Page 200 - Engineering Electromagnetics, 8th Edition

182 ENGINEERING ELECTROMAGNETICS

The law of Biot-Savart is sometimes called Amp` ere’s law for the current element,
butwe will retain the former name because of possible confusion with Amp`ere’s
circuital law, to be discussed later.
In some aspects, the Biot-Savart law is reminiscent of Coulomb’s law when that
law is written for a differential element of charge,
dQ 1 a R12
dE 2 =
4π 0 R 2 12
Both show an inverse-square-law dependence on distance, and both show a linear
relationship between source and field. The chief difference appears in the direction
of the field.
It is impossible to check experimentally the law of Biot-Savart as expressed by (1)
or (2) because the differential current element cannot be isolated. We have restricted
our attention to direct currents only, so the charge density is not a function of time.
The continuity equation in Section 5.2, Eq. (5),
∂ρ ν
∇ · J =−
∂t
therefore shows that
∇ · J = 0
or upon applying the divergence theorem,

J · dS = 0
s
Thetotalcurrentcrossinganyclosedsurfaceiszero,andthisconditionmaybesatisfied
only by assuming a current flow around a closed path. It is this current flowing in a
closed circuit that must be our experimental source, not the differential element.
It follows that only the integral form of the Biot-Savart law can be verified
experimentally,

IdL × a R
H = (3)
4πR 2
Equation (1) or (2), of course, leads directly to the integral form (3), but other
differential expressions also yield the same integral formulation. Any term may be
added to (1) whose integral around a closed path is zero. That is, any conservative field
could be added to (1). The gradient of any scalar field always yields a conservative
field, and we could therefore add a term ∇G to (1), where G is a general scalar field,
without changing (3) in the slightest. This qualification on (1) or (2) is mentioned
to show that if we later ask some foolish questions, not subject to any experimental
check, concerning the force exerted by one differential current element on another,
we should expect foolish answers.
The Biot-Savart law may also be expressed in terms of distributed sources, such
as current density J and surface current density K. Surface current flows in a sheet of
vanishingly small thickness, and the current density J, measured in amperes per square