Page 100 - Engineering Electromagnetics, 8th Edition

82 ENGINEERING ELECTROMAGNETICS

One of the pitfalls in evaluating line integrals is a tendency to use too many minus
signs when a charge is moved in the direction of a decreasing coordinate value. This is
taken care of completely by the limits on the integral, and no misguided attempt should
be made to change the sign of dL. Suppose we carry Q from b to a (Figure 4.2b).
We still have dL = dρ a ρ and show the different direction by recognizing ρ = b as
the initial point and ρ = a as the final point,

a ρ L d ρ Qρ L b
W =−Q = ln
b 2π 0 ρ 2π 0 a
This is the negative of the previous answer and is obviously correct.

D4.2. Calculate the work done in moving a 4-C charge from B(1, 0, 0) to
A(0, 2, 0) along the path y = 2 − 2x, z = 0in the field E = (a)5a x V/m;
(b)5xa x V/m; (c)5xa x + 5ya y V/m.

Ans. 20 J; 10 J; −30 J

D4.3. We will see later that a time-varying E field need not be conservative.
(If it is not conservative, the work expressed by Eq. (3) may be a function of the
path used.) Let E = ya x V/m at a certain instant of time, and calculate the work
required to move a 3-C charge from (1, 3, 5) to (2, 0, 3) along the straight-line
segments joining: (a)(1, 3, 5) to (2, 3, 5) to (2, 0, 5) to (2, 0, 3); (b)(1, 3, 5) to
(1, 3, 3) to (1, 0, 3) to (2, 0, 3).

Ans. −9J;0

4.3 DEFINITION OF POTENTIAL
DIFFERENCE AND POTENTIAL
We are now ready to define a new concept from the expression for the work done
by an external source in moving a charge Q from one point to another in an electric
field E, “Potential difference and work.”

final
W =−Q E · dL
init
In much the same way as we defined the electric field intensity as the force on a
unit test charge, we now define potential difference V as the work done (by an external
source) in moving a unit positive charge from one point to another in an electric field,

final
E · dL (9)
Potential difference = V =−
init