Quadrupole term for uniformly charged sphere (where did I go wrong?)

[snatchingthepi]

Homework Statement

Derive the quadrupole term for a general 3d potential

Relevant Equations

Multiple expansion

So I got an assignment returned to me with fewer marks than I had expected. One part in particular is confusing to me. The professor is only available on Monday for a tutorial, but I'd like to see what is wrong before then.

Can anyone spot why this is incorrect?

 

[Vanadium 50]

Can you write what you have done, using LaTex, and explain what your reasoning is at each step?

 

[snatchingthepi]

Actually I think I may have found the mistake as I confused a part of my notation with something that should have been unprimed. Thanks.

 

[kuruman]

You have also made a mistake in writing
$P_2(\cos\theta ')=\dfrac{3}{2}\cos\theta '-\dfrac{1}{2}$.
The correct Legendre polynomial for $n=2$ is
$P_2(\cos\theta ')=\dfrac{3}{2}\cos^2\theta '-\dfrac{1}{2}$.

 

[snatchingthepi]

You have also made a mistake in writing
$P_2(\cos\theta ')=\dfrac{3}{2}\cos\theta '-\dfrac{1}{2}$.
The correct Legendre polynomial for $n=2$ is
$P_2(\cos\theta ')=\dfrac{3}{2}\cos^2\theta '-\dfrac{1}{2}$.

Thank you!