Electricity and Magnetism

Maxwell's Equations and the Foundations of Electrodynamics

As mentioned, Maxwell's Equations form the foundation of Electrical Engineering. This page explores the fundamental nature of the system of PDEs that govern electromagnetic phenomena. First, we will examine each equation that helped develop the final system of equation, roughly in historical order. At the end, we will combine them to reconstruct what is known today as Maxwell's Equations.

Permeability vs. Permittivity

Throughout this section you will see two common constants. The first constant is

\[ \epsilon_0 \]

This is the permittivity of free space, and is used in electric fields. The other common constant is

\[ \mu_0 \]

This is the permeability of free space, and is used in magnetic fields.

These two quantities are related to each other through the speed of light via the equation

\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

Where \(c\) is the speed of light.

Coulomb's Law (1785)

In 1785, Charles-Augustin de Coulomb measured the force between static electric charges and discovered an inverse-square relationship governing electrostatics. This is very similar to the formula governing the gravitational force between two celestial bodies.

\[ F = k_e \frac{q_1 q_2}{r^2} \]

Where \(k_e = \frac{1}{4\pi \varepsilon_0}\) is Coulomb's constant. The key takeaway is that this law is essentially analogous to Newton's Law of Gravitation, but for electric charges.

Electric Fields Defined by Coulomb's Law

There are two ways to look at an electric field. If we consider an electric monopole \(q_1\) (a single point charge), then we might want to have a graphical representation of its effect on any point in space. Denoted \(\textbf{E}\), the electric field is simply a vector field where each point in space is assigned a vector representing the electric force per unit charge.

\[ \textbf{E} = k_{e}\frac{q_1}{r^2}\hat{r} = k_{e}\frac{q_1}{r^3}\vec{r} \]

In three-dimensional coordinates, \(\vec{r}\) is the vector difference between the position of the field point \((x,y,z)\) and the position of the source charge \((x_0,y_0,z_0)\).

Electric Field Simulator

Here is a live simulator I programmed to get the feel for electric fields of mono and dipoles.

Monopole Field

click and drag the charge • right click to flip polarity

Dipole Field

click and drag the charges • right click to flip polarity

Ampère's Law (1820-1825)

Between 1820 and 1825, André-Marie Ampère studied the magnetic field produced by electric currents. In its standard integral form:

\[ \oint_C \textbf{B}\cdot d\textbf{l} = \mu_0 I_{\text{enc}} \]

This states that an electric current induces a magnetic field around itself. For an infinite straight current carrying line, the magnitude is:

\[ B = \frac{\mu_0 I_{enc}}{2\pi r} \]

Gauss's Laws (1830s-1845)

Gauss's Law for electricity in differential form:

\[ \nabla \cdot \textbf{E} = \frac{\rho}{\epsilon_0} \]

And Gauss's Law for Magnetism, stating there are no magnetic monopoles:

\[ \nabla \cdot \textbf{B} = 0 \]

Faraday's Law of Induction (1831)

A changing magnetic field induces a circulating electric field:

\[ \nabla \times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t} \]

Ampère-Maxwell Law (1861-1862)

Maxwell extended Ampère's Law to include the displacement current:

\[ \nabla \times \textbf{B} = \mu_0 \textbf{J} + \frac{1}{c^2} \frac{\partial \textbf{E}}{\partial t} \]

Maxwell's Synthesis (1860s)

\[ \begin{aligned} \nabla \cdot \textbf{E} &= \frac{\rho}{\epsilon_0} \\ \nabla \cdot \textbf{B} &= 0 \\ \nabla \times \textbf{E} &= -\frac{\partial \textbf{B}}{\partial t} \\ \nabla \times \textbf{B} &= \mu_0 \textbf{J} + \frac{1}{c^2}\frac{\partial \textbf{E}}{\partial t} \end{aligned} \]

Light

In free space, these equations lead to the electromagnetic wave equation:

\[ \nabla^2\mathbf{E} - \frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2} = 0 \]

This confirms that disturbances in the electromagnetic field propagate as waves at speed \(c\).

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