Answer to Question #138427 in Electricity and Magnetism for Omar

Question

Find the total charge within a volume defined by:
1 ≤ r ≤ 5, π/6 ≤ θ ≤ π/2, π/4 ≤ φ ≤ 3π/4
if the charge density is given as:
Pr= rcos φ

Answer 1

Solution

Total charge can be written as

$Q=\int \rho (r) d\tau$

$Q=\iiint r\cos\phi \space r^2 \sin\theta dr d\theta d\phi$

Applying given limits

1 ≤ r ≤ 5, π/6 ≤ θ ≤ π/2, π/4 ≤ φ ≤ 3π/4

and integrate

$Q=\intop^5 _1 r^3 dr\intop^{\pi/2}_{\pi/6}sin\theta d\theta\intop^{3\pi/4}_{\pi/4}\cos\phi d\phi$

then total charge becomes

$Q=0$