Page 94 - Engineering Electromagnetics, 8th Edition

76 ENGINEERING ELECTROMAGNETICS

4.1 ENERGY EXPENDED IN MOVING A POINT
CHARGE IN AN ELECTRIC FIELD
The electric field intensity was defined as the force on a unit test charge at that point
at which we wish to find the value of this vector field. If we attempt to move the test
charge against the electric field, we have to exert a force equal and opposite to that
exerted by the field, and this requires us to expend energy or do work. If we wish to
move the charge in the direction of the field, our energy expenditure turns out to be
negative; we do not do the work, the field does.
Suppose we wish to move a charge Q a distance dL in an electric field E. The
force on Q arising from the electric field is

F E = QE (1)

where the subscript reminds us that this force arises from the field. The component
of this force in the direction dL which we must overcome is

F EL = F · a L = QE · a L

where a L = a unit vector in the direction of dL.
The force that we must apply is equal and opposite to the force associated with
the field,

F appl =−QE · a L
and the expenditure of energy is the product of the force and distance. That is, the
differential work done by an external source moving charge Q is dW =−QE · a L dL,
or dW =−QE · dL (2)

where we have replaced a L dL by the simpler expression dL.
This differential amount of work required may be zero under several conditions
determined easily from Eq. (2). There are the trivial conditions for which E, Q,or dL
is zero, and a much more important case in which E and dL are perpendicular. Here
the charge is moved always in a direction at right angles to the electric field. We can
draw on a good analogy between the electric field and the gravitational field, where,
again, energy must be expended to move against the field. Sliding a mass around with
constant velocity on a frictionless surface is an effortless process if the mass is moved
along a constant elevation contour; positive or negative work must be done in moving
it to a higher or lower elevation, respectively.
Returning to the charge in the electric field, the work required to move the charge
a finite distance must be determined by integrating,

final
W =−Q E · dL (3)
init