Introduction to Electrodynamics/Chapter 2

2.1 The Electrostatic Field

2.1.1 Introduction

The question of Electrodynamics is, given point charges distributed and moving around, what effect will they have on a test charge placed somewhere in space?

This is not an easy question to answer. To do so, we have to tackle the problem in stages. First, we'll consider what happens when the charges are not moving around.

2.1.2 Coulomb's Law

<math> \mathbf{F} = {1 \over 4 \pi \epsilon_0} {q Q \over \mathbf{r}^2 } \hat \mathbf{r} </math>

Where:

• <math>\mathbf{F}</math> is the force on charge Q
• <math>\epsilon_0 = 8.85 \times 10^{-12} {C^2 \over N \cdot m^2}</math>, the permittivity of free space.
• <math>q</math> is the charge of the other charge.
• <math>Q</math> is the charge of the charge in question.
• <math>\mathbf{r}</math> is the vector from <math>q</math> to <math>Q</math>

2.1.3 The Electric Field

When you have multiple charges pushing or pulling on one charge, it makes sense to pull the charge out and look at each contributing force individually. In math,

<math> \begin{align} \sum \mathbf{F} &= \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 + ... + \mathbf{F}_n \\

 &= {1 \over 4 \pi \epsilon_0} {q_1 Q \over \mathbf{r}_1^2 } \hat \mathbf{r}_1 +

{1 \over 4 \pi \epsilon_0} {q_2 Q \over \mathbf{r}_2^2 } \hat \mathbf{r}_2 + {1 \over 4 \pi \epsilon_0} {q_3 Q \over \mathbf{r}_3^2 } \hat \mathbf{r}_3 + ... + {1 \over 4 \pi \epsilon_0} {q_n Q \over \mathbf{r}_n^2 } \hat \mathbf{r}_n \\

&= \sum {1 \over 4 \pi \epsilon_0} {q_i Q \over \mathbf{r}_i^2 } \hat \mathbf{r}_i \\
&= Q \left ( \sum {1 \over 4 \pi \epsilon_0} {q_i \over \mathbf{r}_i^2 } \hat \mathbf{r}_i

\right ) \\

&= Q \mathbf{E} \\

\end{align} </math>

where <math>\mathbf{E}(\mathbf{x}) = \sum {1 \over 4 \pi \epsilon_0} {q_i \over \mathbf{r}_i^2 } \hat \mathbf{r}_i</math>.

Note that <math>\mathbf{E}</math> needs a position, <math>\mathbf{x}</math>.