electric field

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Electric field due to a single point charge

Suppose we have a charge q'\, situated at the origin \mathbf{r}' = 0\,. The force exerted on a test charge q\, with position \mathbf{r}\, is given by Coulomb's law

\mathbf{F}(\mathbf{r}) = \frac{1}{4\pi \epsilon_0} \frac{q q'}{r^2} \hat \mathbf{r}\,.

We are free to place the test charge q\, anywhere and measure the force required to hold it in place. We see therefore that the existence of q'\, allows us to assign a vector to any point \mathbf{r}\,, determined by measuring the force per unit charge exerted on a test charge: \tfrac{\mathbf{F}(\mathbf{r})}{q}\,. Such an assignment is called a vector field, and we define the define the electric field \mathbf{E}\, to be a vector field

\mathbf{E}(\mathbf{r}) = \lim_{q\to0}\frac{\mathbf{F(\mathbf{r})}}{q}\,.

In taking q\to0\, we imagine that we have a very feeble test charge so as not to disturb the field we are measuring. Then, the electric field \mathbf{E}\, due to the existence of a charge q'\, at \mathbf{r}' = 0\,, is

\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \frac{q'}{r^2}\hat\mathbf{r}\,.

we can rewrite this as

\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} q'\frac{\mathbf{r}}{r^3}\,.

Electric field due to a number of point charges

If we have a number of charges q'_i\, at positions \mathbf{r}'_i\,, the forces add, and the electric field is

\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \sum_i q_i' \frac{\mathbf{r}'_i - \mathbf{r}}{|\mathbf{r}'_i - \mathbf{r}|^3}\,

back to Coulomb's law
on to Gauss' law
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