Maxwells equations

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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.

The Equations

Charge Density and the Electric Field

∇·E = ρ/εo

E is the electric field, ρ is the charge density (in C / m3), and εo is the permittivity of free space.

Equivalent integral form: ∫AE·dA = Qenclosed / εo

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

The Structure of the Magnetic Field

∇·B = 0

B is the magnetic field.

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 bacwards to their source or forward to their terminus ultimately leads back to the starting position. This basically means that there are no magnetic monopoles. If a monopole were to be discovered, this equation would need to be modified.

A Changing Magnetic Field and the Electric Field

∇×E = -∂B/∂t

Equivalent Integral Form: ε = -dφB/dt where φB = ∫AB·dA

φB 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.

The Source of the Magnetic Field

∇×B = μoj + μoεoE/∂t

μo is the permeability of free space, and j</b> is the current density (defined by: j = ∫ρqvdV 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 ρq).

Equivalent integral form: ∫sB·ds = μoIencircled - μoεoA (∂E/∂t)·dA

s is the edge of the open surface A (any surface with the curve s as its edge will do), and Iencircled is the current encircled by the curve s (the current through any surface is defined by the equation: Ithrough 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

  • ∇·E = ρ/εo
  • ∇·B = 0
  • ∇×E = -∂B/∂t
  • ∇×B = μoj + μoεo∂<b>E/∂t

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-1B/∂t
  • ∇×B = c-1E/∂t + 4πc-1j

The symmetry is more apparent when the electromagnetic field is considered in a vacuum. The equations take on the following form:

  • ∇·E = 0
  • ∇·B = 0
  • ∇×E = -c-1 (∂B/∂t)
  • ∇×B = -c-1 (∂E/∂t)

Many theoretical physicists like this symmetry so much that they use it despite the fact that it doesn't fit the standard.


All variables that are in bold represent vector quantities.


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