## Electric fields and flux

### Faraday's idea with field lines

• The number electric field lines is $\frac{q}{\epsilon_0}$
• The quantity$\frac{q}{\epsilon_0}$ is called the electric flux (number of lines exiting the surface)
• (With some example shown in lecture) The electric
• The number of lines getting out of a closed surface (flux) is denoted by $\Phi$
• $\Phi _{closed\ surface} = \frac{q _ {enclosed}}{\epsilon_0}$
• In general, elecric field lines like to stay as far away from each other as possible

### Electric fields

• $\vec{E} = \frac{\Phi}{A}$
• The electric field can be thought of as field lines per unit area
• The area vector $\vec{a}$ is a vector that is perpendicular to the surface and whose magnitube is equal to the surface area
• Then, the electric flux $\Phi$ can b written as $\Phi = \vec{E} \cdot\vec{a}$
• In the infinitessimal case, $$\ \Phi_S = \int_{S}{\vec{E} \cdot d\vec{a}}$$
• Electric field is proportional density of field lines

### Derivations from Gauss' law

• For a spherically symmetric charge distribution: $$field\ lines = \frac{q}{\epsilon_0}\ \\area = 4\pi r^2\\ Field = lines \ / \ area \\ \vec{E} = \frac{q}{4\pi\epsilon_0 r^2 }$$
• The same logic can be applied for thin, infinitely long lines
• In general, the electric field for any surface can be calculated using the charge densities ($\lambda$ for the linear charge density, and $\sigma$ for the surface charge density)

### Solid angle

The solid angle $\Omega$ is given by: $$\ \Omega = \int \frac{dA}{r^2}$$

### Usage of Gauss' law

• Gauss' law is only useful when there is some form of symmetry (spherical, cylindrical etc.)

#### Spherical symmetry

• For a spherically symmetric electric field: $$\ \vec{|E|}\cdot 4\pi r^2 = \frac{Q_{in}}{\epsilon_0} \\ \vec{E} = \frac{\rho \vec{r}}{3\epsilon_0}$$
• For the above equation, the difficulty lies in calculating $Q_{in}$
• It is applicable even when the charge distribution is non-uniform. In which case, an integral over the distance would usually be involved (where the volumetric charge density is given as a function of the distance/radius)

#### Cylindrical symmetry

• For a cylindrically symmetric electric field: $$\ \vec{|E|}\cdot 2\pi rh = \frac{Q_{in}}{\epsilon_0} \\ \vec{E} = \frac{\rho \vec{r}}{2\epsilon_0}$$
• The denominator is 2 instead of 3 (as in the shperical case) because of some properties of averages