Potential inside a uniformly charged solid sphere

[vanhees71]

Isn't it now at the time to give the solution?

 

[BvU]

I expect @agnimusayoti already has the solution (from the Gauss approach, problem 2.21) and wants to see the integral yields the same result.

 

[vanhees71]

That's what I meant: Shouldn't we just show, how to get the integral now? I think it's good to finally have the solution in the thread, even if the OP couldn't completely solve it him/herself.

 

[agnimusayoti]

Ok, should I integrate from r' = 0 to z and from r' = z to R ?
Yeah, I can calculate the potential using Gauss Law and want to determine the same potential using another method. ..
I really appreciate if I can find the answer in this thread...

 

[agnimusayoti]

 

[vela]

You made an algebra mistake when you changed variables from $\theta$ to $u$.

 

[agnimusayoti]

You made an algebra mistake when you changed variables from $\theta$ to $u$.

Shoot, I forget to bring r'^2. Thanks now I have the solution pals!

Is it a general approach to place the point of analysis in z Axis for spherical simmetry? Thanks

 

[vanhees71]

Yes, because the $z$ axis is a preferred axis (the polar axis) in spherical coordinates. Also note that spherical coordinates are singular along the entire $z$-axis, because Jacobian for the volume element is $\sqrt{g}=r^2 \sin \vartheta$, which vanishes for $r=0$ as well as for all $r$ if $\vartheta \in \{0,\pi \}$.

 

[BvU]

Well done @agnimusayoti .
Shall we, to top it off, reproduce the result with the much more effective approach proponed by @vanhees71 ?

 

[vela]

Thanks now I have the solution pals!

It looks like on the last line you may have accidentally moved the factor of $R$ in the first term out of the denominator. The way I like to write the final result is
$$V(r) = \frac{Q}{4\pi\epsilon_0 R}\left(\frac 32-\frac 12 \frac{r^2}{R^2}\right).$$ The factor out front is the potential at the surface, and it's multiplied by a unitless factor that varies from 3/2 at the origin to 1 at $r=R$.

Is it a general approach to place the point of analysis in z Axis for spherical symmetry?

You might try set up the integral for an arbitrary point. You'll find you end up with an integral that's much more complicated to evaluate, and you'll see why choosing a point where $\theta = 0$ makes a lot of sense.

 

[agnimusayoti]

This thread shows, why differential equations are so much preferable to using general solutions. After the confusion is hopefully solved in this thread, I strongly recommend to solve the problem, using the local form of electrostatics,i .e.,
$$\Delta \Phi=-\frac{1}{\epsilon_0} \rho,$$
making use of the simplifying fact that in this problem $\rho=\rho(r)$ only and thus also $\Phi$ should be a function of $r$ only. Expressing then the Laplacian in spherical coordinates and applying it to this highly symmetric case, makes the solution a no-brainer!

Wait, phi is potential, isn't it?

 

[agnimusayoti]

It looks like on the last line you may have accidentally moved the factor of $R$ in the first term out of the denominator. The way I like to write the final result is
$$V(r) = \frac{Q}{4\pi\epsilon_0 R}\left(\frac 32-\frac 12 \frac{r^2}{R^2}\right).$$ The factor out front is the potential at the surface, and it's multiplied by a unitless factor that varies from 3/2 at the origin to 1 at $r=R$.You might try set up the integral for an arbitrary point. You'll find you end up with an integral that's much more complicated to evaluate, and you'll see why choosing a point where $\theta = 0$ makes a lot of sense.

Oops. My bad. Yes there is 1/R but I write it at numerator...

 

[vanhees71]

Wait, phi is potential, isn't it?

Yes!