# Work done in electrostatics

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1. Jan 9, 2016

### Yashbhatt

1. The problem statement, all variables and given/known data
A metallic sphere is placed inside a hollow spherical shell. The potential on the inner and outer spheres is 10 V and 5 V respectively. What is the potential at the center?(The spheres are concentric.)

2. Relevant equations
$$V =\frac{kq}{r}$$

3. The attempt at a solution
The obvious answer I thought of was 15 V. But the answer in my text was 10 V. I know that it would be 15 V in case of two concentric spherical shells. Does having a metallic sphere make a difference?(I don't think so because for a metallic sphere the charge resides on the surface.)

2. Jan 9, 2016

### Staff: Mentor

What would happen if the potential inside the metallic sphere were different from the potential at its surface?

3. Jan 9, 2016

### fireflies

how did you calculate 15 V?

4. Jan 9, 2016

### Yashbhatt

Then some work would be done in bringing the charge from the surface of the metallic sphere to its center. But I don't see how that could be the case.

5. Jan 9, 2016

### Yashbhatt

That's quite simple. You can just assume some quantity of charge on the spheres and then assume that the charges are concentrated at the center. That would yield the work done and the potential easily.

6. Jan 9, 2016

### Staff: Mentor

Would the charge carriers in the metallic sphere remain static (stationary, unmoving) if the potential varied anywhere on or in it? Isn't this a problem in electrostatics?

And I don't see the connection between your thread title and your problem statement. You haven't introduced any mechanism where work will be done, or gone through the exercise of calculating the work required to move a charge from infinity to the center. As far as I can tell you just guessed at an answer (15 V).
How do you know this? Can you demonstrate?

7. Jan 9, 2016

### fireflies

Maybe you summed both the potentials.. it's okay. For a metallic sphere potential at any place is same as to the surface. But is it same for a hollow sphere? I am not sure. You may do one thing, be sure about this point first then cone to the rest part.

8. Jan 9, 2016

### cnh1995

Can you find the equipotential part in this system?

9. Jan 9, 2016

### Yashbhatt

The potential is constant inside a hollow sphere.

10. Jan 9, 2016

### cnh1995

When there is net charge present on the sphere, potential due to that charge inside that sphere is constant everywhere.

11. Jan 9, 2016

### Yashbhatt

Doesn't it follow from the superposition principle?

12. Jan 9, 2016

### Yashbhatt

Doesn't this follow clearly from the superposition principle?

Yes. I used the fact in this case.

13. Jan 9, 2016

### cnh1995

Consider this situation. There is a charge +q on the surface of the inner sphere and outer sphere is neutral. The potential of outer sphere is kq/r=5V. What will be the potential at the center then?

14. Jan 9, 2016

### Yashbhatt

$$5 + \frac{kq}{r_i}$$ ?

15. Jan 9, 2016

### cnh1995

Charge is on the surface of the inner sphere. What will be the potential at it's center then?

16. Jan 9, 2016

### Staff: Mentor

If it is properly applied, yes. But you haven't shown how you applied it to arrive at a 15 V potential inside the 10 V sphere.

17. Jan 10, 2016

### Yashbhatt

$$\frac{kq}{r}$$

I simply added the potential due to the two spheres.

18. Jan 10, 2016

### cnh1995

When the charge is on the surface of the sphere, potential at the surface and inside the sphere is same. The sphere is at 10V potential. Potential inside the sphere everywhere should be same. This implies E field inside the sphere is 0.

19. Jan 10, 2016

### Yashbhatt

So?

20. Jan 10, 2016

### cnh1995

So, the potential at the center will be equal to the potential of the inner sphere, won't it? The inner sphere is at 10V. So the potential at the center is also 10V.