Page 122 - Engineering Electromagnetics, 8th Edition

104 ENGINEERING ELECTROMAGNETICS

The question of where the energy is stored in an electric field has not yet been
answered. Potential energy can never be pinned down precisely in terms of physical
location. Someone lifts a pencil, and the pencil acquires potential energy. Is the energy
stored in the molecules of the pencil, in the gravitational field between the pencil and
the earth, or in some obscure place? Is the energy in a capacitor stored in the charges
themselves, in the field, or where? No one can offer any proof for his or her own
private opinion, and the matter of deciding may be left to the philosophers.
Electromagnetic field theory makes it easy to believe that the energy of an electric
field or a charge distribution is stored in the field itself, for if we take Eq. (44), an
exact and rigorously correct expression,

1
W E = 2 D · E dv
vol
and write it on a differential basis,
1
dW E = D · E dv
2
or
dW E = D · E (45)
1
dv 2
1
we obtain a quantity D · E, which has the dimensions of an energy density, or joules
2
per cubic meter. We know that if we integrate this energy density over the entire field-
containing volume, the result is truly the total energy present, but we have no more
justification for saying that the energy stored in each differential volume element dv
1
is D · E dv than we have for looking at Eq. (42) and saying that the stored energy is
2
1 ρ ν Vdv. The interpretation afforded by Eq. (45), however, is a convenient one, and
2
we will use it until proved wrong.
D4.11. Find the energy stored in free space for the region 2 mm < r < 3
200
mm, 0 <θ < 90 ,0 <φ < 90 ,given the potential field V = :(a) V;
◦
◦
r
300 cos θ
(b) V.
r 2
Ans. 46.4 µJ; 36.7 J

REFERENCES

1. Attwood, S. S. Electric and Magnetic Fields.3d ed. New York: John Wiley & Sons,
1949. There are a large number of well-drawn field maps of various charge distributions,
including the dipole field. Vector analysis is not used.
2. Skilling, H. H. (See Suggested References for Chapter 3.) Gradient is described on
pp. 19–21.
3. Thomas, G. B., Jr., and R. L. Finney. (See Suggested References for Chapter 1.) The
directional derivative and the gradient are presented on pp. 823–30.