Electric Fields
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Table of Contents
- Electric Fields
#UCLA #Y1Q3 #Physics1B
Electric Fields
Key Definitions
Field - a physical quantity with a value at every point in space-time
- Scalar Field - scalar assigned to every point (e.g. temperature)
- Vector Field - vector assigned to every point (e.g. air current)
Electric Field - a field created by an which imposes effects on other charges in the field
Electric Field Formula
\(\vec{E}\equiv\frac{\vec F_E}{q_1}=k_e\frac{q_2}{r^2}\hat r\)
\(\vec F_E=q_1\vec E\)
Where $\vec F_E$ is given by :
\(\vec F_E=k_e\frac{q_1q_2}{r^2}\hat r=q_1 \left(k_e\frac{q_2}{r^2}\hat r\right)\)
E.g. 
Superposition Principle
Electric fields obey the superposition principle (multiple charges/fields can be summed) i.e. Generalized :
\(\vec E=\sum_{i=1}^n \vec E_i\)
Discrete Charge Distribution:
\(\vec E=k_e\sum_{i=1}^n \frac{q_i}{r^2}\hat r\)
Continuous Charge Distribution:
\(\vec E=k_e\int \frac{dq}{r^2}\hat r\)
Charge Density
Electric charges follow the superposition principle s.t. we observe charge densities:
\(Q_{net}=\int dq\)
Linear Charge Density:
\(\lambda=\frac{dq}{dL}\quad\therefore\quad dq=\lambda dL\)
Surface Charge Density:
\(\sigma=\frac{dq}{dA}\quad\therefore\quad dq=\sigma dA\)
Volumetric Charge Density:
\(\rho=\frac{dq}{dV}\quad\therefore\quad dq=\rho dV\)
Thus, you can find the electric field for different objects/surfaces using charge density.
Common Fields
Circulars
Ring
\(\vec E = k_e\frac{\lambda 2\pi Rz}{(z^2+R^2)^{3/2}}\hat z\)
Disk
\(\vec E=2\pi\sigma k_ez\left (\frac{1}{z}-\frac{1}{\sqrt{z^2+R^2}} \right )\hat z\)
Annulus
\(\vec E=2\pi\sigma k_ez\left (\frac{1}{\sqrt{r^2+z^2}}-\frac{1}{\sqrt{R^2+z^2}} \right )\hat z\)
Infinite Plane
Single Plane
\(\vec E=\frac{\sigma}{2\epsilon_0}\hat x\)
Two Planes
\(\vec E = \frac{\sigma}{\epsilon_0}\)
Hollow Sphere
Inside ($r < R_0$)
\(\vec E = 0\)
Outside ($r > R$)
\(\vec E = k_e\frac{Q}{r^2}\hat r\)
In ($R_0>r>R$)
\(\vec E = k_e\int \frac{dq}{r^2}\hat r=k_e\int \frac{\rho dV}{r^2}\hat r\)
Electric Field Lines
Rules
