In the world of electrical engineering,
engineers often need to deal with how
changes in electric or magnetic fields
affect the motion or any other aspect of
an electric circuit, such as resistance
current and voltage. In order to do so,
they can use multivariable equations
such as the Maxwell equations or any
other electromagnetic law such as Lenz's
law or Ohm's law which analyzes
multivariable and multi-dimensional
properties. In the last video, I explained
the multivariable aspects and
applications of Faraday's law, but in
this video we're going to be looking at
an electromagnetism problem which
applies Faraday's law and utilizes
multivariable calculus.
Here's the problem, which consists of three parts:
Let's begin with Part A. It asks us to
use Lenz' law alongside other
conservation/physics laws to
qualitatively describe what will happen
if we remove the magnetic field the
wheel is currently under. If you remember
from the last video, one of the Maxwell
equations is Faraday's law, and it states
that a change in magnetic field will
induce an electric current, as evident by
the partial derivative of the magnetic
field being equal to the curl of the
electric field.
Turning off the magnetic induction
causes a change in magnetic field which,
according to Faraday's law, will cause an
electric field to curl around the axis of the wheel.
The electric field that is induced
exerts a force on the charges glued to
the edge of the rim and the wheel starts
to turn. Lenz' law states that an
induced current flows in a direction
such that the current opposes the change
that induced it, so in accordance with
Lenz' law the, current and thus the
direction in which the wheel will rotate
will be counterclockwise so that the
charges' induced magnetic field opposes
the initial change. Now, let's move on to
Part B, where we go through the math to
support our answer in Part A.
In the last video, we talked about Faraday's law in
its differential form, but for this video
we'll talk about it in its integral form,
which involves the contour integral of
the electric field over changes in
length and the rate of change over time
of the magnetic flux described by the
integral of the dot product here.
Magnetic flux, by the way, is the
measurement of a magnetic field in a given area.
It's a different form, but it
means essentially the same thing:
a change in magnetic field will induce a
current on a closed loop. We can begin to
simplify the right side of the equation
by calculating magnetic flux. This can be
done using a double integral, since
magnetic flux measures the magnetic
field in a certain area. Using polar
coordinates, we can set up an integral
that takes the area of the circular
region that the magnetic field is
initially in, with the outer integral
from 0 to 2 pi with respect to theta and
the inner integral from 0 to the smaller
radius 'a' with respect to radius r.
Inside of the integral is B, the magnetic field,
times r. Performing this double
integral leaves us with pi a squared, the
area of the circle. Now that we've
calculated flux, we can replace it in the
Faraday's law equation while assigning
the derivative operator to the magnetic
field B since the area stays constant.
Next, we're going to consider torque,
which is the rotational kinematics
equivalent of force and is defined as
the cross product between the torque arm
and force. In this equation, 'r' represents
the radius or torque arm, and 'E lambda dl'
represents the force exerted on the
charges on the rim. Next, we're going to
integrate the torque over the rim. The
first step in this process is to
evaluate the torque cross product. We can
replace r with x which represents the
radial direction. It's also important to
note that while x and dl are vectors,
E and lambda are not. The x vector goes
along the radius of the wheel, while the
dl vector goes along the circumference
perpendicular to the radius, so their
cross product will result in a vector
pointing upward in the z direction.
The magnitude of the vector of the cross
product is then the product of the
magnitude of vector x, which is the
radius 'b', and the magnitude of dl, which
is E lambda, which leaves us with this
vector. Note that the e sub z here is
meant to represent the direction of the
vector. Now it's time to integrate.
Since b and lambda are constants, you can take
them out of the integral. We're left with
the contour integral of E dl which, as we
found earlier, is equal to
negative pi a square dB/dt, so we can substitute that
in here. Now we are left with this equation.
Angular momentum is defined as
the integral of torque, so now, in order
to determine what effect changing the
magnetic field has on the angular
velocity of the wheel, we are going to
integrate the equation we found.
Everything that is not the magnetic
field B can be considered a constant, so
we can take everything out of the
integral, and the dt in the definition
of angular momentum cancels with the dt
in dB/dt, so we are left with
lambda b pi a squared times the integral from B0,
the initial state of the magnetic field,
to 0, the final state, with respect to the
magnetic field. Taking this integral,
we're left with this equation. Angular
momentum is also defined as L equals I omega,
where I is the moment of inertia
or rotational kinematic equivalent to
mass, and omega is the angular velocity.
Using this equation, we can substitute L
with the integral we found and then
divide by I to find the angular
velocity and this is the equation we're
left with. This equation tells us that
the angular velocity of the wheel does
not depend on the way the B field was
turned off, since it only takes into
account B0, the initial magnetic
field, instead of the change in magnetic
field dB/dt. Part C only asks us to
identify the source the motion of the
wheel and the agent responsible for the
motion. According to the law of
conservation of angular momentum, the
system's total angular momentum cannot
be different before or after the motion
unless acted on by an outside torque, so
since the wheel has acquired a net
quantity of angular momentum with its
rotation without an outside torque, the
electromagnetic fields must have lost an
equal amount of momentum. Therefore, the
source of the motion was the initial
angular momentum present in the
electromagnetic fields, and the agent for
the motion of the source is the electric
field induced by the cancellation of the
magnetic field B. Now that we've gone
through a rather math-intensive problem,
we're left with one question: how would
an electrical engineer actually use this
information? Well, by going through the
problem, we have discovered that a
changing magnetic field would cause a
wheel of these specifications to turn. We
also found out that the way we turn off
the field has no effect on the wheel's
angular velocity, but the magnitude and
properties of the initial magnetic field
does. If we had actual numbers instead of
letters, we would be able to calculate
the angular velocity, flux, momentum or
any other properties we derived in this
problem. An engineer can use all this
information to construct some sort of
device that plays off of the principles
of electromagnetic induction, a spinning
wheel powered only by the switching off of a magnetic
field. Doing such problems allows
engineers to be creative with their
designs, while adhering to all the proper
physical and mathematical principles.
