Maxwell's Equations

© 2013 by  Fernando Caracena

James Clerk Maxwell was a theoretical physicist who had a good grasp of the mathematical theory of fluid dynamics. Using some of the mathematics from that field, he was able to put together Michael Faraday's observations on electrodynamics using a mathematical theory that was similar to that of fluids. In the process, he introduced an extra term into the equations, which incorporated the idea of conservation of charge in electric currents. In the next blog, we shall show that this extra term also allowed the electromagnetic field to carry electromagnetic waves, the vacuum velocity of which was specified by a combination of fundamental constants.

Here we discuss Maxwell's equations in a vacuum. They have to be modified slightly when they are applied to spaces that contain electrically and magnetically polarizable media, which reflect the effects of charges bound in neutral matter and the magentic effects of molecules of material media. See the following Wikipedia article for a more complete treatment: wiki. These effects are important from an engineering perspective. Most of the particle experiments involve vacuum conditions, which simplify the application of Maxwell's equations to the ones we discuss here.

James Clerk Maxwell  realized what Michael Faraday did not, that his imagined lines of force implied a whole mathematical theory of electricity and magnetism. Referring back to the the electric lines of force blog (EML), rewrite (EML.2b) as

E(x,y,z,t) = u_r*Q/(4 π ε₀ r²),                                                    (1a)

where the constant has been chosen as the conventional 1/ε₀   for SI units (see blog on units of measure). Note that (1a) evaluated on the surface of a sphere of radius r gives an electric field of constant strength, but having a direction, which is variable along the surface and parallel to the local normal to the surface of the sphere.

Rearranging terms in (1a), we get

E(x,y,z,t)•u_r 4 π r²  = Q/ε₀ .                                                           (1b)

Suppose that the charge, Q, is spread out as a small fuzzy ball centered at r=0 and is entirely contained at the distance, r, from the center. Call the distribution of charge a charge density [ρ(x,y,z)], which is centered on the origin (r=0).  In that case, we can write the total charge as a volume integral over the distribution out to a sphere of radius, r,

Q = ∫∫∫ _r d³V ρ(x,y,z,t).                                                  (2a)

Suppose that ρ(x,y,z) is non zero only over a very small range (0<r<δ), where that range is very close to zero, δ≈0. The distance of each element of charge in this case  is very close to a distance r from the surface of the sphere. In that case, the electric field departs only infinitesimally from being spherically symmetric, i. e., it is a function of the radius only, and not the angles in spherical coordinates. In that case, the term on the right hand side of (1b) can be written as a result of integration of the normal component of the electric field vector over a spherical surface of radius, r,

E•u_r 4 π r²  = ∫∫_closed E•u_r dS,                                            (2b)

which from the blog on advanced calculus (AC)

∫∫∫ dvol div V = ∫∫_closed dSrf n • V,    (AC.4)

can be written as

as

E(x,y,z,t)•u_r 4 π r²  = ∫∫∫ _r ∇ •E(x,y,z,t)d³V                             (2c)

Using (2a) and (2c), we can write (1b) as

∫∫∫ _r ∇ •E(x,y,z,t) d³V =∫∫∫ _r d³V ρ(x,y,z,t)/ε₀ ,                        (2d)

or

∫∫∫ _r d³V [∇ •E(x,y,z,t) -ρ(x,y,z,t)/ε₀]=0.                                  (2e)

For (2e) to hold for every value of E and ρ  then the bracketed quantity in (2e) must vanish identically, or

∇ •E(x,y,z,t) = ρ(x,y,z,t)/ε₀  ,                                          (3)

which constitutes one of Maxwell's equations, known as Gauss's Law.

Note that Gauss's Law (3) was generalized form (1a), which applies to a point charge to a more general case involving a charge distribution that is given by the charge density, ρ.

The meaning of (3) is that the electric field originates from any distribution of electric charge; and that in any closed volume devoid of charge, the electric flux integrated over the entire boundary, vanishes. The latter property corresponds to Faraday' s idea of lines of force that branch only in volumes of space where there are charges.

In applying Gauss's Law to the magnetic field, Maxwell introduced the idea that there  are no magnetic charges (or monopoles),

∇ •B(x,y,z,t) =0.                                                             (4)

The magnetic field, B, is said to be solenoidal, because it is generated in closed loops around electric current densities,

J = ρ v,                                                                            (2d)

and changing electric fields , ∂ E(x,y,z,t)/∂t, which Maxwell wrote as follows

∇ X B(x,y,z,t) = μ₀   J(x,y,z,t)+μ₀ ε₀ ∂ E(x,y,z,t)/∂t .       (5a)

Maxwell introduced the time derivative term in (5a),  ∂ E/∂t, for symmetry with Faraday' s Law of induction, which he wrote as

∇ X E(x,y,z,t) =-∂ B(x,y,z,t)/∂t ,                                     (6)

which reflects Faraday' s observation that a changing magnetic magnetic flux inside a coil induces an electromotive force in the coil, which can cause a current to flow. This is the basis of the operation of electrical generators in the modern world.

The set (3) through (6) constitutes all the information that we need to specify the entire behavior of the electromagnetic field in a vacuum. Only a few loose ends remain. First, we need to specify the force of the electromagnetic field on matter, called the Lorentz force, which is exerted on a point charge, q,

F = q(E + v X B) .                                                          (7a)

Note that according to (7a), a static charge experiences a force only from the electric component of the electromagnetic field; but responds to the presence of the magnetic field, only if it is moving with a velocity, v, that has a component at right angles to the magnetic field vector, B.

The generalization of (7a) for the force density of a fluid that carries a charge density of ρ is

f = ρ(E + v X B) .                                                          (7b)

In addition to the force of gravity, solar flares are subject to electromagnetic forces (7b), which cause them to move in broad arcs in the Sun's magnetic field.

There is more information embedded in (3) and (5a) that is extracted by taking the divergence of both sides of (5a)

0 = ∇ •J(x,y,z,t)+ε₀∇ • ∂ E(x,y,z,t)/∂t

or

∇ •J(x,y,z,t)+ε₀  ∂∇ •E(x,y,z,t)/∂t = 0.

The above equation is simplified by substituting for ∇ •E(x,y,z,t) using (3)

∇ •J(x,y,z,t)+∂ρ(x,y,z,t)/∂t = 0,                                 (8)

which is called the equation of continuity for charge. All charge that accumulates in a volume, according to (8), is accounted for  by the amount transported into that volume across its closed, bounding surface. This conclusion is demonstrated by applying the divergence theorem (AC.4) from the blog on advanced calculus (AC) to a volume integral of (8),

∫∫∫_closed  dx dy dz ∇ •J(x,y,z,t)+∫∫∫_closed  dx dy dz ∂ρ(x,y,z,t)/∂t = 0,

which becomes,

∫∫_closed dSrf n •J(x,y,z,t) + ∂q(t)/∂t = 0,                                   (9a)

;where,

q(t) = ∫∫∫_closed  dx dy dz ρ(x,y,z,t),

is the total electric charge contained in the designated volume.

Alternatively, write (9a) as

∂q(t)/∂t =- ∫∫_closed dSrf n •J(x,y,z,t),                                         (9b)

which means that the rate of change of the amount of charge within the defined volume is equal to the integrated flux of the amount of charge crossing orthogonally across the bounding surface of that boundary. If the flux is outward, the rate of change of that charge is negative and visa versa.