 #1
 178
 13
Homework Statement:
 Use Eq 2.29 to calculate potential inside a uniformly charged solid sphere of radius R and total charge q. Compare tour answer to Prob 2.21
Relevant Equations:

Eq. 2.29:
$$V(\vec r)=\frac{1}{4 \pi \epsilon_0} \int \frac{\rho (\vec r')}{\mu} d\tau' $$
where ##\mu## is distance from ##d\tau'##
Well, in this problem, I try to use
$$d \tau '= \mu ^2 \sin {\theta} {d\mu} {d\theta} {d\phi}$$
With these domain integration:
$$0<\mu<r$$
$$0<\theta<\pi$$
$$0<\phi<2\pi$$
, I get $$V=\frac{1}{4\pi \epsilon_0} \frac{3Qr^2}{2R^3}$$
This result is wrong because doesn't match with Prob 2.21, which potential is determined with line integral.
I suspect that I made a mistake when define the ##\mu##, which is distance from volume element to point of analysis. Could you please what is wrong and how to fix it? Thanks
$$d \tau '= \mu ^2 \sin {\theta} {d\mu} {d\theta} {d\phi}$$
With these domain integration:
$$0<\mu<r$$
$$0<\theta<\pi$$
$$0<\phi<2\pi$$
, I get $$V=\frac{1}{4\pi \epsilon_0} \frac{3Qr^2}{2R^3}$$
This result is wrong because doesn't match with Prob 2.21, which potential is determined with line integral.
I suspect that I made a mistake when define the ##\mu##, which is distance from volume element to point of analysis. Could you please what is wrong and how to fix it? Thanks