Introduction to Electrodynamics/Chapter 2
Contents
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 the electric field <math>\mathbf{E}</math> needs a position, <math>\mathbf{x}</math>.
Note that the position or charge of Q is irrelevant to the electric field. If you can calculate the electric field at any point, then the force on the particle at that point depends only on its charge and the electric field.
2.1.4 Continuous Charge Distributions
What about the case where you have an infinite number of point charges that come together to charge distributions? We simply use calculus to find the electric field.
