Electric Field of a Point Charge

$\vec{E}=\frac{kq}{r^2}\hat{r}$

where

• $k$ is Coulomb’s Constant
• $\hat{r}$ is the unit vector pointing in direction from the charge to the point we are measuring

Original equation. $\vec{E}=\frac{1}{4\pi {ϵ}_0}\frac{q}{r^2}\hat{r}$

Electric Field of Multiple Point Charges

title: Principle of Superposition
Electric fields obey the *principle of superposition*, which means that **the net electric field is the vector sum of the electric fields due to each charge**. 

For multiple point charges, the net electric field is the superposition of the electric fields due to each individual charge. ${\vec{E}}_{tot}={\vec{E}}_1+{\vec{E}}_2+\cdots +{\vec{E}}_N={\sum }_{n=1}^N{E}_n$

Electric Field of a Continuous Charge Distribution

The charged region we look at must be divided into small charge elements $dq$ so each can act as a point charge. Charge Density

Algorithmic steps in calculation Electric Field

HIGH LEVEL INTUITION: Break electric field down into components and integrate. To solve these problems, here are the steps we should consider: 0. If charge density is not given, calculate the charge density from the charge distribution.

1. Choose the right co-ordinate system and place the origin appropriately. You need to integrate over the complete charge distribution.
2. Go to an arbitrary point A on the charge distribution. Represent the position of point A and the differential element at A in terms of the co-ordinate system. By changing the variables of the position, you should be able to go to any arbitrary point within the charge distribution.
3. Write down the differential Electric field, 𝑑𝐸⃗⃗⃗⃗⃗ , in terms of the charge density and vector 𝑟̂.
4. Simplify. Look for symmetry and cancel things out. If you cannot find symmetry, then you have to write the components of 𝒅𝑬⃗⃗⃗⃗⃗ in all three directions and integrate them separately.
5. Represent $r$ and other terms which vary with position in terms of the parameters of position of point A and integrate the components of 𝑑𝐸⃗⃗⃗⃗⃗ which you think exist due to symmetry to get the value of the Electric field. If you get the answer to be “0”, check whether your symmetry was correct. If you can’t find an error, do the calculation in the other direction.

Related

We can also use Gauss Law to calculate the Electric Field, but it must respect certain conditions (Gaussian Symmetry).

• See Gauss Law