Caryssa Perez

Maxwell's Equations

October 11, 20184 min read ScienceMath

I learn best when I understand the high level overview of a topic. So when I want to learn about how computers work and why they work the way they work I need to start at the beginning: the fundamentals of electromagnetism.

This will be the first part of a series of posts that will cover how we get from physics equations to combinational logic to programming languages to common devices and things we use daily like radios and video games.

The first few posts will cover Maxwell’s equations, with a post for each equation that will explore what they mean and their practical applications.

So let’s begin with James Clerk Maxwell, a Scottish mathematical physicist, who unified the prevailing theories of electromagnetism at the time into a cohesive whole with four equations. The equations describe how electric and magnetic fields behave and how they behave together.

Maxwell’s Equations are a bit horrifying to look at the first time:

Gauss’ law for electricity

EdA=Qencε0\oint \vec{E} \cdotp d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}

Gauss’ law for magnetism

BdA=0\oiint \vec{B} \cdotp d\vec{A} = 0

Faraday’s law of induction

EdL=dϕBdt\oint \vec{E} \cdotp d\vec{L} = - {d \phi_B \over dt}

Ampere’s law

BdL=μ0Iencl+μ0ε0dϕEdt\oint \vec{B} \cdotp d\vec{L} = \mu_0I_{encl} + \mu_0\varepsilon_0{d\phi_E \over dt}

If all of the symbols seem confusing, don’t worry, I cover a brief math review here. In the meantime, we’ll define some variables and explore why the equations matter in the first place.

Definition of Variables

Electric Field

E\vec{E} represents an electric field, a vector field around an electric charge. Individual protons and electrons have their own electric field. Protons have a field that is directed outward from it while electrons have it directed inward. The interaction of these fields causes the attraction between protons and electrons.

Magnetic Field

H\vec{H} represents a magnetic field, a vector field around a magnetic object.

Total Charge

QQ is the total charge. Charge is commonly referred to with qq on general but for total charge we use capital QQ. encenc is used to signify that we want the charge enclosed in a particular surface.

Electric Permittivity

ε0\varepsilon_0 is the electric permittivity in a vacuum, which basically measures how much the atomical structure of some material affects the intensity of an electric field when it travels through that material. In a vacuum, the electric field doesn’t have to travel through anything and so it isn’t affected.

The electric permittivity in a vacuum is 8.8541012Farads/meter8.854 * 10^{-12} Farads/meter and it’s used as the control for relative permittivity of different materials.

Materials with low permittivity include paper and glass because their molecular structure make them a poor conductor of electricity. In contrast, materials with high permittivity include copper and silver for the opposite reason.

Vacuum Permeability

μ0\mu_0 represents permeability in a vacuum, which is how strong the magnetic field is in a vacuum.

The vacuum permeability is 4π×107henries/meter4π×10^{-7} henries/meter and, similar to electric permittivity, it’s used as the control for the degree of magnetization for relative permeability of different materials.

Materials can respond to magnetic fields differently and their permeability describes that affect. Materials with high permeability exhibit stronger magnetic fields through them, where materials with low permeability are the opposite.

Magnetic Flux Density

ϕB\phi_B represents the magnetic flux which is the measure of how much of the magnetic field is passing through a surface. We only care about the components of the magnetic field that are normal to the surface since it describes the true flow of the field out or into a surface.

Electric Current

II represents the electric current which is the flow of electric charge.

Why do these equations matter?

We know that physics equations are abstractions that help us predict how something will behave. Maxwell’s Equations are no different, even though they’re more complex than F=maF= ma.

However, even before Maxwell’s equations were a thing, physicists were experimenting with resistance and conductance and other behaviors of electricity and magnetism. Maxwell’s equations are the abstractions for that experimentation because they describe behaviors and give us the means to predict behavior.

Therefore, if we can figure out the overall distribution of electric charges and how they are moving, then we can calculate and define the subsequent electric and magnetic fields and their behavior.

In the next post, I’ll discuss Gauss’ Law for Electricity.