Potential of a Solid Charged Sphere

[jmtome2]

Homework Statement

A sphere of radius 'a' contains a charge Q, uniformly distributed throughout it's volume. Calculate the total potential energy of this configuration.

Homework Equations

$$U=\frac{1}{2}\int \rho V d\tau$$

with $$\rho$$=charge density
and $$V$$=potential

The Attempt at a Solution

Potential of a Charged Sphere is $$V=\frac{Q}{4\pi\epsilon_0\cdot r}$$
Uniform Charge Density of a Sphere is $$\rho=\frac{Q}{Vol}=\frac{Q}{\frac{4}{3}\pi\cdot a^{3}}$$

Therefore our equation becomes:
$$U=\left(\frac{1}{2}\right)\left(\frac{Q}{\frac{4}{3}\pi\cdot a^{3}}\right)\left(\frac{1}{4\pi\epsilon_0}\right) \int \frac{1}{r}d\tau$$

In spherical coordinates, $$d\tau=r^2sin(\theta)drd\theta\d\phi$$

$$U=\frac{Q^2}{\frac{32}{3}\pi^2\espilon_0\cdot a^3} \int^{a}_{0} rdr \int^{\pi}_{0} sin(\theta) d\theta \int^{2\pi}_{0} d\phi$$

Final: $$U=\left(\frac{3}{16}\right)\left(\frac{Q^2}{\pi\epsilon_0\cdot a}\right)$$

 

[jmtome2]

This result looks a little bizarre to me... did I miss anything?

 

[gabbagabbahey]

Potential of a Charged Sphere is $$V=\frac{Q}{4\pi\epsilon_0\cdot r}$$

Is it really? Even inside the sphere?

 

[jmtome2]

Potential of a Solid Charged Sphere Inside: $$V=\frac{Q}{4\pi\epsilon_0\cdot a}$$

i think...

 

[gabbagabbahey]

Don't guess, calculate...what is the field inside and outside a uniformly charged sphere?

 

[jmtome2]

OK i got the problem worked out, thanks... one question left tho, I get a negative number for the potential and, therefore a negative result for the potential energy

$$V_{in}=\frac{-Q\cdot r^{2}}{8\pi\epsilon_0\cdot a^{3}}$$

and

$$U_{in}=-\left(\frac{3}{80}\right)\left(\frac{Q^2}{\pi\epsilon_0\cdot a}\right)$$

Does it make sense that the potnetial energy inside is negative?

 

[gabbagabbahey]

You seem to be missing a constant term in your potential. In most cases, adding or subtracting a constant doesn't matter, but the formula you are using for $U$ assumes that the potential goes to zero at infinity...

 

[jmtome2]

I thought about the constant term and realized that the potential needs to go to 0 as r goes to infinity, but noticed 2 things...

Firstly, no constant is going to be able to compesate for fact that V_in has to go to infinity as r goes to inifinity

Secondly, the problem later goes on to state that the outside potential energy is five times the potential energy of the inside, which is the result that we get if no constant is added to V_in after integration of E

So... what do you suggest? Maybe there's some way to justify this result or...?

 

[gabbagabbahey]

Firstly, no constant is going to be able to compesate for fact that V_in has to go to infinity as r goes to inifinity

Of course not! Vin is only defined for $r\leq a$, but Vout must go to zero as $r\to\infty$, and the potential must be continuous at $r=a$

 

[jmtome2]

that last condition must imply then that the inside potential must be positive, but how can this be since E_in is postive and is the negative gradient of V_in?

 

[gabbagabbahey]

The gradient of a constant is zero, so adding a constant will not affect the field, even if the constant is large enough to make the potential positive...

What diid you get for the field inside and out? What do you get when you integrate it?

$$V_{\text{in}}(r)=-\int_{\infty}^r\textbf{E}\cdot d\textbf{s}=-\int_{\infty}^a\textbf{E}_\text{{out}}\cdot d\textbf{s}-\int_{a}^r\textbf{E}_{\text{in}}\cdot d\textbf{s}$$

The first term corresponds to your constant!

 

[jmtome2]

Ok...

For E_in, i get...
$$E_{in}=\frac{Q\cdot r}{4\pi\epsilon_0\cdot a^{3}}$$

Integrating, for V_in, i get...
$$V_{in}=\int^{r}_{0}\frac{Q\cdot r}{4\pi\epsilon_0\cdot a^{3}} dr$$

and therefore,...
$$V_{in}=\frac{Q\cdot r^{2}}{8\pi\epsilon_0\cdot a^{3}} + C$$

 

[gabbagabbahey]

Why are you integrating from zero to $r$? And why are you just adding a constant on the end of a definite integral like that?

If your reference point is at infinity, the you need to integrate from infinity to $r$ as shown in my previous post.

 

[jmtome2]

So I get that

$$V_{in}=\frac{Q}{8\pi\epsilon_0\cdot a}\left(3-\frac{r^{2}}{a^{2}}\right)$$

I like where this is going :)

 

[jmtome2]

Last question?

So final V_in and V_out are:

$$V=\frac{Q}{4\pi\epsilon_0\cdot r}$$

and

$$V_{in}=\frac{Q}{8\pi\epsilon_0\cdot a}\left(3-\frac{r^{2}}{a^{2}}\right)$$

Which fixes the continuity of V problem that I had earlier when r=a.


Now upon calculating U_in and U_out, i get:

$$U_{out}=\left(\frac{3}{16}\right)\left(\frac{Q^2}{\pi\epsilon_0\cdot a}\right)$$

and

$$U_{out}=\left(\frac{3}{20}\right)\left(\frac{Q^2}{\pi\epsilon_0\cdot a}\right)$$

However, at the end of the problem, it states that "there is five times more energy outside than inside". I'm I not calculating the energy stored inside and outside correctly?

 

[gabbagabbahey]

The integral,

$$U=\frac{1}{2}\int \rho V d\tau$$

gives the total potential energy of the distribution (Remember, $\rho$ is zero outside the sphere, so using this formula to calculate the enrgy stored in the fields outside the sphere makes no sense at all!)

If you are asked to find the energy stored in the fields inside and out, you'll want to use

$$U=\frac{\epsilon_0}{2}\int E^2 d\tau$$

You should of course find that

$$U_{in}+U_{out}=\frac{1}{2}\int \rho V d\tau$$

 

[jmtome2]

Thank you! It all works out and makes sense! I found that equation hiding at the end of section 2.4.3 in Griffith's right before I got your response but you clarified it perfectly... Feels so good to see the big picture after drudging through this all day

 

[jmtome2]

Thank you! It all works out and makes sense! I found that equation hiding at the end of section 2.4.3 in Griffith's right before I got your response but you clarified it perfectly... Feels so good to see the big picture after drudging through this all day