Page 280 - Engineering Electromagnetics, 8th Edition

262 ENGINEERING ELECTROMAGNETICS

due to the other, move the sheet a differential distance against this force, and equate
the necessary work to the change in energy. If we did, we would be wrong, because
Faraday’s law (coming up in Chapter 9) shows that there will be a voltage induced
in the moving current sheet against which the current must be maintained. Whatever
source is supplying the current sheet turns out to receive half the energy we are putting
into the circuit by moving it.
In other words, energy density in the magnetic field may be determined more
easily after time-varying fields are discussed. We will develop the appropriate expres-
sion in discussing Poynting’s theorem in Chapter 11.
An alternate approach would be possible at this time, however, for we might
define a magnetostatic field based on assumed magnetic poles (or “magnetic
charges”). Using the scalar magnetic potential, we could then develop an energy
expression by methods similar to those used in obtaining the electrostatic energy
relationship. These new magnetostatic quantities we would have to introduce would
be too great a price to pay for one simple result, and we will therefore merely present
the result at this time and show that the same expression arises in the Poynting the-
orem later. The total energy stored in a steady magnetic field in which B is linearly
related to H is
1
W H = B · H dν (46)
2 vol
Letting B = µH,wehave the equivalent formulations
1
2
W H = µH dν (47)
2 vol
or
1 B 2
W H = dν (48)
2 vol µ
It is again convenient to think of this energy as being distributed throughout the
1
3
volume with an energy density of B · H J/m , although we have no mathematical
2
justification for such a statement.
In spite of the fact that these results are valid only for linear media, we may use
them to calculate the forces on nonlinear magnetic materials if we focus our attention
on the linear media (usually air) which may surround them. For example, suppose
that we have a long solenoid with a silicon-steel core. A coil containing n turns/m
with a current I surrounds it. The magnetic field intensity in the core is therefore
nIA · t/m, and the magnetic flux density can be obtained from the magnetization
curve for silicon steel. Let us call this value B st . Suppose that the core is composed of
2
two semi-infinite cylinders that are just touching. We now apply a mechanical force
to separate these two sections of the core while keeping the flux density constant. We
apply a force F overa distance dL, thus doing work FdL.Faraday’s law does not

2 A semi-infinite cylinder is a cylinder of infinite length having one end located in finite space.