Page 298 - Engineering Electromagnetics, 8th Edition

280 ENGINEERING ELECTROMAGNETICS

This is one of Maxwell’s four equations as written in differential, or point, form,
the form in which they are most generally used. Equation (5) is the integral form of
this equation and is equivalent to Faraday’s law as applied to a fixed path. If B is not
a function of time, (5) and (6) evidently reduce to the electrostatic equations

E · dL = 0 (electrostatics)
and

∇× E = 0 (electrostatics)
As an example of the interpretation of (5) and (6), let us assume a simple magnetic
field which increases exponentially with time within the cylindrical region ρ< b,

kt
B = B 0 e a z (7)
where B 0 = constant. Choosing the circular path ρ = a, a < b,in the z = 0 plane,
along which E φ must be constant by symmetry, we then have from (5)
kt
emf = 2πaE φ =−kB 0 e πa 2
kt
2
2
The emf around this closed path is −kB 0 e πa .Itis proportional to a because
the magnetic flux density is uniform and the flux passing through the surface at any
instant is proportional to the area.
If we now replace a with ρ, ρ< b, the electric field intensity at any point is
1 kt (8)
E =− kB 0 e ρa φ
2
Let us now attempt to obtain the same answer from (6), which becomes
1 ∂(ρE φ )
kt
(∇× E) z =−kB 0 e =
ρ ∂ρ
Multiplying by ρ and integrating from 0 to ρ (treating t as a constant, since the
derivative is a partial derivative),
1 kt 2
− kB 0 e ρ = ρE φ
2
or
1 kt
2
E =− kB 0 e ρa φ
once again.
If B 0 is considered positive, a filamentary conductor of resistance R would have
a current flowing in the negative a φ direction, and this current would establish a flux
within the circular loop in the negative a z direction. Because E φ increases exponen-
tially with time, the current and flux do also, and thus they tend to reduce the time rate
of increase of the applied flux and the resultant emf in accordance with Lenz’s law.
Before leaving this example, it is well to point out that the given field B does
not satisfy all of Maxwell’s equations. Such fields are often assumed (always in ac-
circuit problems) and cause no difficulty when they are interpreted properly. They