Page 132 - Electromagnetics

3.2 Electrostatics
3.2.1 The electrostatic potential and work

The equation

E · dl = 0 (3.29)

satisfied by the electrostatic field E(r) is particularly interesting. A field with zero
circulation is said to be conservative. To see why, let us examine the work required to
move a particle of charge Q around a closed path in the presence of E(r). Since work is
the line integral of force and B = 0, the work expended by the external system moving
the charge against the Lorentz force is

W =− (QE + Qv × B) · dl =−Q E · dl = 0.

This property is analogous to the conservation property for a classical gravitational field:
any potential energy gained by raising a point mass is lost when the mass is lowered.
Direct experimental verification of the electrostatic conservative property is difficult,
aside from the fact that the motion of Q may alter E by interacting with the sources of
E. By moving Q with nonuniform velocity (i.e., with acceleration at the beginning of the
loop, direction changes in transit, and deceleration at the end)we observe a radiative
loss of energy, and this energy cannot be regained by the mechanical system providing
the motion. To avoid this problem we may assume that the charge is moved so slowly,
or in such small increments, that it does not radiate. We shall use this concept later to
determine the “assembly energy” in a charge distribution.

The electrostatic potential. By the point form of (3.29),

∇× E(r) = 0,
we can introduce a scalar field = (r) such that

E(r) =−∇ (r). (3.30)

The function carries units of volts and is known as the electrostatic potential. Let us
consider the work expended by an external agent in moving a charge between points P 1
at r 1 and P 2 at r 2 :

P 2 P 2
W 21 =−Q −∇ (r) · dl = Q d (r) = Q [ (r 2 ) − (r 1 )] .
P 1 P 1
The work W 21 is clearly independent of the path taken between P 1 and P 2 ; the quantity

P 2
W 21
V 21 = = (r 2 ) − (r 1 ) =− E · dl, (3.31)
Q
P 1
called the potential difference, has an obvious physical meaning as work per unit charge
required to move a particle against an electric field between two points.

© 2001 by CRC Press LLC