Page 130 - Electromagnetics

density ρ 0 (r) is introduced at time t = 0. The charge density must obey the continuity
equation

∂ρ(r, t)
∇· J(r, t) =− ;
∂t
since J = σE,wehave
∂ρ(r, t)
σ∇· E(r, t) =− .
∂t
By Gauss’s law, ∇· E can be eliminated:
σ ∂ρ(r, t)
ρ(r, t) =− .
∂t
Solving this differential equation for the unknown ρ(r, t) we have

ρ(r, t) = ρ 0 (r)e −σt/ . (3.22)
The charge density within a homogeneous, isotropic conducting body decreases exponen-
tially with time, regardless of the original charge distribution and shape of the body. Of
course, the total charge must be constant, and thus charge within the body travels to
the surface where it distributes itself in such a way that the field internal to the body
approaches zero at equilibrium. The rate at which the volume charge dissipates is deter-
mined by the relaxation time /σ; for copper (a good conductor)this is an astonishingly
small 10 −19 s. Even distilled water, a relatively poor conductor, has /σ = 10 −6 s. Thus
we see how rapidly static equilibrium can be approached.
3.1.3 Steady current

Since time-invariant fields must arise from time-invariant sources, we have from the
continuity equation

∇· J(r) = 0. (3.23)
In large-scale form this is

J · dS = 0. (3.24)
S
A current with the property (3.23)is said to be a steady current. By (3.24), a steady
current must be completely lineal (and infinite in extent)or must form closed loops.
However, if a current forms loops then the individual moving charges must undergo
acceleration (from the change in direction of velocity). Since a single accelerating particle
radiates energy in the form of an electromagnetic wave, we might expect a large steady
loop current to produce a great deal of radiation. In fact, if we superpose the fields
produced by the many particles comprising a steady current, we find that a steady current
produces no radiation [91]. Remarkably, to obtain this result we must consider the exact
relativistic fields, and thus our finding is precise within the limits of our macroscopic
assumptions.
If we try to create a steady current in free space, the flowing charges will tend to
disperse because of the Lorentz force from the field set up by the charges, and the
resulting current will not form closed loops. A beam of electrons or ions will produce
both an electric field (because of the nonzero net charge of the beam)and a magnetic field
(because of the current). At nonrelativistic particle speeds, the electric field produces
an outward force on the charges that is much greater than the inward (or pinch)force
produced by the magnetic field. Application of an additional, external force will allow

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