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Introduction

Alright so today's lesson is an introduction to something called an "Electric Field". It may seem extremely intuitive, but don't judge a book by its cover!

The Easy(ish) Definition

You may have heard of an [electromagnetic field][https://en.wikipedia.org/wiki/Electromagnetic_field] and so you may have guessed that there is a correlation between it and our topic of the day. If you thought so, then you are correct! (Aren't you a smart cookie :D ) The electric field is a component of the electromagnetic field. Furthermore, an electric field is a form of a vector field that's generated in two ways:

1.) Electric charges (pretty intuitive)

2.) Time varying [magnetic fields][https://en.wikipedia.org/wiki/Magnetic_fields] which are described by Maxwell's equations

Taking all of this into consideration, the British supergenius Michael Faraday and his successors came up with the following formal/somewhat-general definition:

The electric field $\mathbf{E}$ at a given point is defined as the electric force $\mathbf{F}$ that would be exerted on a stationary test particle of unit charge by electromagnetic forces. A particle of charge $q$ would be subject to a force $\mathbf{F}=q\mathbf{E}$

Widening the Definition through Lorentz

However the question arises...

Since an electric field varies based off of your reference point, the more general definition for an electric field can be made through Lorentz Force Law The electric field can be defined as the electromagnetic force per unit charge in the rest frame of the charge. For all you math and physics nerds/geeks/lovers/fans/etc., here is another way to define it:

The force $\mathbf{F}$ acting on a particle of electric charge $q$ with instantaneous velocity $\mathbf{v}$, due to an external electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ can be defined as

$\mathbf{F} = q\left[\mathbf{E} + (\mathbf{v} \times \mathbf{B})\right]$

Where $\times$ is defined as the cross product of vectors. You may also see this written as shown in hyperphysics. Or more explicitly

$\mathbf{F}(\mathbf{r},\mathbf{\dot{r}},t,q) = q[\mathbf{E}(\mathbf{r},t) + \mathbf{\dot{r}} \times \mathbf{B}(\mathbf{r},t)]$

where $\mathbf{r}$ is the position vector of the charged particle, $t$ is time, and the overdot is a time derivative.

I'll leave it up to you all to decide on what this implies :)

The Energy in an Electric Field

Ok, I will admit that it is quite difficult to define the energy in a given volume of an electric field explicitly, so here is a more roundabout way of doing it.

Suppose that the magnetic field B is nonzero, then he total energy per unit volume stored by its electromagnetic field can be expressed as

$j_{EM} = \frac{\varepsilon}{2} |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{B}|^2$

where $\varepsilon$ defines the permittivity of the medium where our field exists, $\mu$ defines its magnetic permeability, while $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic field vectors.

In reality, it would be unjust to split this expression into two separate "electric" and "magnetic" parts since they are coupled. However, let us consider the exception of when the field is in a steady-state. At this point, the two fields are no longer coupled. Thus it makes sense that we can express the electrostatic energy per unit volume as

$j_{ES} = \frac{1}{2} \varepsilon |\mathbf{E}|^2$

Aha! So now we can define the total energy $J$ stored in the electric field in a given volume $V$ as

$J_{ES} = \frac{1}{2} \varepsilon \int_{V} |\mathbf{E}|^2 \, \mathrm{d}V$

Concluding Remarks

With this our introduction to electric fields concludes. If you found this difficult, as I said before don't judge a book by its cover! I hope you could understand everything, and that you had fun!

An electric field is a region of space in which an electric charge experiences an electric force. Theoretically, an infinite electric field will be created in free space by any electric charge unless the charge is located in a conductor.