The Electric Field

Introduction

In principle, Coulomb’s law can be used together with the superposition principle to describe the electrostatic forces between many point charges. However, this process becomes very complicated and tedious as the number of charges increases and is outright impossible to use when dealing with continuous charge distributions.

The Model of the Electric Field

It turns out that we can paraphrase Coulomb’s law in a way to make this simpler. We imagine that space is filled with a vector field, i.e. that each point $\mathbf{r}$ is assigned a vector $\mathbf{E}(\mathbf{r})$. This vector depends on the spatial configuration of all electric charges and can be used to calculate the total electrostatic force which would act on a point charge if it were placed at rest at $\mathbf{r}$.

Definition: The Electric Field

The electric field at the point $\mathbf{r}$ is defined to be the vector $\mathbf{E}$ such that if a point charge $q$ is positioned at $\mathbf{r}$ and is at rest, then the total electrostatic force $\mathbf{F}$ which acts on $q$ is given by $q$ and the electric field $\mathbf{E}$ at $\mathbf{r}$ as follows:

$$\mathbf{F}=q\mathbf{E}$$

NOTATION

We usually seek to express $\mathbf{E}$ as a function of $\mathbf{r}$ and thus write $E(\mathbf{r})$. If $\mathbf{r}$ is apparent from context, we can drop it.

TIP

If the charge $q$ is positive, then $\mathbf{F}$ points in the same direction as $\mathbf{E}$. Conversely, if $q$ is negative, then $\mathbf{F}$ points in the exact opposite direction.

Coulomb’s law and the concept of the electric field are equivalent - we can use the former to derive the latter and the latter to derive the former. They are two equivalent models of the same thing. However, the electric field is generally a lot easier to work with, especially when multiple charges are involved. More importantly, it can even be used to calculate the total electromagnetic force on a moving charge.

Sources of Electric Fields

The electric field can be generated by electric charges or by a time-varying magnetic field.

Empirical Law: Principle of Superposition for Electric Fields

To determine the total electric field at a point, one can treat each source separately, as if the other sources do not exist, and just sum up the end results.

Point Charges

Definition: Electric Field due to Point Charge

Let $Q$ be a point charge located at the point ${\mathbf{r}}_Q$.

The electric field ${\mathbf{E}}_Q(\mathbf{r})$ generated by $Q$ at some point $\mathbf{r}$ is defined as

$${\mathbf{E}}_Q(\mathbf{r})\stackrel{\text{def}}{=}\frac{1}{4\pi \epsilon }\frac{Q}{r^2}{\hat{\mathbf{r}}}_{{\mathbf{r}}_Q\to \mathbf{r}},$$

where ${\mathbf{r}}_{{\mathbf{r}}_Q\to \mathbf{r}}=\mathbf{r}-{\mathbf{r}}_Q$ is the vector which points from ${\mathbf{r}}_Q$ to $\mathbf{r}$, $r=||{\mathbf{r}}_{{\mathbf{r}}_Q\to \mathbf{r}}||$ is the distance between ${\mathbf{r}}_Q$ and $\mathbf{r}$, and ${\hat{\mathbf{r}}}_{{\mathbf{r}}_Q\to \mathbf{r}}$ is the unit vector which points in the direction of ${\mathbf{r}}_{{\mathbf{r}}_Q\to \mathbf{r}}$.

Tip: Direction of the Electric Field

If $Q$ is positive, then the ${\mathbf{E}}_Q(\mathbf{r})$ always points radially away from $Q$. Conversely, if $Q$ is negative, then ${\mathbf{E}}_Q(\mathbf{r})$ towards $Q$.

So defined, the electric field yields the same results as Coulomb’s law.

Continuous Charge Distributions

TODO

Bibliography