Maxwell's equations

The set of four equations by James Maxwell that describe the behavior of both the electric and magnetic fields. Maxwell's equations provided the basis for the unification of electric field and magnetic field, the electromagnetic description of light, and ultimately, Albert Einstein's theory of relativity. The elegant mathematical formulations of Maxwell's equations were not developed by Maxwell. In 1884, Oliver Heaviside reformulated Maxwell's equations using vector calculus. This change reinforced the perception of physical symmetries between the various fields with a more symmetric mathematical representation.

see also electricity, magnetism

The Equations

Charge Density and the Electric Field

∇·D = ρ

where ρ is the electric charge density (in units of C/m³), and D is the electric displacement field (in units of C/m²) which is related to the electric field E via a materials-dependent constant called the permittivity, ε. The permittivity of free space is referred to as ε₀, resulting in the equation for free space:

∇·E = ρ/ε₀

where, again, E is the electric field (in units of V/m), ρ is the charge density, and ε₀ (approximately 8.854 pF/m) is the permittivity of free space.

Equivalent integral form: ∫_AE·dA = Q_enclosed / ε₀

dA is the area of a differential square on the surface A with an outward facing surface normal defining its direction, Q_enclosed is the charge enclosed by the surface.

Note: the integral form only works if the integral is over a closed surface. Shape and size do not matter. The integral form is also known as Gauss's Law.

This equation corresponds to Coulomb's law.

The Structure of the Magnetic Field

∇·B = 0

B is the magnetic flux (in units of tesla, T).

Equivalent integral form: ∫_AB·dA = 0

dA is the area of a differential square on the surface A with an outward facing surface normal defining its direction.

Note: like the electric field's integral form, this equation only works if the integral is done over a closed surface.

This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Sturcturally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. This implies that there are no magnetic monopoles. If a monopole were to be discovered, this equation would need to be modified to read

∇·B = ρ_m

where ρ_m would be the density of magnetic monopoles.

A Changing Magnetic Field and the Electric Field

∇×E = -∂B/∂t

Equivalent Integral Form: ε = -dφ_B/dt where φ_B = ∫_AB·dA

φ_B is the magnetic flux through the area A described by the second equation, ε is the Electromotive Force around the edge of the surface A.

Note: this equation only works of the surface A is not closed because the net magnetic flux through a closed surface will always be zero, as stated by the previous equation. That, and the electromotive force is measured along the edge of the surface; a closed surface has no edge. Some textbooks list the Integral form with an N (representing the number of coils of wire that are around the edge of A) in front of the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.

Note the negative sign; it is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's Law.

This equation relates the electric and magnetic fields, but it also has a lot of practical applications, too. This equation describes how electric motors and electric generators work.

This law corresponds to the Faraday's law of electromagnetic induction.

The Source of the Magnetic Field

∇×H = J + ∂E/∂t

where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ, and j is the current density, defined by: J = ∫ρ_qvdV where v is a vector field called the drift velocity that describes the velocities of that charge carriers which have a density described by the scalar function ρ_q.

In free space, the permeability μ is the permeability of free space, μ₀, which is defined to be exactly 4π×10^-7 W/Am. Thus, in free space, the equation becomes:

∇×B = μ₀J + μ₀ε₀∂E/∂t

Equivalent integral form: ∫_sB·ds = μ₀I_encircled - μ₀ε₀∫_A (∂E/∂t)·dA

s is the edge of the open surface A (any surface with the curve s as its edge will do), and I_encircled is the current encircled by the curve s (the current through any surface is defined by the equation: I_{through A} = ∫_AJ·dA).

Note: unless there is a capacitor or some other place where ∇·J ≠ 0, the second term on the right hand side is generally negligable and ignored. Any time this applies, the integral form is known as Ampere's Law.

Summary

• ∇·D = ρ
• ∇·B = 0
• ∇×E = -∂B/∂t
• ∇×H = J + ∂D/∂t

For free space, eliminating the nonphysical quantities D and H, this reduces to:

• ∇·E = ρ/ε₀
• ∇·B = 0
• ∇×E = -∂B/∂t
• ∇×B = μ₀J + μ₀ε₀∂E/∂t

Simplifying further, by considering the case in the absence of imposed current or electric charge, gives the propagation equation for electromagnetic waves in free space:

• ∇·E = 0
• ∇·B = 0
• ∇×E = -∂B/∂t
• ∇×B = μ₀ε₀∂E/∂t

Note that μ₀ε₀ = c^-2, thus relating the speed of light to the permittivity and permeability of free space.

A Final Note on Unit Systems

The above equations are all in a unit system called mks (short for meter, kilogram, second; also know as the International System of Units (or SI for short). In a related unit system, called cgs (short for centimeter, gram, second), the equations take on a more symmetrical form, as follows:

• ∇·E = 4πρ
• ∇·B = 0
• ∇×E = -c^-1 ∂B/∂t
• ∇×B = c^-1 ∂E/∂t + 4πc^-1J

Where c is the speed of light in a vacuum. The symmetry is more apparent when the electromagnetic field is considered in a vacuum. The equations take on the following, highly symmetric form:

• ∇·E = 0
• ∇·B = 0
• ∇×E = - ¹/_c ^∂B/_∂t
• ∇×B = ¹/_c ^∂E/_∂t

---

Note: All variables that are in bold represent vector quantities.