Page 118 - Engineering Electromagnetics, 8th Edition

100 ENGINEERING ELECTROMAGNETICS

The dipole moment p will appear again when we discuss dielectric materials.
Since it is equal to the product of the charge and the separation, neither the dipole
moment nor the potential will change as Q increases and d decreases, provided the
product remains constant. The limiting case of a point dipole is achieved when we let
d approach zero and Q approach infinity such that the product p is finite.
Turning our attention to the resultant fields, it is interesting to note that the
potential field is now proportional to the inverse square of the distance, and the
electric field intensity is proportional to the inverse cube of the distance from
the dipole. Each field falls off faster than the corresponding field for the point charge,
but this is no more than we should expect because the opposite charges appear to
be closer together at greater distances and to act more like a single point charge
of zero Coulombs.
Symmetrical arrangements of larger numbers of point charges produce fields
proportional to the inverse of higher and higher powers ofr. These charge distributions
arecalledmultipoles,andtheyareusedininfiniteseriestoapproximatemoreunwieldy
charge configurations.

D4.9. An electric dipole located at the origin in free space has a moment
p = 3a x − 2a y + a z nC · m. (a) Find V at P A (2, 3, 4). (b) Find V at r = 2.5,
θ = 30 , φ = 40 .
◦
◦
Ans. 0.23 V; 1.97 V

D4.10. A dipole of moment p = 6a z nC · mis located at the origin in free
space. (a) Find V at P(r = 4,θ = 20 , φ = 0 ). (b) Find E at P.
◦
◦
Ans. 3.17 V; 1.58a r + 0.29a θ V/m

4.8 ENERGY DENSITY IN THE
ELECTROSTATIC FIELD

We have introduced the potential concept by considering the work done, or en-
ergy expended, in moving a point charge around in an electric field, and now we
must tie up the loose ends of that discussion by tracing the energy flow one step
further.
Bringing a positive charge from infinity into the field of another positive charge
requires work, the work being done by the external source moving the charge. Let
us imagine that the external source carries the charge up to a point near the fixed
charge and then holds it there. Energy must be conserved, and the energy expended in
bringing this charge into position now represents potential energy, for if the external
source released its hold on the charge, it would accelerate away from the fixed charge,
acquiring kinetic energy of its own and the capability of doing work.
In order to find the potential energy present in a system of charges, we must find
the work done by an external source in positioning the charges.