graph, with a vertex for each element of A and an arrow for each pair in R. For example, the following diagram represents the relation {(a,b),(b,e),(b,f),(c,d),(g,h),(h,g),(g,g)}: Using these diagrams, we can describe the three equivalence relation properties visually: 1. reflexive (∀x,xRx): every node should have a self-loop. Using the notation from the definition, they are: = {a ∈ Z | a ≡ 0 (mod 3)}, 2 classes: must have 2 elements in 1 and the last in the other. Every class has a feature that every pair in this class has the same sum.$$[(1,1)]=\{(1,1)\} \qquad \text{and} \qquad [(5,6)]=\{(5,6), (6,5)\}.$$. How do I find complex values that satisfy multiple inequalities?$$[(1,5)]=\{(1,5), (5,1), (3,3), (2,4), (4,2)\}. Well, R has to be transitive and $\,bRc\,\,,\,cRe\,$, so... You only know that ${(a,d),(b,c),(e,a),(c,e)}\subset R$. Is it normal to need to replace my brakes every few months? How many different equivalence classes of R are there? In class 11 and class 12, we have studied the important ideas which are covered in the relations and function. Asset classes and asset class categories are often mixed together. Origin of “Good books are the warehouses of ideas”, attributed to H. G. Wells on commemorative £2 coin? Why is an early e5 against a Yugoslav setup evaluated at +2.6 according to Stockfish? Let X = {1,2,3,4,5} and Y = {3,4}. without explicitly listing the partition of $S × S$. Let Z=nZ be the set of equivalence classes of Z under ˘ n. Then every x 6= 0 2Z=nZ admits an inverse under multiplication if and only if n is prime. Discrete Structures Objective type Questions and Answers. You need to apply the rules for equivalence relationships to extrapolate enough relationships. Verify that R is an equivalence relation and describe the equivalence class E 5. 315k 87 87 gold badges 785 785 silver badges 969 969 bronze badges. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Equivalence relation and equivalence classes, Equivalence relation - Equilavence classes explanation, Compute equivalence classes of equivalence relation, How many equivalence classes does a set have. This is a result of an important theorem, given here without proof. How many distinct equivalence classes are there? Application of the (2×6 2) B2,3 symmetry permutations produces 36288 (28×6 4) equivalence classes, each of size 72. 27 1 1 silver badge 7 7 bronze badges. That means, that you have 11 classes. The problem is: how many equivalence classes are there? the equivalence classes of R form a partition of the set S. More interesting is the fact that the converse of this statement is true. How many classes does the equivalence relation partition the set? Describe the equivalence classes of $R$ You need to work on the transitivity to get the answer right. (3,4). For an element a ∈ A, let [a] denote the set {b ∈ A given aRb}. Comparing method of differentiation in variational quantum circuit. The bothering part is how to write an efficient and non-naive "equal" operator. It is beneficial for two cases: When exhaustive testing is required. The main effect of there being three species in solution at this point is to buffer the pH around the second equivalence point. The equivalence class could equally well be represented by any other member. CRL over HTTPS: is it really a bad practice? Equivalence Partitioning. Instead of trying to write down all the pairs in $R$ in a list, it is better to draw a diagram: Each line connects two elements that you explicitly know are related. Figure out what else it has to have, and then we can talk. Asking for help, clarification, or responding to other answers. Solution for How many (distinct) equivalence classes does the relation R- (1.1), (2.2), (3,3)(4.4), (1.2), (2,1). Is it possible to assign value to set (not setx) value %path% on Windows 10? The partition forms the equivalence relation $$(a,b)\in R$$ iff there is an $$i$$ such that $$a,b\in A_i$$. Anyone who’s played a game with dice (e.g. So for finding the equivalence class of $(1,5)$ we ask ourselves: what are all other elements $(c,d) \in S \times S$ such that $(1,5) \sim (c,d)$? What does "Drive Friendly -- The Texas Way" mean? (There are infinitely many matrices in the pictured class, but we've only got room to show two.) Figure out what else it has to have, and then we can talk. Misc 3 Important Not in Syllabus - CBSE Exams 2021. Theorem 3.6: Let F be any partition of the set S. Define a relation on S by x R y iff there is a set in F which contains both x and y. How many equivalence classes does $R$ have? MHF Hall of Honor. Equivalence class testing is a black box software testing technique that divides function variable ranges into classes/subsets that are disjoint. Forums. This plays an essential role in many situations, such as when selecting input parameters. See answer pereTow7tmat0ta is waiting for your help. So it is larger than you thought it was. In other words, in this case there is just one equivalence class, everything. Notice that there are 2 ''connnected'' components, one containing elements 4 and 8 and the other, elements 2, 6 and 10. There are ve distinct equivalence classes, modulo 5: [0];[1];[2];[3], and [4]. In class 11 and class 12, we have studied the important ideas which are covered in the relations and function. the class $[(1,5)]$ can be associated to the sum $6$ and class $[(1,1)]$ can be associated with the sum $2$ and so on. Brainly User Brainly User Figure out what else it has to have, and then we can talk. This set is the equivalence class of a under R and we call a a representative of the set [a]. a=a, since anything is equal to itself (i.e., by reflexivity). You're told that $R$ is an equivalence relation. How does Shutterstock keep getting my latest debit card number? $a=c$ since $c=e$ by (4), $e=a$ by (3), and transitivity, $a=b$ since $a=c$, and $c=b$ by (2) and transitivity again. How many distinct equivalence classes are there? Asking for help, clarification, or responding to other answers. Is the Gelatinous ice cube familar official? In general if $f:X\to Y$ is some function then the relation $\sim$ on $X$ defined by $a\sim b\iff f(a)=f(b)$ is an equivalence relation. Next: Misc 18→ Chapter 1 Class 12 Relation and Functions; Serial order wise; Miscellaneous. On the other hand, it is elementary to construct a convergent sequence for (√2 + √3) from any of the convergent sequences for √2 and √3. – lhf May 20 '14 at 11:49. It looks as if your possibilities are: {1,2} {1,2,3} {1,2,4} {1,2,3,4} >c) How many equivalence classes are there? One important heuristic for effective testing is to increasingly test along the boundaries of a permitted value range because these are the areas where most errors occur. Add your answer and earn points. Equivalence Class Testing This week we turn to equivalence class testing. What you need to do is make deductions like this: If we know that $aRd$, then we must have $dRa$ since we are told that $R$ is an equivalence relation, and hence is symmetric. Equivalence relations have equivalence classes and Rubik’s cube is not an equivalence relation. What happens to a Chain lighting with invalid primary target and valid secondary targets? They are traded on stock exchanges such as the NYSE or NASDAQ. Theorem 3.6: Let F be any partition of the set S. Define a relation on S by x R y iff there is a set in F which contains both x and y. In effect, you'd be writing an integer (say 18) as "2 and 6/6" instead of as "3 and 0/6". So the number of distinct classes is the number of distinct sums. Suppose you are from a joint family, or you all cousins have gathered at some cousin's house. It only takes a minute to sign up. If you keep doing things like that, you'll soon see the answer. Colleagues don't congratulate me or cheer me on, when I do good work? There are numerous examples of categorical equivalences from many areas of mathematics. Answer Choices: A. distinct horizontal lines in the plane. rev 2021.1.7.38271, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. The answer to (Right? Consider the set $S = \{1, 2, 3, 4, 5, 6\}$. How many equivalence classes in the equivalence relation. THANK YOU & STAY SAFE Explanation: ATTACHMENT PREVIEW Download attachment WhatsApp Image 2020-05-03 at 9.38.15 AM.jpeg. How many equivalence classes does the following equivalence relation have? Misc 1 Not in Syllabus - CBSE Exams 2021. How many equivalence classes does $R$ have? Louis Alain P. asked • 11/04/20 There are as many equivalence classes as there are which of the following? *Response times vary by subject and question complexity. There are as many equivalence classes as there are which of the following? Justify your answer. the equivalence classes of R form a partition of the set S. More interesting is the fact that the converse of this statement is true. How many equivalence classes are there for the equivalence relation R = { (a, b) | a ≡ b (mod 6) } on the set of integers? My try: Suppose $R$ is an equivalence relation on $A$. Last updated at Jan. 28, 2020 by Teachoo. For the equivalence relation on Z, (mod 2), there are two equiv-alence classes, [0], which is the set of even integers, and [1], which is the set of odd integers. If construct the minimal DFA M' equivalent to M, then all the equivalent states belong to one class, and number of equivalence classes is the number of states in M'. MA1100 e.g. is Wrong. Thread starter godelproof; Start date Jun 26, 2011; Tags classes equivalent; Home. This is equivalent to (a/b) and (c/d) being equal if ad-bc=0. ( or routers ) defined subnet every equivalent relation there is just one equivalence class of a collection of how many equivalence classes are there. Of connected components in a range [ 2,12 ] Shutterstock keep getting my debit. Evaluated at +2.6 according to Stockfish that satisfy multiple inequalities doing things like that, agree! Professionals in related fields misc 17 - Chapter 1 class 12 relation and ;! Have many di erent names sets and natural numbers, but it was warehouses of ”. R are there on a set with five elements 10 15 25 30 A. distinct lines. 'S Fear effect ] denote the set $S × S$ there.. Than you thought it was suggested to me by a concrete problem chapters like probability,,. N such that 2 n = 1 5 the following the simple way put... { x, Y, z, w, h }... element of a collection of subsets x. & STAY SAFE Explanation: ATTACHMENT PREVIEW Download ATTACHMENT WhatsApp image 2020-05-03 at 9.38.15 AM.jpeg relationships to extrapolate enough.... Identify how many equivalence classes of this relation is reflexive, symmetric and. Related fields class 11 and class 12 relation and its equivalence classes, that... * Response times vary by subject and question complexity mathematics Stack Exchange Inc ; user contributions licensed cc. And deep cabinet on this wall safely than 30 feet of movement dash when affected Symbol! The fact that if a language L is regular if it has to have, and so on species solution. Is, in this case there is one and only one reduced echelon form matrix in each equivalence... Answer site for people studying math at any level and professionals in related fields ) defined?! Let R be an equivalence class testing this week we turn to equivalence.... For from MA 1100 at National University of Singapore 5 the last examples above illustrate a very property. Badges 785 785 silver badges 969 969 bronze badges H. G. Wells on commemorative £2 coin: notes... Have studied the important ideas which are covered in the plane clicking “ Post your answer how many equivalence classes are there, attributed H.. Occurs when HTyr ] is a question and answer site for people studying math at any level and in! Avoid redundancy explain how to write an efficient and non-naive  equal '' operator: //you.tube/teachoo to our Youtube -. Of R as the NYSE or NASDAQ such that 2 n = 1 5 gives rise to Jan.,! Is re & pm ; exive, symmetric, and then we can use x denote... On this wall safely a canonical form for row equivalence class may have many di erent.... Get more help from Chegg is beneficial for two cases: when exhaustive testing is a question and answer for! Size 4 0 ) and how many equivalence classes are there ( 2 ) B2,3 symmetry permutations produces 36288 ( 28×6 4 equivalence... Geodataframe with Polygon integration, and in some cases a negative correlation, different. Some argument about exactly how many equivalence classes does $R$ is early. Back them up with references or personal experience 12 ( from ( 6,6 $. Relation there is just one equivalence class of this relation is not,... What does  Drive Friendly -- the Texas way '' mean cl ( 0 ) and cl ( )... Level and professionals in related fields 's easy to see, the partition has to within! While Ossof 's was n't in the relations and function different classes of R the! Horizontal lines in the pictured class, everything, how many equivalence classes there... Three different equivalence classes as there are infinitely many matrices in the Chernobyl series that ended in the relations Functions. 30 feet of movement dash when affected by Symbol 's Fear effect strong... ) have 1 silver badge 7 7 bronze badges the main effect of there three...$ bRc $,$ R $have the pictured class, everything di erent names pairs that satisfy condition. To cast this problem in more familiar terms every class has a feature that every a ∈ a,,...$ ( 1,1 ) $not an equivalence relation of all Functions of random variables implying independence Teachoo... ] denote the set$ S $∈ a belongs to at least one equivalence class may have di. }$ œ a, we have studied the important ideas which are covered in the.. An element a ∈ a belongs to at most one equivalence class in Java, 2003 with?... A cardinal number so if in partition we have seen, there is no n such that 2 =... Legend from an attribute in each layer in QGIS joint family, or 3 equivalence classes and ’! Functions of random variables implying independence are asking it correctly element a ∈ a given aRb } every. Dash when affected by Symbol 's Fear effect the fact that if a language L regular. In Syllabus - CBSE Exams 2021 - Chapter 1 class 12 relation and ;... Paste this URL into your RSS reader identity, there is an equivalence,... This set is the equivalence relation partition the set a = { 1 2. That number of distinct classes is the equivalence classes of assets there are which the! Partition has to be done … there is usually very little correlation, in... Minimal deterministic finite automaton from them R $is an equivalence relation and Functions by clicking “ your... Me on, when I do Good work x œ a, b, c d! Nov 8, 2018 in mathematics by Afreen ( 30.7k points ) relations and Functions here, an class... Any student x œ a, b, c, d, e\ }$ to represent each of arrows... That an equivalence class for each natural number corresponding to bit strings with that number of equivalence classes there! User contributions licensed under cc by-sa see our tips on writing great answers as many classes! In some cases a negative correlation, between different asset classes and asset class categories are mixed! Random variables implying independence way, each of size 72 but not every element for element. Many areas of mathematics the pairs that satisfy multiple inequalities to implement equivalence class could equally well represented. Learned to make diagrams of equivalence classes, each of size 72 @ HHH remembered... Room to show two. it has to have, and so on an!: you are from a to a that the relation is an relation... And Functions ; Serial order wise ; Miscellaneous first question can seem very strange, but it was 're that. Second is quite difficult and I … Johannes Link, in fact the pH around the second is quite and... Many different equivalence relations with exactly three different equivalence classes, I Good. Suggested to me by a concrete problem members are elements of $R$ have }. Each one into a state how many equivalence classes are there accepting states and how to convert DataFrame GeoDataFrame... ) equivalence classes does $R$ is incomplete ; $R$ have from.... Family, or responding to other answers three species in solution at this point is to buffer the around. And how many different classes of R are there was n't opinion back. Layers in the other a canonical form for row equivalence class testing seem very strange, we. Is equal to itself ( i.e., by reflexivity ) class in Java called a cardinal number this. Graph, equivalence relation and its equivalence classes of R are there on a set with five 10! Is it possible to assign value to set ( not setx ) %! Relation have ; start date Jun 26, 2011 ; Tags classes equivalent ; Home multiple. Really only three distinct equivalence classes, namely that an equivalence involves demonstrating strong similarities between mathematical! An attribute in each layer in QGIS ; 2 ; first Prev 2 of 2 Go to.... Share  node_modules '' folder between webparts in paint seems to slowly getting longer, . For from MA 1100 at National University of Singapore 5 that it 's easy to see, the will! Classes having each class size = 1 5 [ x ] problem is: how many equivalence.... At National University of Singapore 5 cc by-sa { 3,4 } so, no of classes... Testing is a black box software testing technique that divides function variable ranges into classes/subsets that are disjoint there... Layer in QGIS over HTTPS: is it really a bad practice clicking Post! Warehouses of ideas ”, attributed to H. G. Wells on commemorative £2 coin to quite make.. Of $R$ have 2^6 \$ subsets such as when selecting input parameters attributed to H. Wells! Early e5 against a Yugoslav setup evaluated at +2.6 according to Stockfish elements 10 15 25 30 important ideas are! 'S Fear effect vibrational specra { 1,2,3,4,5 } and Y = { x, Y z. Relations and function the transitivity to get the remaining equivalence classes Ossof 's was n't a under and! } are a concrete problem the start state and accepting states and how many equivalence,...


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Mon, Mar 5, 7:20 p.m. - For diffraction, why does the intensity of light change based on the number maxima you are at?

Are you asking about a single slit? If so, intensity goes down because the phasors continue to wrap in on themselves, making each subsequent "max" more wrapped and thus lower total amplitude. Try playing around with the simulation (scroll down for a single slit) that is posted on the lecture 12 calendar page to see this effect. If you're talking about a diffraction grating, the answer is MUCH more complicated, and beyond the scope of what we've learned in PHYS212-it's something students learn about in our upper division Optics course!

Sun, Feb 18, 3:12 a.m. - How does the Earth have magnetic poles? Why are the magnetic poles on a magnet opposite of the magnetic poles on Earth? Why is the South Pole at the North Pole? Why didn't they just make it so the names were the same?

Geographic "North" was established before we knew anything about magnets. Magnetic "north" was so named because the north pole of a compass magnet is the side that points toward geographic north. The Earth generates a magnetic field in its spinning metallic liquid outer core. As we learned in Unit 1, the motion of charged particles (aka current) creates magnetic fields. In the Earth's outer core, heat escapes from the core via convection. These convective currents contain charged particles and thus create the Earth's magnetic field.

Tue, Feb 6, 11:11 p.m. - I'm really confused on chapter 20 #36. I started the problem by first calculating the force using F=qE Then I used the F value to calculate the acceleration using F=ma I know the initial and final velocities but I don't know how the distance fits in here

You actually don't want to calculate the acceleration. Since the electric field is uniform, the electric force is constant for the proton, so the work is $\vec{F} \cdot \Delta\vec{r}$. You can then use the work-kinetic energy theorem to find $\Delta\vec{r}$.

Tue, Feb 6, 8:02 p.m. - For the previous year exam, question #7, how do you determine that F(mag) points up?

The protons are traveling at a constant speed in a straight line. Thus, $\vec{a} = 0$ The electric field in this region points downward, which means that there is a downward electric force (since $\vec{F} = q\vec{E}$). Since there is no net force on the proton, the magnetic force must be balancing the electric force (to make $\sum F_y=0$). Thus, the magnetic force must point upwards.

Fri, Feb 2, 1:10 p.m. - How do you know when to use Biot-Savart or Amperes law?

Start by drawing the B-field. To use Ampere's Law you need to have a B-field with symmetry that allows you to simplify the circulation $\oint \vec{B} \cdot d\vec{l}$. This means having an amperian loop where each side of the loop either has a line integral of 0 ($\vec{B} \perp d\vec{l}$) or $\vec{B}=0$) or where $\vec{B}$ can be pulled out of the line integral (because it is constant on that particular segment of the loop). Biot-Savart should be used for everything else. And the special cases of a long straight wire or a single loop are fine for you to use if justified.

Sun, Feb 4, 6:00 p.m. - For a solenoid made out of a fixed amount of wire and a fixed amount of current is the magnetic field the same in the center regardless of the amount of turns?

For an ideal solenoid, the # of turns per meter and the current are what determine the magnetic field in the center of the solenoid. So, depending on how you wind the wire, you could get different B-fields.

Fri, Feb 2, 7:26 p.m. - For CH 26 question 32, can I ask a verification question. When it says that the B-field 1.2 cm from the wire's axis is 67 microT, does it mean that the entire wire has a radius of 1.2 cm? Or after I find the enclosed current in 1.2 cm, would I need to use the ratio to find an equation to express the current of the wire?

Assume that the wire has a radius $<<$ 1.2 cm.

Fri, Feb 2, 12:07 a.m. - I am really confused about CH 26 #31 (me again). The length is 2.2m. Dividing 2.2m by 2*pi*(.025m), where .025 m is the radius (.05 m diameter; 5 cm diameter given in the book). The answer is 14 (n = 14 coils). Please explain!

N = 14 coils but we don't have a length for the solenoid itself. $n$ should be the total # of coils divided by the length of the solenoid. But for problem #31, we don't have the length of the solenoid (the length of the wire is not the same thing)!

Mon, Jan 29, 12:53 p.m. - p. 481 Figure 26.23a (Wolfson). How did we get the direction for B? If I try placing my thumb along the bottom along the direction of I, doesn't my hand curl towards the opposite of the currently denoted direction of B?

For this figure, the magnetic field isn't created by the loop. So, the direction of $\vec{B}$ is given.

Thu, Feb 1, 6:25 p.m. - Isn't CH 26 #31 a solenoid? Why can I not use B = (mu not)*n*I?

It's not really a solenoid, because it doesn't have a length, so it is more just a case of multiple loops. In the equation for a solenoid, n, is the # of turns per unit length. For # 31, we don't know n!

Mon, Jan 29, 12:57 p.m. - Why does a dipole experience a net force in a nonuniform field? Is this because a uniform field (I am assuming magnetic) implies that the forces cancel one another out?

Yes! As we discussed in CT#1 in lecture05, the force on the right side of the loop is equal in magnitude but opposite in direction to the force on the left side of the loop. If the field weren't uniform, it would be possible for the magnitudes of the forces to be different because $I\vec{L}\times\vec{B}$ could have a different value of B for each side.

Tue, Jan 30, 6:24 p.m. - When talking about two parallel wires and their magnetic fields and forces, the reading quiz for Tuesday states that "the current in one wire produces a magnetic field at the other wire, and that causes a magnetic force on the current in the second wire)." How does the magnetic field from one wire cause a force on that wire towards the other wire? Is this found by using one of the right hand rules?

It helps to consider the two wires separately. Take a look at CT#5 from the lect05 notes. Let's just consider the top wire for a second. All by itself, this current carrying wire produces a magnetic field, and we could figure out the strength of this magnetic field for a distance, $y$, away from the wire. Now, let's add back in the bottom wire. This bottom wire is sitting in the magnetic field created by the top wire. And we can calculate the force on the bottom wire as $I_{bottom}\vec{L}\times\vec{B}$ where $\vec{B}$ is the magnetic field created by the top wire! This is sort of like charges, where we talked about source charges and test charges, but in this case we have source currents (the top wire) and test currents (the bottom wire).

Tue, Jan 30, 6:20 p.m. - How do you know which formula to use when trying to find the magnetic field? For example, when trying to calculate the magnetic field for ch.26 number 31, how would you know which equation to use? B=UoI/2a or dB=UoIsin(theta)/4pir^2? Or are you not supposed to use either of these? I'm a little (lot) lost..

For 26.31, the easiest way is to use $B_{loop} = \frac{mu_0 I}{2 a}$. This is the expression for the B-field due to a single loop of current. So you need to multiply it by the number of loops that you get if you wrap the given length of wire into loops of the specified radius (hint: one loop requires a length of wire equal to the circumference of the loop)

Mon, Jan 29, 12:41 p.m. - Wait, how does a permanent magnet have moving charge? It doesn't shock you...

We'll talk about this in the quantum mechanics unit, but essentially, electrons and protons have magnetic moments because they are rotating charges!

Sat, Jan 27, 6:11 p.m. - Let's say we're trying to calculate the potential difference across a wall. Should we be calculating for $|\Delta V|$ or $\Delta V$ (with or without absolute value)?

This depends on what the question is asking. In general we use $|\Delta V|$ when dealing with circuits, ohms law, electrical power etc. basically things where the direction of current or sign of $\Delta V$ doesn't make a difference. If, however, a question is asking about the direction of the electric field or how the energy of a charged particle will change as it moves across $\Delta V$, then the sign matters!

Sat, Jan 27, 9:05 a.m. - What is the relationship between energy and electric field and voltage and electric field?

For a case where the electric force is the only force we're dealing with: $q\Delta V_{AB} = \Delta U_{AB} = -W_E = -\int \vec{F} \bullet \vec{dr} = -\int q \vec{E} \bullet \vec{dr}$. If we're in a spatially uniform electric field: $q\Delta V_{AB} = \Delta U_{AB} = -q \vec{E} \bullet \Delta\vec{r}$

Sat, Jan 27, 9:14 a.m. - Is the whole point of A12a to see that no light is produced unless you complete the circuit with the silver wire? Isn't the nichrome wire just a short? Should it be giving extra resistance? I don't understand A12b.

for 12a) You are comparing two completed circuits. The first circuit has the two bulbs connected in series to the batteries. The second circuit has two bulbs and a length of nichrome wire connected in series to the batteries. The bulbs should glow in both cases, allowing you to compare the brightness. for 12b) the nichrome wire doesn't short out the circuit for 2 reasons: 1) nichrome has fairly high resistivity (compared to a normal wire) and 2) it is connected in series, so isn't providing a direct current path between the two terminals of the battery.

Sat, Jan 27, 11:36 p.m. - I am still VERY confused about work on versus work by. An electron is being slowed down. Is work on q*delta-V? Or work by? How about kinetic energy? Is W=delta-K (work-energy) where W is work on or work by? I am thoroughly confused. I asked about this on problem session on Friday, and I still can't seem to wrap my head around this. I really would appreciate some clarification (thank you in advance).

The work done on a charge by the electric force is equal to $\Delta K$, so, if the particle sped up, positive work was done *on* it by the electric force. When we write: $\Delta U_{AB} = -W_E = -\int \vec{F} \bullet \vec{dr} = -\int q \vec{E} \bullet \vec{dr}$, $W_E$ is the work done on the charge by the electric field.
So, let's take an example case. Let's consider a uniform electric field, and we're going to take an electron and a proton and move them along (in the same direction as) the electric field.

• If a proton moves along $\vec{E}$, then $\int q \vec{E} \bullet \vec{dr}$ is positive, $W_E$ is positive, and the proton will have sped up (makes sense) because the electric force did positive work on the proton.
• If an electron moves along $\vec{E}$, then $\int q \vec{E} \bullet \vec{dr}$ is negative, (because $q$ is negative), $W_E$ is negative, and the electron will have slowed down because the electric force did negative work on the proton.

Thu, Jan 25, 1:55 p.m. - When using the right hand rule for number A18, I get that the force due to the magnetic field is downwards, and I believe that the bar is moving to the left. Because F and R are perpendicular, why isn't the work done by the magnetic force equal to 0? I'm pretty sure it cannot be because otherwise the final speed of the bar would be equal to 0 by work-kinetic energy.

To get the direction of the force, you'll need to use $\vec{F} = I\vec{l} \times \vec{B}$. $\vec{l}$ for the bar points $\uparrow$ (the direction the current is flowing through the bar) and $\vec{B}$ points into the page, so the force is not downwards!

Sun, Jan 21, 2:30 p.m. - I am having problems with CH 20: 57. Could you point me in the right direction regarding how to replace r with a constant? Would it need to be replaced by a constant?

First off, there is improved wording for #57 posted on the homework sheet and the calendar page. That might clear things up. Remember that r needs to be expressed in only constants (like d, or L) and/or the integration variable. r should be the distance between point P (which for the improved wording is at x=d) and your tiny piece of charge $dq$ (which is at x=x!).

Sun, Jan 21, 12:27 p.m. - I don't understand how the electric force (exerted by the charge) causes the balloon to expand/contract? I KNOW that like charges repel. But WHY does this mean that the balloon is changing in size? This is what I am not able to understand force-wise.

Draw a balloon. Imagine that charge is uniformly distributed over the surface of the balloon. Pick any charge at random, let's call this charge Fred. If you think about the positions of all of the other like charges pushing on Fred, they will contribute a net force outward on Fred. Each charge on the balloon experiences such an outward force, thus the balloon expands a little bit.

Sat, Jan 20, 4:27 p.m. - Regarding CH 20 #55 (Wednesday's Assigned Problems), I'm a bit confused on how the signs work out for Coloumb's law. I end up getting q1 = q2 after solving the quadratic. But, my expression for the the force on q1 before has a negative sign because q1 and q2 have opposite signs (attracted to each other). But, the value of this expression is 2.5N [F before on q1] = -(k|q1||q2|)/r^2 = -(k|q1||q2|)/1^2 = -k|q1||q2|=2.5N -k|q1||q2|=2.5N k can't be negative (neither can the pairs of absolute value symbols); therefore, the left side has to be negative. However, 2.5N is positive. The two sides are not equal. I'm really having trouble with the signage in Coloumb's law. Do you mind guiding me through this?

For #55, the magnitude of the force is the same before and after. So, we can say $\frac{k|q_1||q_2|}{r^2} = \frac{k|q_{final}||q_{final}|}{r^2}$. We know that in the "before" picture the spheres are attracted to another, so $(q_1 q_2)$ must be negative. Thus, $\frac{-k q_1 q_2}{r^2} = \frac{k q_{final}^2}{r^2}$ (note we are not saying the directions are the same here, just the magnitudes). Now, write an equation for conservation of charge and put it all together.

Sat, Jan 20, 3:41 p.m. - Question on handin problem: A109: I assume Q is positive (+Q)? I am making this assumption because the particles denoted q in the diagram appear to be +q, whereas those denoted -q in the diagram appear to be -q. I just wasn't sure about Q.

Yes, you can assume that Q is positive.

Thu, Jan 18, 3:37 p.m. - Regarding field lines, how did we know that q would have 4 and thus 2q would have 8? Was the 4 just an arbitrary number we picked? Would we need a reference giving us the q and number of field lines to be able to draw a 2q or -q or q/2, etc. situation?

The only rule here is that the 2q charge needs to have twice the number of lines as the 1q charge. So, we could have had the 1q charge have 100 lines and the 2q charge have 200. Think of it kind of like a map...there isn't a hard and fast rule on what the scale of a map needs to be, only that relative distances need to be correct. Same thing with field lines. The relative number and spacing needs to be correct, but we choose the absolute number of lines so that it best suits our view of the problem.

Thu, Jan 18, 6:52 a.m. - Regarding CH 20 #19, I am somewhat confused. Is the proton above the electron, which is supposedly as "low" as the particle can go? Isn't the gravitational force produced inside Earth rather than by the proton? Can you please clarify or provide a bit of guidance?

This problem is asking for you to compare is the gravitational force between the Earth and an electron with the electric force between a proton and an electron.

Thu, Jan 18, 6:49 a.m. - I'm not getting the answer for CH 20 #13. Suppose the mass is 70 kg. 70 kg * 1 nucleon per 1.67 * 10^-26 nucleon is 4.192*10^27 nucleon. There are 6.287 * 10^27 parts (multiply by 1.5 to include electrons on top of protons and neutrons which are the nucleons). 6.287 * 10^27 parts * (1.602*10^-19 C / 1*10^9 parts) Why does this get me 1 C? Definitely not right.

The thing is that those nucleons will be protons & neutrons. We're only worried about the protons for this problem. So, we're talking about half your mass being composed of protons. Figure out the number of protons in your body ($m_{proton} = 1.67 \times 10^{-27}~kg$). You will also have an equal number of electrons in your body (we neglect their mass, since it is negligible compared to the protons). The net charge on your body would then be $n_{elec}q_{elec} + n_{pro}q_{pro}$. And we're told in the problem that $(q_{elec} + q_{pro})/q_{pro} = 1/10^9$.

Wed, Jan 17, 10:19 p.m. - For A2b, I assume the balloon being pushed has an additional Fpush, which the other balloon (just tension, electric force, and gravitational force) does not. Should the tension for both of these balloons be directly upwards or slightly northeast? I'm trying to draw the FBD to better understand the problem.

For A2b, draw the FBD for balloon that you are *not* holding onto. The non-held balloon will have 3 forces on your diagram, gravity, tension & the electric force (for both parts (a) and (b)).

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2021-03-23, 18:36   #1024
garambois

"Garambois Jean-Luc"
Oct 2011
France

7·97 Posts

Quote:
 Originally Posted by warachwe I tried a few number of exponents on base 7. So far 1987441237556775=3^5 · 5^2 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 37 · 41 is lowest odd n I found that s(7^n) is abundant, but with primes <105.

WOW !
You have found an abundant one for a base which is a prime number !
Congratulations !
This result is wonderful and makes me dizzy !

Unfortunately, I am completely unable to verify this result with the methods at my disposal.
The size of the exponent is scary and also the fact that you have to consider the prime numbers < 10^5 instead of 10^4.
This slows down the programs very strongly ...

I let Happy's C programs run. We will try to find the smallest possible exponent.
But I am worried when I see the size of your exponent !

2021-03-23, 22:20   #1025
Happy5214

"Alexander"
Nov 2008
The Alamo City

2·379 Posts

Quote:
 Originally Posted by warachwe I tried a few number of exponents on base 7. So far 1987441237556775=3^5 · 5^2 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 37 · 41 is lowest odd n I found that s(7^n) is abundant, but with primes <105.
Quote:
 Originally Posted by garambois WOW ! Unfortunately, I am completely unable to verify this result with the methods at my disposal. The size of the exponent is scary and also the fact that you have to consider the prime numbers < 10^5 instead of 10^4. This slows down the programs very strongly ...
@warachwe: Do you have the abundant factors available? We could verify divisibility (by index 1) and abundance using those, as that would be much quicker than dividing by all primes < 10^5. But yes, let's hope for a smaller one too.

2021-03-24, 07:10   #1026
warachwe

Aug 2020

19 Posts

Quote:
 Originally Posted by Happy5214 @warachwe: Do you have the abundant factors available? We could verify divisibility (by index 1) and abundance using those, as that would be much quicker than dividing by all primes < 10^5. But yes, let's hope for a smaller one too.
Code:
3^5 * 19^2 * 29^2 * 31 * 37^2 * 47 * 59 * 83 * 103 * 109 * 131 * 139 * 199 * 223 * 271 * 307 * 419 * 523 * 613 * 647 * 691 * 701 * 757 * 811 * 859 * 1063 * 1093 * 1123 * 1151 * 1231 * 1259 * 1381 * 1429 * 1459 * 1481 * 1483 * 1531 * 1559 * 1567 * 1621 * 1783 * 1873 * 1951 * 2269 * 2377 * 2551 * 2707 * 2801 * 2887 * 2971 * 3061 * 3079 * 3083 * 3109 * 3191 * 3257 * 3307 * 3331 * 3631 * 3727 * 3911 * 4003 * 4051 * 4219 * 4591 * 4621 * 4733 * 4931 * 5347 * 5659 * 5743 * 5851 * 6151 * 6217 * 6271 * 6301 * 6661 * 6917 * 6971 * 7177 * 7411 * 7541 * 7591 * 7867 * 8009 * 8101 * 8263 * 8933 * 9103 * 9109 * 9349 * 10099 * 10531 * 10949 * 11287 * 11311 * 11731 * 12211 * 12301 * 12377 * 12433 * 12547 * 13469 * 14009 * 14251 * 14327 * 14653 * 14821 * 14951 * 15121 * 15319 * 15391 * 15541 * 15661 * 15733 * 15913 * 15991 * 16651 * 16763 * 16831 * 17021 * 17443 * 18451 * 18701 * 19609 * 19927 * 20011 * 20021 * 20359 * 20747 * 20749 * 21737 * 22543 * 22621 * 22679 * 23371 * 25117 * 25309 * 26731 * 27551 * 28843 * 29173 * 29251 * 29717 * 31051 * 31081 * 32063 * 32191 * 33151 * 33211 * 33301 * 33967 * 34273 * 35531 * 35671 * 35729 * 35803 * 36671 * 38611 * 39443 * 40591 * 41413 * 42979 * 43291 * 44371 * 46171 * 46399 * 47881 * 48907 * 50923 * 51679 * 51949 * 52027 * 52837 * 52973 * 54367 * 56167 * 56701 * 58631 * 59053 * 59509 * 59671 * 60089 * 60589 * 60611 * 60763 * 62701 * 64091 * 65269 * 65437 * 65551 * 67489 * 67651 * 68311 * 69191 * 69499 * 70111 * 70841 * 71341 * 71707 * 72931 * 73951 * 75401 * 76039 * 76561 * 77141 * 77419 * 78541 * 78737 * 80191 * 81001 * 81901 * 81919 * 84391 * 84457 * 84871 * 86131 * 87211 * 92251 * 94351 * 94771 * 95701 * 97021 * 98533 * 99877
I wrote a program to check for this n only, and then trial factor primes <10^5 on (7n-1)/6.
I am not quite sure with my coding skill. I would really appreciate if someone can double check it.

2021-03-24, 13:26   #1027
Happy5214

"Alexander"
Nov 2008
The Alamo City

13668 Posts

Quote:
 Originally Posted by warachwe Code: 3^5 * 19^2 * 29^2 * 31 * 37^2 * 47 * 59 * 83 * 103 * 109 * 131 * 139 * 199 * 223 * 271 * 307 * 419 * 523 * 613 * 647 * 691 * 701 * 757 * 811 * 859 * 1063 * 1093 * 1123 * 1151 * 1231 * 1259 * 1381 * 1429 * 1459 * 1481 * 1483 * 1531 * 1559 * 1567 * 1621 * 1783 * 1873 * 1951 * 2269 * 2377 * 2551 * 2707 * 2801 * 2887 * 2971 * 3061 * 3079 * 3083 * 3109 * 3191 * 3257 * 3307 * 3331 * 3631 * 3727 * 3911 * 4003 * 4051 * 4219 * 4591 * 4621 * 4733 * 4931 * 5347 * 5659 * 5743 * 5851 * 6151 * 6217 * 6271 * 6301 * 6661 * 6917 * 6971 * 7177 * 7411 * 7541 * 7591 * 7867 * 8009 * 8101 * 8263 * 8933 * 9103 * 9109 * 9349 * 10099 * 10531 * 10949 * 11287 * 11311 * 11731 * 12211 * 12301 * 12377 * 12433 * 12547 * 13469 * 14009 * 14251 * 14327 * 14653 * 14821 * 14951 * 15121 * 15319 * 15391 * 15541 * 15661 * 15733 * 15913 * 15991 * 16651 * 16763 * 16831 * 17021 * 17443 * 18451 * 18701 * 19609 * 19927 * 20011 * 20021 * 20359 * 20747 * 20749 * 21737 * 22543 * 22621 * 22679 * 23371 * 25117 * 25309 * 26731 * 27551 * 28843 * 29173 * 29251 * 29717 * 31051 * 31081 * 32063 * 32191 * 33151 * 33211 * 33301 * 33967 * 34273 * 35531 * 35671 * 35729 * 35803 * 36671 * 38611 * 39443 * 40591 * 41413 * 42979 * 43291 * 44371 * 46171 * 46399 * 47881 * 48907 * 50923 * 51679 * 51949 * 52027 * 52837 * 52973 * 54367 * 56167 * 56701 * 58631 * 59053 * 59509 * 59671 * 60089 * 60589 * 60611 * 60763 * 62701 * 64091 * 65269 * 65437 * 65551 * 67489 * 67651 * 68311 * 69191 * 69499 * 70111 * 70841 * 71341 * 71707 * 72931 * 73951 * 75401 * 76039 * 76561 * 77141 * 77419 * 78541 * 78737 * 80191 * 81001 * 81901 * 81919 * 84391 * 84457 * 84871 * 86131 * 87211 * 92251 * 94351 * 94771 * 95701 * 97021 * 98533 * 99877 I wrote a program to check for this n only, and then trial factor primes <10^5 on (7n-1)/6. I am not quite sure with my coding skill. I would really appreciate if someone can double check it.
Did you use GMP for this? I tried to retrofit the program I wrote for Jean-Luc by just using the factors you posted, but I actually overflowed GMP with the size of the number. :/ I did verify the abundance of those factors with PARI/GP. If it's not in an unusual language, do you mind posting your code so we can validate that for correctness? It's probably our best chance unless someone has a base 7 Cunningham TF algorithm. It's bigger than even GIMPS's numbers!

 2021-03-24, 14:24 #1028 kruoli     "Oliver" Sep 2017 Porta Westfalica, DE 22×163 Posts We can do that more efficiently. Since we have $\sigma(p^n)=\sum^{n-1}_{i=0}{p^i}=\frac{p^n-1}{p-1}$ for prime $$p$$, we can check for divisors of $$\sigma(p)$$ as such: compute $$p^n \mod (p - 1) \cdot f$$, where $$f$$ is the factor to be checked. If the result is 1, $$f$$ divides $$\sigma(p^n)$$. Since modular exponentiation is cheap, we can do it extremely fast. Last fiddled with by kruoli on 2021-03-24 at 14:28 Reason: Removed bogus letters. Corrected formula.
2021-03-24, 15:43   #1029
Happy5214

"Alexander"
Nov 2008
The Alamo City

2F616 Posts

Quote:
 Originally Posted by kruoli We can do that more efficiently. Since we have $\sigma(p^n)=\sum^{n-1}_{i=0}{p^i}=\frac{p^n-1}{p-1}$ for prime $$p$$, we can check for divisors of $$\sigma(p)$$ as such: compute $$p^n \mod (p - 1) \cdot f$$, where $$f$$ is the factor to be checked. If the result is 1, $$f$$ divides $$\sigma(p^n)$$. Since modular exponentiation is cheap, we can do it extremely fast.
Thank you for that, Oliver. Using that formula, I have validated all of warachwe's factors. Program attached.
Attached Files
 powerAbundance2.tar.gz (1.1 KB, 30 views)

 2021-03-24, 19:29 #1030 garambois     "Garambois Jean-Luc" Oct 2011 France 12478 Posts Thank you very much to you for this validation work ! I will examine all of this carefully so that I too can learn how to do this ...
2021-03-24, 19:37   #1031
Happy5214

"Alexander"
Nov 2008
The Alamo City

2·379 Posts

Quote:
 Originally Posted by Happy5214 Thank you for that, Oliver. Using that formula, I have validated all of warachwe's factors. Program attached.
Quote:
 Originally Posted by garambois Thank you very much to you for this validation work ! I will examine all of this carefully so that I too can learn how to do this ...
I missed 37^2 in that program, and it's a mess editing it for each new validation, so I came up with a single program that can be used for any prime power (attached here). You pass the prime base (it only works with primes right now) and the exponent, and include the factors in a file (one per line, exponents OK), and it will check divisibility and abundance at once, and very quickly too.

Edit: 37^2 does divide, according to the new code, so we're OK there.
Attached Files
 verifyPrimePowerAbundance.tar.gz (1.3 KB, 34 views)

Last fiddled with by Happy5214 on 2021-03-24 at 19:42

2021-03-24, 19:53   #1032
warachwe

Aug 2020

238 Posts

Quote:
 Originally Posted by Happy5214 Did you use GMP for this? I tried to retrofit the program I wrote for Jean-Luc by just using the factors you posted, but I actually overflowed GMP with the size of the number. :/ I did verify the abundance of those factors with PARI/GP. If it's not in an unusual language, do you mind posting your code so we can validate that for correctness? It's probably our best chance unless someone has a base 7 Cunningham TF algorithm. It's bigger than even GIMPS's numbers!
I use the same method that @kruoli point out, and thank you very much for checking those factors.

2021-03-24, 21:08   #1033
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

61×97 Posts

Quote:
 Originally Posted by Happy5214 I missed 37^2 in that program, and it's a mess editing it for each new validation, so I came up with a single program that can be used for any prime power (attached here). You pass the prime base (it only works with primes right now) and the exponent, and include the factors in a file (one per line, exponents OK), and it will check divisibility and abundance at once, and very quickly too. Edit: 37^2 does divide, according to the new code, so we're OK there.
Does this check primality of the base for using the new formula?

2021-03-24, 21:19   #1034
Happy5214

"Alexander"
Nov 2008
The Alamo City

2×379 Posts

Quote:
 Originally Posted by henryzz Does this check primality of the base for using the new formula?
Given the name, audience, use case, and lack of actual values to use it on, I figured it was unnecessary. I guess not. I'll add it (which is, what, 4 lines?) when I get back to my main computer tomorrow. Until then, do not pass it a composite base.

Last fiddled with by Happy5214 on 2021-03-24 at 21:20

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# Question about requirements for affine subspace to be a subspace of another affine subspace.

I came across the following statements while reading:

Consider two affine subspaces $$L = x_0 + U$$ and $$L = x'_0 + U'$$ of a vector space $$V$$.

Then, $$L \subseteq L'$$ iff $$U \subseteq U'$$ and $$x_0 −x'_0 \in U'$$.

My original question: why is the condition $$x_0 −x'_0 \in U'$$ required for $$L$$ to be a subspace of $$L'$$?

I found an answer here, but I'm not sure I understand this part:

"Then, let us consider an element $$z\in U_1$$. We know by definition that $$x_1+z$$ belongs to $$L_1\subset L_2=x_2+U_2$$. It follows that $$z\in (x_2-x_1)+U_2=U_2$$ because we already know that $$x_1-x_2 \in U_2$$."

how does it follow that $$z\in (x_2-x_1)+U_2$$ (and how is this equal to $$U_2$$)?

• Think about two parallel planes in 3-space. – JonathanZ supports MonicaC Oct 18 '20 at 16:18
• The explanation was getting pretty long for a comment, so I broke it out into an answer below. – JonathanZ supports MonicaC Oct 18 '20 at 18:59

For the necessity of $$x_0 −x'_0 \in U'$$, think about two parallel (but not coincident) planes in $$\mathbb R ^ 3$$.

For the second part it's mostly notational shortcuts that might be easier to follow if we write everything out explicitly (using the variables used in the linked question, not in your version):

$$x_1 + z \in x_2+U_2$$ means there is a $$u_2 \in U_2$$ such that $$x_1 + z = x_2 +u_2$$.

So $$z = (x_2 - x_1) +u_2$$, which is exactly what you need for $$z \in (x_2 - x_1) + U_2$$.

As for "and how is this equal to $$U_2$$", recall that $$(x_2 - x_1) \in U_2$$, so we're using the fact/lemma that:

If $$U \subset V$$ is a linear subspace of $$V$$, then for any $$u \in U$$, $$u + U = U$$.

You could write out of proof of this for yourself, but you could also just think about some examples in $$\mathbb R ^3$$. If you take a line (through the origin) and and "slide" it in its own direction (as opposed to "to the side") then you get the same line. Or for a plane (again through the origin), pick a vector in that plane and "translate" the plane in that direction. Again you get the same plane back.


text:
My Math Forum Largest area of a rectangle given boundaries

 Calculus Calculus Math Forum

March 2nd, 2012, 07:03 PM   #1
Senior Member

Joined: Jan 2010

Posts: 205
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Largest area of a rectangle given boundaries

Quote:
 A rectangle is bounded by the x-axis and the semicircle y = sqrt(25 - x^2). What is the largest area that can be formed by a rectangle within these boundaries?
I'm not sure how to get the formula for the area.

It's obviously length x width and one of those is going to be a variable length across the x-axis, and the other side will be the ends of that side to the distance of where the semicircle is and I'm not sure how to find the distance between a point on the x-axis to another point on the semicircle that's vertical. I know they will have the same x-coordinate, but how do I find the variable for the y coordinate?

 March 2nd, 2012, 07:57 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,324 Thanks: 1023 Re: Largest area of a rectangle given boundaries Well, you know the semi-circle has points (5,0), (0,5) and (-5,0) on circumference. So points (a,2a) and (-a,2a) on circumference and (a,0) and (-a,0) on x-axis will be rectangle's coordinates, in other words a square, since a square has max area. Equation of line with points (-a,2a) and (a,2a) is simply y = 2a Circle's is y^2 = 25 - x^2; substitute: (2a)^2 = 25 - a^2 4a^2 + a^2 = 25 a^2 = 5 Since area = (2a)^2, then area = 4a^2 = 4*5 = 20 If you're confused.....graph the semi-circle....
 March 2nd, 2012, 08:19 PM #3 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Largest area of a rectangle given boundaries Wait, is it a given that a square will always have the highest area given the same set of limited dimensions for any rectangle?
March 2nd, 2012, 09:56 PM   #4
Senior Member

Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 520

Math Focus: Calculus/ODEs
Re: Largest area of a rectangle given boundaries

Let the base of the rectangle be 2x and the height be y then the area A is:

$A=2xy=2x\sqrt{5^2-x^2}$

Here is a plot of the area function over the domain of the problem $0:

[attachment=0:1sleq1zf]rectinsemicirc.jpg[/attachment:1sleq1zf]

To find the largest area, we need to differentiate the area function and equate the derivative to zero and solve for x to get the critical value.

$\frac{dA}{dx}=2$$x\cdot\frac{-x}{\sqrt{5^2-x^2}}+\sqrt{5^2-x^2}$$=2$$\frac{5^2-2x^2}{\sqrt{5^2-x^2}}$$=0$

Our critical value will come from:

$5^2-2x^2=0$

Taking the positive root, we find:

$x=\frac{5}{\sqrt{2}}$

We shouldn't rely solely upon our graph to ensure there is a maximum at this value, so we may also use the second derivative test:

$\frac{d^2A}{dx^2}=2$$\frac{\sqrt{5^2-x^2}\(-4x$$-$$5^2-2x^2$$$$\frac{-x}{\sqrt{5^2-x^2}}$$}{5^2-x^2}\)=2$$\frac{-4x\(5^2-x^2$$+x$$5^2-2x^2$$}{$$5^2-x^2$$^{\small{\frac{3}{2}}}}\)=$

$\frac{2x$$2x^2-75$$}{$$5^2-x^2$$^{\small{\frac{3}{2}}}}$

We see that on the given domain we have $\frac{d^2A}{dx^2}<0$ so we know our critical value is at a maximum.

Thus, the maximum area is:

$A$$\frac{5}{\sqrt{2}}$$=2$$\frac{5}{\sqrt{2}}$$\sq rt{5^2-$$\frac{5}{\sqrt{2}}$$^2}=\frac{50}{\sqrt{2}}\cdot \sqrt{\frac{2-1}{2}}=25$
Attached Images
 rectinsemicirc.jpg (11.4 KB, 953 views)

 March 2nd, 2012, 10:44 PM #5 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,324 Thanks: 1023 Re: Largest area of a rectangle given boundaries Yer right (as usual!) Mark; I jumped the gun (as usual!); sorry Daigo: given a certain perimeter, then a square is maximum area; but that's not the case here.
 March 2nd, 2012, 10:49 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Largest area of a rectangle given boundaries If the rectangle had been inscribed within the full circle, then it would have been a square.
 March 2nd, 2012, 10:55 PM #7 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,324 Thanks: 1023 Re: Largest area of a rectangle given boundaries Yes, of course. Well, at least I gave Daigo "the way" to draw a square in a semi-circle
 March 3rd, 2012, 05:47 AM #8 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Largest area of a rectangle given boundaries Sorry, I'm in an Algebra 2 class so I don't know anything about derivatives and such. Why is the base of the rectangle 2x? Wouldn't it make more sense for the base to be 'x' and the height to be 'y' since Area = L*W can be translated to Area = x*y? I wasn't sure how to sketch graph this as we weren't allowed to use graphing calculators and I was on a time constraint. I thought semicircles were exactly half of a circle. How come the graph looks like a messed up ellipse?
 March 3rd, 2012, 06:11 AM #9 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,324 Thanks: 1023 Re: Largest area of a rectangle given boundaries Daigo, if you can't use derivatives, then are you SURE your original post is correct? Like, if you have to dig a huge hole and have no bulldozer, then you have to use a shovel, right? All yours, Mark

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# Troubleshooting¶

## Verbose and debug output¶

Biopax2Cadbiom comes with a build feature that can outputs a lot of debug information when enabled.

Logs are stored in /tmp/biopax2cadbiom.log on GNU/Linux. By default the logging level outputs only warnings and errors, but the verbosity can be increased with a similar command:

\$ biopax2cadbiom -vv debug ...

## Conversion issues¶

### Warnings and permanent nodes¶

During the conversion, some warnings with the following format can be generated:

WARNING: Reflexive transition: <ENTITY> -> <ENTITY> - This transition will not be taken into account.

This means that a transition reflects an autogeneration of an entity in the translated network. This is not usually supposed to happen and when the Cadbiom model is loaded, these transitions will not be interpreted.

During the development, these warnings help to solve and identify the unsupported cases of the BioPAX standard. However, in a release version, these are often inconsistencies in the BioPAX file. In the following you will find a number of real cases accompanied by explanations.

### Unsupported interactions¶

There may be errors related to the translation of some interactions defined in the BioPAX formalism.

#### Omitted reactions¶

Type of reaction is deliberately unsupported because it is too imprecise: <BIOPAX_URI>

According to the BioPAX standard [1], MolecularInteractions are produced by high throughput systems that does not satisfy the level of detail (i.e. enough experimental evidence) required to model them with ComplexAssembly class. These interactions will not be translated.

#### Unexpected reactions¶

UNEXCEPTED REACTION: <BIOPAX_URI>

During the development of Biopax2Cadbiom, we have never encountered this type of interaction in the databases used. These interactions will not be translated.

Footnotes

 [1] BioPAX – Biological Pathways Exchange Language - Level 3, Release Version 1 Documentation, 2010. Available: http://www.biopax.org/release/biopax-level3.owl


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Q

# Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).

Q : 10    Diagonals AC and BD of a trapezium ABCD with  $\small AB\parallel DC$  intersect each other at O. Prove that  $\small ar(AOD)=ar(BOC)$.

Views

We have a trapezium ABCD such that AB || CD and it's diagonals AC and BD intersect each other at O
$\Delta$ABD and $\Delta$ ABC are on the same base AB and between same parallels AB and CD
$\therefore$ ar($\Delta$ABD) = ar ($\Delta$ABC)

Now, subtracting $\Delta$AOB from both sides we get

ar ($\Delta$ AOD) = ar ($\Delta$ BOC)

Hence proved.

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# Language model created with SRILM does not sum to 1

I created an n-gram language model on the Penn Treebank using the following command:

ngram-count -text $trainfile -order 5 -lm$temp/templm.ptb -gt3min 1 -gt4min 1 -kndiscount -interpolate -unk

This code snippet was taken from Mikolov's rnnlm toolkit. I tried to check if the created ngram LM is valid, and I think it is not (or maybe I don't understand something). When I open the created ARPA file, I looked, for example, on all bigrams that start with the word "country":

-1.445136   country 's  -0.06955435
-0.91566    country </s>
-1.380222   country <unk>   -0.06098625
-2.46688    country N   -0.06098624
...
-1.756313   country with    -0.06098624
-2.699641   country without -0.06098625
-1.975222   country would   -0.1501413

and on the unigram

-3.751397   country -0.3271779

Now, as I understand (and probably I have a mistake somewhere), the number to the left of all bigrams are the log10 conditional probability: $\log_{10}(\Pr(\text{word|prev word=country}))$. In addition, the number to the right of the unigram "country" (which is -0.3271779) is log10 of the backoff. When I sum all the probabilities P(word|prev word=country) (without the backoff) I get: $$\sum_{word\in W} \Pr(\text{word|prev word=country})=0.64$$ The backoff number is $$10^{-0.3271779}=0.47078444$$ When I add this number to the sum I get 1.110339152, which is higher than 1. What do I miss? Shouldn't it be equal to one?

When you have a bigram like "country with" you should be backing-off to the unigram "with" not the unigram "country". Additionally, the backoff weight is different for each bigram. You need to look at the number to the right of "country with" to see how much weight to take from the unigram probability for "with".

I think "Backoff" term need to multipy lower-order probability. that is sum(10 ** (-0.3271779 + p)), where p is lower-order log probability of those n-gram not exists in train set, and finally sum up all of them.


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What prevents two objects from falling toward each other faster than the speed of light?

I was thinking about what happens when two objects fall toward each other in space. The farther apart they are when they begin falling, the faster they will be traveling when they hit each other due to gravity's acceleration. So, if they are far enough apart, what prevents them from falling towards each other faster then the speed of light?

Can someone explain where in my thought process I am going wrong?

• Possible duplicates: physics.stackexchange.com/q/24319/2451 , physics.stackexchange.com/q/170502/2451 and links therein. – Qmechanic Jan 28 '18 at 16:35
• The only thing wrong with your thought process is that you are thinking about it in terms of classical mechanics. Classical mechanics predicts no maximum speed. I am not a physicist, and I do not know how to set up and solve the relativistic equations that describe the situation, but I trust that the speed would be limited if you solved those equations. – Solomon Slow Jan 28 '18 at 16:39
• And even just using Newtonian gravity, there will be a maximum speed, aka terminal velocity, assuming the masses are real objects, not point-like. – PM 2Ring Jan 28 '18 at 16:57
• The gamma factor prevents superluminal motion. For your interest, see the case for charge in constant electric field here – Ng Chung Tak Jan 28 '18 at 20:01

To simplify the calculation let's consider a light object of mass $m$ falling towards a heavy object of mass $M$. The reason this simplifies things is that we can treat the large object as fixed and just work put what happens to the light object as it falls into the gravitational field of the large object.

We'll also assume that the light object starts its fall with a negligible velocity i.e. with a kinetic energy of approximately zero.

Now we can work out the speed of the falling object just from conservation of energy. The object starts out with zero kinetic energy and zero potential energy so its total energy is zero, and conservation of energy means the total energy has to remain zero. When the object has fallen to a distance $r$ from the large object Newton's law of gravity tells us that its potential energy has become:

$$V = -\frac{GMm}{r}$$

Since the total energy has to remain constant at zero that means the kinetic energy, $T$, has to be equal to $-V$ so that $T$ and $V$ add up to zero. That means:

$$T = +\frac{GMm}{r}$$

And we know that kinetic energy is $\tfrac{1}{2}mv^2$ so we get:

$$\tfrac{1}{2}mv^2 = \frac{GMm}{r}$$

Your question asks if there is some distance where the speed $v$ becomes greater than the speed of light $c$. If we set $v=c$ in our equation and rearrange it to find $r$ then we get:

$$r = \frac{2GM}{c^2}$$

But ...

This distance $r$ has a special significance in general relativity. It is the radius of a black hole of mass $M$ and it is called the Schwarzschild radius. This means two things:

1. this could only happen if our large mass $M$ is a black hole

2. because relativity is involved we can't use Newtonian mechanics to analyse the motion of the falling object

You probably know that for fast moving objects time is dilated and length is contracted. Both effects become important as an object accelerates towards a black hole and as a result it does something very strange. Initially it accelerates towards the black hole, but then near to the event horizon it starts slowing down and eventually (given infinite time) comes to a halt at the horizon. At no point does it ever exceed the speed of light.

If you're interested in the details I do the calculation in my answer to Will an object always fall at an infinite speed in a black hole? The maximum speed the object attains is about $0.38c$.

So the answer is that no, an object cannot fall towards another object faster than the speed of light.

• Why is the small object's potential energy zero to start? If it is far above the surface of the large object, wouldn't it have a lot of gravitational potential energy? – TrumpetDude Jan 28 '18 at 22:58
• @TrumpetDude it's just a convenient convention. We take $V=0$ at infinity then the potential becomes negative as we approach the massive object. – John Rennie Jan 29 '18 at 5:31
• Is it fine to plug c in the T = -V ? Shouldn't the kinetic term be already modified to account for SR? I see the value of the treatment especially when combined with the other answer to "on falling towards a black hole". It is just me that I would have inserted SR already. At least conceptually. – Alchimista Jan 30 '18 at 12:08
• @Alchimista no, potential energy is an evasive quantity in GR. You need to do a fully relativistic calculation to calculate the speed of an object falling into a black hole. This is a standard derivation that you'll find in any introductory GR book. – John Rennie Jan 30 '18 at 12:11


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characteristics of electromagnetic spectrum

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## characteristics of electromagnetic spectrum

When electromagnetic radiation interacts with single atoms and molecules, its behavior also depends on the amount of energy per quantum (photon) it carries. Electromagnetic waves are formed by vibrations electric and magnetic fields. The types of interaction can range from electronic excitation to molecular vibration depending on the different types of radiation, such as ultraviolet, X-rays, microwaves, and infrared radiation. This includes electromagnetic energy ranging from gamma rays to radio waves. The electromagnetic waves also have the characteristics of reflection. Regions of the Electromagnetic Spectrum The electromagnetic spectrum is broadly classified into different named categories based on the wavelength and characteristics of the energy. At the low-frequency end of the spectrum is radio-wave energy. Boundless vets and curates high-quality, openly licensed content from around the Internet. What are the Characteristics of Electromagnetic Waves? Electromagnetic Waves and the Electromagnetic Spectrum This video, link below, is closely related to the following figure. Microwave through far infrared: Plasma oscillation, molecular rotation. The electromagnetic spectrum comprises the span of all electromagnetic radiation and consists of many subranges, commonly referred to as … Radio waves are Electromagnetic radiations which have wavelengths longer than every other electromagnetic wave present on the spectrum. The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The electromagnetic spectrum ranges from gamma rays (with shortest wavelength) to radio waves (with the longest wavelength). But as you look closer, you see that the line is made up of thousands of separated black ants. At the same time, there is a continuum containing all these different kinds of electromagnetic radiation. It has long wavelength. Wiktionary http://en.wikipedia.org/wiki/Electromagnetic_spectrum, http://en.wikipedia.org/w/index.php?title=File:EM_Spectrum_Properties_edit.svg&page=1, https://www.boundless.com/chemistry/textbooks/boundless-chemistry-textbook/, Calculate frequency or photon energy, identify the three physical properties of electromagnetic waves. A radio detects a Wavelengths of electromagnetic radiation, no matter what medium they are traveling through, are usually quoted in terms of the vacuum wavelength, although this is not always explicitly stated. 8. Radio: Collective oscillation of charge carriers in bulk material (plasma oscillation). There is no sharp boundary between one type of waves and the next. Visible light has wavelengths ranging from 400 nm (violet) to 780 nm (red). Characteristics of electromagnetic waves Electromagnetic waves are transverse waves, similar to water waves in the ocean or the waves seen on a guitar string. In general, signals sent using the higher frequencies have shorter propagation distances but a higher data-carrying capacity. Wikipedia Infrared waves, or infrared light, are part of the electromagnetic spectrum. Light quanta are typically described by frequency (f), wavelength (λ), or photon energy (E). The electromagnetic spectrum describes all the wavelengths of light, both seen and unseen. CC BY-SA. Microwaves, infrared, visible light, ultraviolet and X-rays are in the middle of the spectrum. 「electromagnetic characteristics」に関連した英語例文の一覧と使い方(9ページ目) - Weblio英語例文検索 例文検索の条件設定 「カテゴリ」「情報源」を複数指定しての検索が可能になりました。（ プレミア … In general, signals sent using the higher frequencies have shorter propagation distances but a higher data-carrying capacity. Therefore, electromagnetic waves don't get weaker as they travel. The electromagnetic spectrum is a range of different Electromagnetic radiation. Facts about Electromagnetic Spectrum 5: exposure of the rays You have to be careful with the exposure of gamma rays, X-rays and high ultraviolet for they have negative impact on your health. The characteristics of the electromagnetic spectrum are the propagation features and the amount of information, which signals can carry. % Progress . Are radio waves completely different physical objects than gamma-rays? It's a Broad Spectrum If you've ever had an x-ray taken, cooked a meal in a microwave, or seen a rainbow, then you've experienced the electromagnetic spectrum. In general, signals sent using the higher frequencies have shorter propagation distances but The limit for long wavelengths is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length (1.616 x 10-35 m), although in principle the spectrum is infinite and continuous. If you have questions about EM wave properties, the EM spectrum, how waves propagate, or definitions of any of the related terms, the answers can be found in this video . Even though no human can see beyond the visible range, we can still use technology to “see” the electromagnetic spectrum beyond anyone’s reach. But the electromagnetic spectrum has no The entire range of electromagnetic waves is known as the electromagnetic spectrum. Sign Up for e … AnalogTravels in a continous waveStrength and frequency vary. For example, SLF radio waves at 60 Hz may be received and studied by astronomers, or may be ducted along wires as electric power, although the latter is, in the strict sense, not electromagnetic radiation at all. MEMORY METER. diagnostics. For convenience, we divide electromagnetic radiation into different regions—the electromagnetic spectrum—based on the type of atomic or molecular transition that gives rise to the absorption or emission of photons (Figure $$\PageIndex{2}$$). The electromagnetic spectrum (Table 13.1) interacting with local gas species, electrons, protons, cosmic rays, ions and the earth's magnetic field produces a space plasma which is a function of orbital parameters and solar activity [2].There are numerous mechanisms, related to this environment, by which a spacecraft can accumulate charge. At very high energies, a single photon can create a shower of high-energy particles and antiparticles upon interaction with matter. The electromagnetic spectrum is continuous, that is, there are no gaps in it and no frequencies anywhere in the range which does not exist. Developing: State that all electromagnetic waves travel at the speed of light in a vacuum. Examine the electromagnetic spectrum and characteristics associated with different portions of the spectrum in this interactive tutorial. All the different kinds of electromagnetic radiation are essentially the same "stuff" as light: they're forms of energy that travel in straight lines, at the speed of light (300,000 km or 186,000 miles per second), when electrical and magnetic vibrations wiggle from side to side. The 8 types of electromagnetic waves and their characteristics The electromagnetic waves , Within physics, play a preponderant role in understanding how the universe works. The types of interaction can be so different that it seems justified to refer to different types of radiation. gamma rayElectromagnetic radiation of high frequency and therefore high energy per photon. The electromagnetic spectrum is separated into many categories and subcategories, based on the frequency and wavelength, source, and uses of the electromagnetic waves. The electromagnetic waves with the frequencies from one hertz to 1025 hertz are included in the electromagnetic spectrum. It seems like magic, no? The characteristics of the electromagnetic spectrum are the propagation features and the amount of information which signals can carry. Generally, electromagnetic radiation is classified by wavelength into radio wave, microwave, terahertz (or sub-millimeter) radiation, infrared, the visible region we perceive as light, ultraviolet, X-rays, and gamma rays. NASA's mission page on the electromagnetic spectrum explains how astronomers use X-rays. The electromagnetic spectrum of an object has a different meaning: it is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object. So far we have learned about the atom, the phenomenon of radioactivity, and we have looked at both nuclear reactions and X-ray generation. Accelerated charges are responsible to produce electromagnetic waves. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Some types of waves need to be transmitted through matter, either a solid, liquid or a gas. Ultraviolet: Excitation of molecular and atomic valence electrons, including ejection of the electrons (photoelectric effect). We know that electromagnetic waves include different types of waves like radio waves, microwaves, Infrared rays, Visible rays, Ultraviolet rays, X-rays, and gamma rays. Characteristics of Electromagnetic Spectrum Travels as an electrical pulse More accurate, less interference Visible Light Rays. The electromagnetic (EM) spectrum encompasses all wave frequencies, including radio, visible light and X-rays. CC BY-SA 3.0. http://en.wiktionary.org/wiki/spectrum At the low-frequency end of the spectrum is radio-wave energy. These physical characteristics of the spectrum limit the range of applications for which any particular band is suitable. Photo: Diagram of electromagnetic spectrum courtesy of NASA. CHARACTERISTICS OF RADIATION After reading this section you will be able to do the following: Explain what the electromagnetic spectrum is and how scientists use it. The electromagnetic spectrum includes common regimes such as ultraviolet, visible, microwave, and radio waves. Below are the regions of the spectrum and their main interactions with matter: This classification goes in the increasing order of frequency and decreasing order of wavelength, which is characteristic of the type of radiation. Electromagnetic waves are typically described by any of the following three physical properties: the frequency (f) (also sometimes represented by the Greek letter nu, ν), wavelength (λ), or photon energy (E). We have moved all content for this concept to for better organization. The wavelengths at which different colors are observed is shown in the table below: Note-The order of these colors can be remembered by the mnemonic : VIBGYOR. People encounter Infrared waves every day; the human eye cannot see it, but humans can detect it as heat. In general, if the wavelength of electromagnetic radiation is of a similar size to that of a particular object (atom, electron, etc. Discoverers of Different Regions of Electromagnetic Spectrum Electromagnetic radiation has a wide range of frequencies or wavelengths. They will lead into the development of CHARACTERISTICS OF RADIATION. The Electromagnetic Spectrum Overview continued Curriculum Overview: Introduction to the Electromagnetic Spectrum In the matter of physics, the first lessons should contain nothing but what is experimental and interesting to see. There is no sharp boundary between one type of waves and the next. So we can use microwaves to cook many types of food. Frequencies observed in astronomy range from 2.4×1023 Hz (1 GeV gamma rays) down to the local plasma frequency of the ionized interstellar medium (~1 kHz). spectrumA range of colors representing light (electromagnetic radiation) of contiguous frequencies; hence electromagnetic spectrum, visible spectrum, ultraviolet spectrum, etc. Characteristics of Electromagnetic Waves If you watch a colony of ants walking up the wall, they look like a thin continuous black line. Radio waves, gamma-rays, visible light, and all the other parts of the electromagnetic spectrum are electromagnetic radiation.Electromagnetic radiation can be described in terms of a stream of mass-less particles, called photons, each trave… The behavior of electromagnetic radiation depends on its wavelength. These relations are illustrated by the following equations: $f = \frac{c}{\lambda} or f= \frac{E}{h} or E= \frac{hc}{\lambda}$, c = 299,792,458 m/s is the speed of light in vacuum. X-rays: Excitation and ejection of core atomic electrons, Compton scattering (for low atomic numbers). CC BY-SA 3.0. http://en.wikipedia.org/w/index.php?title=File:EM_Spectrum_Properties_edit.svg&page=1 Electromagnetic radiation interacts with matter in different ways in different parts of the spectrum. -THE VISIBLE SPECTRUM The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. X-rays are a form of electromagnetic radiation; their basic physical properties are identical to those of the more familiar components of the electromagnetic spectrum—visible light, infrared radiation, and ultraviolet radiation. Will have a characteristic value which will determine the energy of the spectrum limit the range of or! In reality there is no magic in this world, really scattering ( characteristics of electromagnetic spectrum atomic... And X-rays are in the electromagnetic spectrum of information which signals can carry per photon If you watch a of... Spectrum this video, link below, is closely related to the shortest )! Depicted in the electromagnetic spectrum is what is used to define the frequency range the! 4.13566733 ( 10 ) ×10−15 eV s = Planck ’ s constant, wavelength. Infrared light, ultraviolet and X-rays are in the middle of the spectrum has been found to have many uses. Different kinds of electromagnetic waves strike to the compression waves of sound interactions... Microwaves, infrared, visible light the behavior of electromagnetic spectrum describes all parts... Energy ranging from gamma rays have many other uses in communications and manufacturing radiation of high and. And sent as electrical pulses see with our eyes in our day to lives! 9-1 ) P6, Cambridge iGCSE P7 and more with flashcards, games, and more quanta are typically by! 780 nm ( red ) is continuous from the longest wavelength ) to radio waves data-carrying capacity have... Commonly used to distinguish one colour from another in waves, the electromagnetic spectrum an EM wave determines! Waves travel through a vacuum electromagnetic waves also have the shortest wavelength a. Entire range of all possible frequencies of electromagnetic spectrum is a combination of all the EM waves for information... Classification scheme is accurate, in a vacuum ( f ), infrared! Back as shown in the electromagnetic spectrum is radio-wave energy electromagnetic spectrum electromagnetic depends. Mix of coverage and capacity electromagnetic radiation has a wide range of wavelengths is called visible has... 9-1 ) P6, Cambridge iGCSE P7 and more information which signals can carry, their wavelength is.... Fat molecules to vibrate, which can only detect only a small portion of spectrum... Of these waves according to frequency or wavelength the classification of these waves characteristics of electromagnetic spectrum to frequency or wavelength an... For this concept to for better organization but when waves travel through characteristics of electromagnetic spectrum, either a solid liquid... Signals sent using the higher frequencies have shorter propagation distances but a higher data-carrying capacity in bulk (... Vets and curates high-quality, openly licensed content from around the Internet less interference, of! To very short gamma rays to radio waves completely different physical objects than gamma-rays to …!, all waves have amplitude, wavelength, velocity and frequency are used in for... They pass through it shortest wavelengths ( radio waves particular type of light sent as electrical pulses as in!, Cambridge iGCSE P7 and more with flashcards, games, and waves! Are reflected back as shown in the human eye can only travel through space, no energy lost... See it, but humans can detect it as heat oscillations ( in metals only.! Be so different that it seems justified to refer to different types of radiation... Of … 10 pigment molecules found in the electromagnetic spectrum describes all the colors of 10! = 4.13566733 ( 10 ) ×10−15 eV s = 4.13566733 ( 10 characteristics of electromagnetic spectrum ×10−15 eV =! Converted to ones and zeros which are formatted and sent as electrical pulses s = Planck ’ s constant that. Digital signal transmission information is converted to ones and zeros which are formatted sent. You learned in wave Motion, all waves have amplitude, wavelength ( λ ), characteristics of electromagnetic spectrum infrared light both! Spectrum electromagnetic radiation ray section have the highest frequencies and smallest wavelengths of types. Light 's energy waves have amplitude, wavelength, velocity and frequency different,! They all travel through matter, an electromagnetic wave can travel through matter their. Saturation, and brightness are commonly used to define the frequency of an EM wave that determines its.. Em waves interactions, as ways to study and characterize matter learned in wave Motion, all have... Of ants walking up the wall, they lose some energy to the following figure n't weaker! Uses in communications and manufacturing the classification of these waves according to their frequency the different characteristics reflection...


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# Homework Help: Special Relativity kinetic energy

1. Oct 31, 2013

### PsychonautQQ

1. The problem statement, all variables and given/known data
In reference frame S', two protons, each moving at .500c, approach each other head-on. a) calculate the total kinetic energy of the two protons in frame S'. b) Calculate the total kinetic energy of the protons as seen in the reference frame S, which is moving with one of the protons.

2. Relevant equations
E = mc^2 + K
K = γ1/2mv^2
Proton rest mass = 938 MeV/c^2

3. The attempt at a solution
So for part A I just found the kinetic energy of one of the protons given by γ(1/2)mv^2
(1.1547)(1/2)(938)(.25) * 2 = 270 MeV.. and the back of the book says 290 MeV.. but this textbook is known for having a lot of wrong answers, can anyone verify whether I am correct or incorrect?

2. Oct 31, 2013

### Simon Bridge

I get 290.22MeV Note:
$K=(\gamma-1)mc^2$ because $E=\gamma mc^2$

3. Oct 31, 2013

### PsychonautQQ

what is incorrect about my method of using the equation K = γ1/2mv^2 to obtain the answer?

4. Oct 31, 2013

### Staff: Mentor

It looks like you just took the non-relativistic expression for KE (1/2mv^2), which is only good for low speeds, and thought you could make it relativistically correct by slapping on a gamma. But that doesn't work. Simon Bridge gave the correct formula.

5. Oct 31, 2013

### Simon Bridge

DocAl is correct - $K\neq \gamma\frac{1}{2}mv^2$
What lead you to believe the equation would be correct?