Maxwell's Equations

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What follows is a survey of the various forms that these equations take, beginning with the most utilitarian and progressing to the most elegant. Which form you prefer depends on your scientific outlook, and perhaps your taste in T-shirts. The various \nabla \cdot \mathbf{E} and \nabla \times \mathbf{E} symbols appearing in some of the equations are the divergence and curl operators, respectively.


Name differential form integral form
Coulomb's law of electrostatics, or Gauss's Law: \nabla \cdot \mathbf{D} = \rho \oint_S  \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V
Absence of magnetic monopoles: \nabla \cdot \mathbf{B} = 0 \oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0
Faraday's Law of Induction: \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} \oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}  = -  \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}
Ampère's Law, or the Biot-Savart Law, plus displacement current: \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t} \oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
 \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d} \mathbf{A}

In these, E denotes the electric field, B denotes the magnetic field, D denotes the electric displacement field, and H denotes the magnetic field strength or auxiliary field. J denotes the free current density, and ρ denotes the free electric charge density.


Integral Form



What the Four Equations mean

Coulomb's Law


F = \frac{q_1 q_2}{4 \pi \epsilon\ d^2}

In this case, the constant defining the strength of the electric force is 4πε in the denominator. More about that presently.

In SI units the charges are measured in Coulombs, the force in Newtons, the distance in Meters, and the value of ε is 8.854 \times 10^{-12} Coulombs2 per Newton meter2, or Farads per meter.


E = \frac{q_1}{4 \pi \epsilon\ d^2}


F = q2E

Now consider a sphere of radius d with the charge at the center. If ρ is the charge density in Coulombs per cubic meter (Maxwell's equations are in terms of densities), the total charge in some volume is the integral, over that volume, of ρ.

So we have

q = \int_V \rho\, \mathrm{d}V

Now the field at the surface of the sphere is

\frac{q_1}{4 \pi \epsilon\ d^2}, or \frac{1}{4 \pi \epsilon\ d^2} \int_V \rho\, \mathrm{d}V

That field is directly outward, perpendicular to the sphere's surface, and is uniform over the surface. The integral of the field over the surface is d2 times that (the surface area of the sphere is d2; this is why we have the pesky factor of in various formulas; remember that d is the distance, and hence is the sphere's radius, not its diameter), so

\int_V \frac{\rho}{\epsilon}\, \mathrm{d}V = \oint_S  \mathbf{E} \cdot \mathrm{d}\mathbf{A}

But, by Gauss's Theorem,

\oint_S  \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \int_V \nabla \cdot \mathbf{E}\ \mathrm{d}V


\int_V \nabla \cdot \mathbf{E}\ \mathrm{d}V = \int_V \frac{\rho}{\epsilon}\, \mathrm{d}V

Since this is true for any volume, we have

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon}

Now D = \epsilon\ E in the straightforward case (more about that later), so

\nabla \cdot \mathbf{D} = \rho

Absence of Magnetic Monopoles



Faraday's Law


The Unification of Light with the Theory of Electromagnetism

It is possible, by means of experiments with electrical charges and currents, to find such a figure c2 that has turned out to be the square of the velocity of propagation of electromagnetic effects. It could be demonstrated from static measurements of forces between two "unite" charges and between two "unit" currents that c ≈ 3.00 x 10 8ms-1. When Maxwell had performed this calculation with his equations for the first time, he could conclude that the system of electric and magnetic field will be propagated by this velocity and he realized the intriguing equivalence of this constant with the already experimentally determined speed of the light. Thus, he suggested that the light is the wave motion of the same medium that causes electric and magnetic phenomena. Electricity and magnetism had been unified by experimental works of Faraday, Ørsted and Ampère. After them, Maxwell carried out one of the greatest unifications in the physical sciences, the light ceased to be "something different" and have become electricity and magnetism in a new form - electromagnetic field propagated across the space.[2]

Other Formulations

There are even more esoteric formulations (leading to more esoteric T-shirts) involving advanced concepts of differential geometry. See [1].


  2. Richard Phillips Feynman, Robert B. Leighton, Matthew Linzee Sands (1966). The Feynman Lectures on Physics, Volume 2, 2, Addison-Wesley, 18-9. 

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