Maxwell's Equations   

Coulomb's law says that if a charge of q is located at a point r = ( x,y,z) and another charge of q_j is located at a point p_j, then the force between them is an inverse square field of the form

F =  1 4pe0   qqj || r-p|| 2   up

where u_p is the unit vector parallel to r-p and where e₀ is the permittivity of free space, which satisfies

1 4pe0 = 10-7c2

where c is the speed of light in a vacuum.

In a charge cloud, there are n charges q₁,¼,q_n, where n is extremely large, and to each charge there corresponds an inverse square field. We often consider the charge q at r = (x,y,z) to be a test charge that can be ''moved around'' in space.

LiveGraphics3d Applet

Suppose now that ¶W is the boundary surface of a solid containing the entire charge cloud, then

ó õ ó õ ¶W  E·dS = n å j = 1  4p(  qj 4pe0  )=   1 e0 n å j = 1  qj

However, åq_j is the total charge Q of the charge cloud. Thus, Gauss' theorem says that

Flux  through  ¶W =   1 e0 Q

That is, we can measure the amount of charge contained within a solid by measuring the flux of the electric field through the boundary surface of the solid.

We obtain even more information by applying the divergence theorem. In particular,

Flux  through  ¶W = ó õ ó õ ó õ W  div( E) dV =  1 e0 Q

However, if r( x,y,z) is the density of the charge cloud, then Q is also a triple integral.

ó õ ó õ ó õ W  div( E) dV =   1 e0 Q  =  1 e0 ó õ ó õ ó õ W  r(x,y,z) dV

That is, we have

ó õ ó õ ó õ W  div( E) dV = ó õ ó õ ó õ W  1 e0 r( x,y,z) dV

Finally, let's notice the triple integrals are the same regardless of the solid W (as long as it is a solid that contains the entire charge). For example, the triple integrals are the same even if W is a sphere with radius R, regardless of what R is or how large it becomes. Since arbitrary solids always produce the same integrals, the integrands must be the same, which implies that

div( E) =   1 e0 r(x,y,z)

(3)

That is, the charge density itself is determined by calculating the divergence of the electric field it produces. Conversely, if r(x,y,z) is known, then the electric field E = áE₁,E₂,E₃ ñ is a solution to the partial differential equation

¶E1 ¶x   +  ¶E2 ¶y  +  ¶E3 ¶z  =   1 e0 r(x,y,z)

Equation (3) is important mathematically, physically, and historically, for it is the first partial differential equation in the set of 4 partial differential equations known as Maxwell's equations. Specifically, Maxwell's equations consider a charge cloud to be made up of both stationary and moving charges. The stationary charges are those that produce an electric field E( x,y,z,t) at time t and have a charge density at time t of r( x,y,z,t) . The moving charges are those that produce a magnetic field B(x,y,z,t) at time t, and each moving charge in the field is assigned a vector J( x,y,z,t) dV which indicates the direction of its motion at time t.

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Maxwell's equations describe the spatial (x,y,z) and temporal (t) dependence of the electric field E( x,y,z,t) and the magnetic field B( x,y,z,t) on the charge density r( x,y,z,t) and the current density J( x,y,z,t) . This set of 4 equations is given by

(1) div( E) =  1 e0  r(x,y,z,t) (2) div( B) = 0 (3) curl( E) = -   ¶B ¶t (4) curl( B) = m0J + e0m0  ¶E ¶t

where m₀ is permeability of free space, which satisfies c^-2 = e₀m₀. Equation (1) is the law we derived above, and the others follow from similar arguments with triple integrals, the divergence theorem, and Stoke's theorem, which we will explore in the next section.