Radius of Half Electric Potential Energy

[NiendorfPhysics]

Homework Statement

There is a solid cylinder of radius a and then empty space then a shell cylinder of radius b. Show that half of the stored potential energy lies within a cylinder of radius $$\sqrt{ab}$$

Homework Equations

In the attempt

The Attempt at a Solution

I'm not sure what they want me to calculate the potential energy with respect to. If I do it wrt infinity it is infinity, same with 0. Let's say they want me to calculate it wrt the cylinder of radius that we have yet to determine. Then the energy is (getting rid of constants since they won't matter later):
$$\frac{1}{4}+\ln{\frac{R}{a}}$$

At least for the solid cylinder of radius a. Now we add the part of the energy stored in the field from the end of a to the fake cylinder. $$ln(\frac{R}{a})$$

Now we take one half of these values and add them together and set that equal to $$ln(\frac{R}{a})$$ and nothing works and I hate it because I know all of this is wrong.

But this problem cannot be this hard. This is just an RHK problem, I must be missing something simple. Can someone put me the on the right track?

 

[ehild]

Homework Statement

There is a solid cylinder of radius a and then empty space then a shell cylinder of radius b. Show that half of the stored potential energy lies within a cylinder of radius $$\sqrt{ab}$$

Homework Equations

In the attempt

The Attempt at a Solution

I'm not sure what they want me to calculate the potential energy with respect to. If I do it wrt infinity it is infinity, same with 0. Let's say they want me to calculate it wrt the cylinder of radius that we have yet to determine. Then the energy is (getting rid of constants since they won't matter later):
$$\frac{1}{4}+\ln{\frac{R}{a}}$$

At least for the solid cylinder of radius a. Now we add the part of the energy stored in the field from the end of a to the fake cylinder. $$ln(\frac{R}{a})$$

Now we take one half of these values and add them together and set that equal to $$ln(\frac{R}{a})$$ and nothing works and I hate it because I know all of this is wrong.

But this problem cannot be this hard. This is just an RHK problem, I must be missing something simple. Can someone put me the on the right track?

Do not mix potential an potential energy. The potential with respect to one surface changes proportionally with ln(R/a). So the potential difference between the surfaces of radii a and b is?
The energy is stored entirely between the cylindrical surfaces of the capacitor. You can use the formula for the energy of a capacitor in terms of voltage and charge.

 

[NiendorfPhysics]

Do not mix potential an potential energy. The potential with respect to one surface changes proportionally with ln(R/a). So the potential difference between the surfaces of radii a and b is?
The energy is stored entirely between the cylindrical surfaces of the capacitor. You can use the formula for the energy of a capacitor in terms of voltage and charge.

So $$U_{total} = \frac{1}{2} C (ln(\frac{b}{a}))^2$$ and $$U_{radial} = \frac{1}{2} C (ln(\frac{R}{a}))^2$$? I know this is wrong but I don't know what is right.

 

[NiendorfPhysics]

Do not mix potential an potential energy. The potential with respect to one surface changes proportionally with ln(R/a). So the potential difference between the surfaces of radii a and b is?
The energy is stored entirely between the cylindrical surfaces of the capacitor. You can use the formula for the energy of a capacitor in terms of voltage and charge.

Nevermind I got it thank you for the help