Electric Fields

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Table of Contents

• Electric Fields
  • Key Definitions
  • Electric Field Formula
    • $\overrightarrow{E}\equiv \frac{{\overrightarrow{F}}_E}{q_1}=k_e\frac{q_2}{r^2}\hat{r}$
    • ${\overrightarrow{F}}_E=q_1\overrightarrow{E}$
    • ${\overrightarrow{F}}_E=k_e\frac{q_1{q}_2}{r^2}\hat{r}=q_1\left(k_e\frac{q_2}{r^2}\hat{r}\right)$
  • Superposition Principle
    • $\overrightarrow{E}=\sum _{i=1}^n{\overrightarrow{E}}_i$
    • $\overrightarrow{E}=k_e\sum _{i=1}^n\frac{q_i}{r^2}\hat{r}$
    • $\overrightarrow{E}=k_e\int \frac{dq}{r^2}\hat{r}$
    • Charge Density
      • $Q_{net}=\int dq$
      • $\lambda =\frac{dq}{dL}\therefore dq=\lambda dL$
      • $\sigma =\frac{dq}{dA}\therefore dq=\sigma dA$
      • $\rho =\frac{dq}{dV}\therefore dq=\rho dV$
  • Common Fields
    • Circulars
      • Ring
        • $\overrightarrow{E}=k_e\frac{\lambda 2\pi Rz}{(z^2+R^2{)}^{3/2}}\hat{z}$
      • Disk
        • $\overrightarrow{E}=2\pi \sigma k_{e}z(\frac{1}{z}-\frac{1}{\sqrt{z^2+R^2}})\hat{z}$
      • Annulus
        • $\overrightarrow{E}=2\pi \sigma k_{e}z(\frac{1}{\sqrt{r^2+z^2}}-\frac{1}{\sqrt{R^2+z^2}})\hat{z}$
    • Infinite Plane
      • Single Plane
        • $\overrightarrow{E}=\frac{\sigma }{2{ϵ}_0}\hat{x}$
      • Two Planes
        • $\overrightarrow{E}=\frac{\sigma }{{ϵ}_0}$
    • Hollow Sphere
      • Inside ($r<R_0$)
        • $\overrightarrow{E}=0$
      • Outside ($r>R$)
        • $\overrightarrow{E}=k_e\frac{Q}{r^2}\hat{r}$
      • In ($R_0>r>R$)
        • $\overrightarrow{E}=k_e\int \frac{dq}{r^2}\hat{r}=k_e\int \frac{\rho dV}{r^2}\hat{r}$
  • Electric Field Lines
    • Rules

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Electric Fields

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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

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Electric Field Formula

$\overrightarrow{E}\equiv \frac{{\overrightarrow{F}}_E}{q_1}=k_e\frac{q_2}{r^2}\hat{r}$

${\overrightarrow{F}}_E=q_1\overrightarrow{E}$

Where ${\overrightarrow{F}}_E$ is given by :

${\overrightarrow{F}}_E=k_e\frac{q_1{q}_2}{r^2}\hat{r}=q_1\left(k_e\frac{q_2}{r^2}\hat{r}\right)$

E.g.

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Superposition Principle

Electric fields obey the superposition principle (multiple charges/fields can be summed) i.e. Generalized :

$\overrightarrow{E}=\sum _{i=1}^n{\overrightarrow{E}}_i$

Discrete Charge Distribution:

$\overrightarrow{E}=k_e\sum _{i=1}^n\frac{q_i}{r^2}\hat{r}$

Continuous Charge Distribution:

$\overrightarrow{E}=k_e\int \frac{dq}{r^2}\hat{r}$

Charge Density

Electric charges follow the superposition principle s.t. we observe charge densities:

$Q_{net}=\int dq$

Linear Charge Density:

$\lambda =\frac{dq}{dL}\therefore dq=\lambda dL$

Surface Charge Density:

$\sigma =\frac{dq}{dA}\therefore dq=\sigma dA$

Volumetric Charge Density:

$\rho =\frac{dq}{dV}\therefore dq=\rho dV$

Thus, you can find the electric field for different objects/surfaces using charge density.

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Common Fields

Circulars

Ring

$\overrightarrow{E}=k_e\frac{\lambda 2\pi Rz}{(z^2+R^2{)}^{3/2}}\hat{z}$

Disk

$\overrightarrow{E}=2\pi \sigma k_{e}z(\frac{1}{z}-\frac{1}{\sqrt{z^2+R^2}})\hat{z}$

Annulus

$\overrightarrow{E}=2\pi \sigma k_{e}z(\frac{1}{\sqrt{r^2+z^2}}-\frac{1}{\sqrt{R^2+z^2}})\hat{z}$

Infinite Plane

Single Plane

$\overrightarrow{E}=\frac{\sigma }{2{ϵ}_0}\hat{x}$

Two Planes

$\overrightarrow{E}=\frac{\sigma }{{ϵ}_0}$

Hollow Sphere

Inside ($r<R_0$)

$\overrightarrow{E}=0$

Outside ($r>R$)

$\overrightarrow{E}=k_e\frac{Q}{r^2}\hat{r}$

In ($R_0>r>R$)

$\overrightarrow{E}=k_e\int \frac{dq}{r^2}\hat{r}=k_e\int \frac{\rho dV}{r^2}\hat{r}$

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Electric Field Lines

Rules