Max Voltage Cylindrical Capacitor w/Dielectric

[bodensee9]

Homework Statement

Hello:
I am asked to find the maximum voltage in a cylindrical capacitor. The capacitor consists of an inner wire and an outer cylindrical shell. The wire has radius $$r_{1}$$ and the cylinder has inner radius $$r_{2}.$$ The space between the wires is filled with a dielectric having dielectric constant $$\kappa.$$

Homework Equations

This is in CGS units (actual calculations have been converted to SI)
So I know that the electric field E in a dielectric is $$E_{no_dielectric}/\kappa$$. So then if my cylindrical capacitor has E field = $$\frac{2\lambda}{r}$$, then my E field inside the dielectric material would be $$\frac{2\lambda}{r\kappa}$$. So then if I am given a value for the dielectric strength of the dielectric (say $$A$$, which would happen at the inner radius of the cylindrical shell which is $$r_{2}$$), would I do
$$A = \frac{2\lambda}{r\kappa}$$, and then I can find the charge density which is
$$\frac{Ar\kappa}{2}$$. And, since the potential between the wire and the shell would be $$2\lambda*ln\frac{r_{2}}{r_{1}}$$, would I just substitute the new value for lambda I got to get the potential? For some reason this was marked wrong?

Thanks!

 

[jbsteele]

The Attempt at a SolutionThe electric field inside a dielectric is found by E_{no_dielectric}/\kappa. So, if you know the electric field outside the dielectric, you can find the electric field inside the dielectric. The electric field outside a cylindrical capacitor with radius r_{1} and inner radius r_{2} is \frac{2\lambda}{r}. So, the electric field inside the dielectric would be \frac{2\lambda}{r\kappa}. Now, if you are given a value for the dielectric strength of the dielectric (say A, which would happen at the inner radius of the cylindrical shell which is r_{2}), then you can solve for the charge density which is \frac{Ar\kappa}{2}. And, since the potential between the wire and the shell would be 2\lambda*ln\frac{r_{2}}{r_{1}}, you can substitute the new value for lambda you got to get the potential.

 

[tomcue]

Your approach to finding the maximum voltage in a cylindrical capacitor with a dielectric is correct. However, there are a few things that need clarification.

Firstly, the dielectric strength of a material is typically measured in units of volts per meter (V/m). In your equation, you have used the dielectric constant (kappa) in place of the dielectric strength. These two are related, but not the same. The dielectric strength is a measure of the maximum electric field a material can withstand before breaking down, while the dielectric constant is a measure of how much the material can affect the electric field. In order to use the dielectric strength in your equation, you would need to convert it to the appropriate units (V/m).

Secondly, the dielectric strength should be used at the point where the electric field is the strongest, which would be at the inner radius of the cylindrical shell (r2 in your equation). This means that your equation should be A = 2λ/(r2κ), instead of r1.

Once you have the correct value for the dielectric strength, you can use it in your equation to find the charge density and ultimately the potential between the wire and the shell. Make sure to use the correct value for lambda, which would be the charge density at the inner radius of the cylindrical shell.

I hope this helps clarify your approach and leads you to the correct solution. Good luck with your homework!