“average electric field in a spherical control volume | 119 R θ x′ x Figure 5.9 Geometry used to evaluate the surface integral in (5.41). and hence ¿ SR dS |x − x | = żπ 0 (2πRsinθ)(cos θ x/ x )(R dθ) (Rcos θ − |x |)2 + (Rsinθ)2 = 2πR2 x |x | ż 1 −1 χ dχ R2 + |x | 2 − 2R|x |χ , (5.42) where χ = cos θ. The integral on the second line of (5.42) may now be evaluated using the indefinite integral ż χ dχ √a − bχ = 2 b2 a − bχ 3 − a a − bχ, and after a little work we obtain ¿ SR dS |x − x | = ⎧ ⎨ ⎩ 4π 3 x, x < R, (5.43a) 4π 3 R3 |x| 3 x, x > R. (5.43b) Thus our integral of E over VR becomes ż VR E(x)dx = VR 4πε0 ż |x|>R ρe(x) (−x) |x | 3 dx − 1 3ε0 ż VR x ρe(x )dx, (5.44) or equivalently ż VR E(x)dx = VREqout(0) − pqin 3ε0. (5.45) We have arrived back at (5.39), albeit after a great deal of integration. Although this particular derivation of (5.39) is rather long-winded, some ofthe results we have deduced on route, such as (5.43a, b), will prove useful when it comesto our discussion of magnetostatics.”