Page 194 - Engineering Electromagnetics, 8th Edition

176 ENGINEERING ELECTROMAGNETICS

Figure 6.13 See Problem 6.21.

6.24 A potential field in free space is given in spherical coordinates as

[ρ 0 /(6 0 )] [3a − r ](r ≤ a)
2 2
V (r) =
(a ρ 0 )/(3 0 r)(r ≥ a)
3
where ρ 0 and a are constants. (a) Use Poisson’s equation to
find the volume charge density everywhere. (b) Find the total charge present.
2 3
6.25 Let V = 2xy z and = 0 .Given point P(1, 2, −1), find. (a) V at P;(b) E at
P;(c) ρ ν at P;(d) the equation of the equipotential surface passing
through P;(e) the equation of the streamline passing through P.( f ) Does V
satisfy Laplace’s equation?

6.26 Given the spherically symmetric potential field in free space, V = V 0 e −r/a ,
find. (a) ρ ν at r = a;(b) the electric field at r = a;(c) the total charge.
2
6.27 Let V (x, y) = 4e 2x + f (x) − 3y in a region of free space where ρ ν = 0.
It is known that both E x and V are zero at the origin. Find f (x) and V (x, y).