Page 73 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 55

Figure 3.3 Applying Gauss’s law to
the field of a point charge Q on a
spherical closed surface of radius a. The
electric flux density D is everywhere
normal to the spherical surface and has
a constant magnitude at every point on it.

where the limits on the integrals have been chosen so that the integration is carried
2
over the entire surface of the sphere once. Integrating gives
2π 2π Q
Q π
−cos θ 0 dφ = dφ = Q
0 4π 0 2π
and we obtain a result showing that Q coulombs of electric flux are crossing the
surface, as we should since the enclosed charge is Q coulombs.

2
2
D3.3. Given the electric flux density, D = 0.3r a r nC/m in free space:
(a) find E at point P(r = 2,θ = 25 , φ = 90 ); (b) find the total charge
◦
◦
within the sphere r = 3; (c) find the total electric flux leaving the sphere r = 4.
Ans. 135.5a r V/m; 305 nC; 965 nC
D3.4. Calculate the total electric flux leaving the cubical surface formed by the
sixplanes x, y, z =±5ifthechargedistributionis:(a)twopointcharges,0.1 µC
at (1, −2, 3) and 1 7 µCat(−1, 2, −2); (b)a uniform line charge of π µC/m at
2
x =−2, y = 3; (c)a uniform surface charge of 0.1 µC/m on the plane y = 3x.
Ans. 0.243 µC; 31.4 µC; 10.54 µC

2 Note that if θ and φ both cover the range from 0 to 2π, the spherical surface is covered twice.