# Electrostatic potential energy - dielectric between concentric spheres

1. May 31, 2013

### Saitama

1. The problem statement, all variables and given/known data
A conducting sphere of radius $a$ carries an initial charge $q_o$. It is surrounded by another concentric sphere of radius $b$. The space between the two spheres is filled with a dielectric of permittivity $\epsilon$ and conductivity $\sigma$. Find the electrostatic potential energy stored in the system at time $t$.

2. Relevant equations

3. The attempt at a solution
The electric field at a distance $r$ from the centre of smaller sphere is $\frac{q_0}{4\pi \epsilon r^2}$. The electrostatic potential energy stored can be calculated by the following integral:
$$U=\int \frac{1}{2}\epsilon E^2dV$$
Replacing dV with $4\pi r^2dr$ and solving, I don't get the right answer. Also, I don't see how can I express U in terms of t and $\sigma$.

Any help is appreciated. Thanks!

2. May 31, 2013

### ehild

That conductivity allows charge to flow from one plate to the other. It is the same as if a resistor was connected across the capacitor.

ehild

3. May 31, 2013

### Saitama

What if there was no conductivity given in the question? Would my approach be correct then?

And I still don't know how to begin with this question. :(

4. May 31, 2013

### AGNuke

You have HCV Part II? There's a formula given for finding out Capacitance of Spherical Capacitor. That's ought to help.

5. May 31, 2013

### Saitama

I don't think that formula would help. A current flows here so I think a good equation to start with would be $J=\sigma E$ (that's the only equation I remember which involves conductivity), where J is current density and E is the electric field but I am not sure if it would work though.

6. May 31, 2013

### AGNuke

The charge will be conducted from Inner Sphere to outer Sphere. The instantaneous charge can be found out by the help of current. Then we can find the energy of the system. Let's see if that helps.

BTW, is your D-Day this year or next? (You know what I mean)

7. May 31, 2013

### ehild

You are on the right track. J=σE and JE is the power dissipated in unit volume in unit time.

You also can find both the capacitance and the resistance of the spherical shell, and treat the problem as discharging a capacitor.

ehild

8. May 31, 2013

### Saitama

How can I calculate the resistance here? Can I have a few hints to start with?

9. May 31, 2013

### AGNuke

Use conductivity as inverse of resistivity to find resistance. Take an elemental spherical shell of thickness dx and radius x. Now use integral calculus to find the overall resistance.

10. May 31, 2013

### Saitama

That's the first thought that came to my mind but what should I use as cross-section area?
We have
$$R=\frac{1}{\sigma}\cdot \frac{\ell}{A}$$
What should be A for the differential element?

11. May 31, 2013

### AGNuke

$$dR=\frac{1}{\sigma}\times \frac{dx}{4\pi x^2}$$

12. May 31, 2013

### Saitama

That gives me,
$$R=\frac{1}{4\pi \sigma}\cdot \left(\frac{1}{a}-\frac{1}{b}\right)$$

The capacitance is
$$4\pi \epsilon\left(\frac{ab}{b-a}\right)$$

From ehild's suggestion,
$$I=I_oe^{-t/RC}$$
where $RC=\epsilon/\sigma$ and how can I find $I_o$?

13. May 31, 2013

### AGNuke

This is what I am coming to$$E=\frac{Q_0^2}{8C}\times (1 - e^{-t/RC})^2$$

It really helps if you have some solution to reach to...

14. May 31, 2013

### Saitama

That's wrong and I am still clueless about what should be $I_o$. :(

15. May 31, 2013

### AGNuke

How about finding the potential at the inner sphere and the outer sphere due to the charge on inner sphere, find their difference and divide by resistance?

16. Jun 1, 2013

### ehild

ehild

17. Jun 1, 2013

### ehild

Conduction means dissipation, loss of electric energy. The energy of the electric field has to decrease.

ehild

18. Jun 1, 2013

### Saitama

Thank you both, I have reached close to the answer but it is still not correct.
$$U=\frac{Q_o^2}{2C}e^{-\frac{2t}{RC}}$$
Substituting RC and C,
$$U=\frac{Q_o^2(b-a)}{8\pi \epsilon ab}e^{-\frac{2\sigma t}{\epsilon}}$$
$$U=\frac{q_o^2(b-a)e^{-2\sigma \epsilon t}}{4\pi \epsilon ab}$$
I am sure that my exponential term is correct. $2\sigma \epsilon t$ is not dimensionless so the given answer is wrong but what about the factor of 1/2?

Thanks!

Last edited: Jun 1, 2013
19. Jun 1, 2013

### ehild

The given answer is wrong. Yours is correct (at least the same, I got) :)

ehild

20. Jun 1, 2013

### Saitama

Thanks a lot ehild!