# Spherical Shell-Potential Energy, Energy density

1. Mar 27, 2017

### Arman777

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
I fount these

Part(a)its $E=\frac {ρa^3} {3ε_0}$ and
$υ=\frac 1 2ε_0E^2$
Part (b)
$dU=4πr^2drυ$
Part (c)
$U=\int_0^a 4πr^2udr$ but it gives me $U=\frac {-Q^2} {8πε_0a}$

This"-" bothers me.

2. Mar 27, 2017

### TSny

How is the charge Q distributed for a metallic sphere?
What is $E$ for points inside the sphere?
What is $E$ for points outside the sphere?

[EDIT: Note that your expression for $E$ does not have the correct dimensions for an electric field.]
The "-" sign should bother you. Your integrand is positive (including the dr). So, the integral must be positive. But are you sure you want to integrate from 0 to $a$? If you can answer my questions above, it should help you see what you should use for the range of integration.

3. Mar 27, 2017

### Monci

Someone was faster :P

4. Mar 27, 2017

### Arman777

Just in surface
E=0
$E=\frac {ρa^3} {3ε_0r^2}$ where $r≥a$ I know in the upside I forget $r^2$
What else it could be.. ?
I thought about that but it still give me nothing...
I am thinking there should be minus cause I am thinking like bringing charges from $r=0$ to $r=a$ but I should bring charges $r=∞$ to $r=a$ but still there appears "-" sign also in $r=∞$ case what should I use the volume...? Or is it make sense ?
The result of integral will be $U=Constant\int_∞^a\frac {1} {r^2}dr=constant\frac {-1} {r}$ I cant get rid of "-".I think its cause of potential energy case.

5. Mar 27, 2017

### Arman777

Is it $U=Constant\int_a^r\frac {1} {r^2}dr$ where r goes to ∞ ?

6. Mar 27, 2017

### TSny

So, all the charge Q is on the surface. So, there is no volume charge density $\rho$.

Yes, $E = 0$ inside the sphere. So, what is the energy density inside the sphere? What would you get if you integrated the energy density over the volume of the inside of the sphere ($0<r<a$)?

There is no volume charge density. For points outside the sphere, try to express E in term of Q and r.

I'm not sure what you are doing here. Follow the outline given in the problem. How would you express the energy contained in a spherical shell of radius r and thickness dr?

7. Mar 27, 2017

### Arman777

Zero ?
$E=\frac {Q} {4πε_0r^2}$
$dU=4πr^2drυ$

8. Mar 27, 2017

### TSny

Yes Yes Yes

9. Mar 27, 2017

### Arman777

soo?

10. Mar 27, 2017

### TSny

What is preventing you from finishing the problem?

11. Mar 27, 2017

### Arman777

Limits of integral

12. Mar 27, 2017

### TSny

You need to add up the energy in every spherical shell for which $E \neq 0$.

13. Mar 27, 2017

### Arman777

from a to ∞ ?

14. Mar 27, 2017

### Arman777

a to ∞ ?

15. Mar 27, 2017

### TSny

Yes

16. Mar 27, 2017

### Arman777

İnteresting... ok thanks a lot

17. Mar 27, 2017

### TSny

Yes, it is very interesting. The potential energy stored in the system can be thought of as stored in the field that extends from the surface of the sphere all the way out to infinity!

18. Mar 27, 2017

### Arman777

Thats just amazing....

19. Mar 27, 2017

### Arman777

I used gaussian law and surface charge density etc to find E

20. Mar 27, 2017

OK