Finding surface charge densities and potential (metal cylinder)

1. Oct 13, 2014

ghostfolk

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

A solid metal cylinder of radius $R$ and length $L$, carrying a charge $Q$, is surrounded by a thick coaxial metal shell of inner radius $a$ and outer radius $b$. The shell carries no net charge.
a) Find the surface charge desnities $\sigma$ at $R$, at $a$, and at $b$.
b) Find the potential at the center using $r=b$ as the reference point.
2. Relevant equations
$\sigma=q/A$, $A_{cyl}=2\pi rh$, $V(r)=-\int_{r_1}^{r_2} \frac{\sigma}{r} da$

3. The attempt at a solution
a) $\sigma_R=\frac{Q}{2\pi RL}$ $\sigma_a=\frac{Q}{2\pi aL}$ $\sigma_b=\frac{Q}{2\pi bL}$

b) $V=-\int_b^a \frac{\sigma_b}{r}da-\int_a^R \frac{\sigma_a}{r}da-\int_R^0 \frac{\sigma_R}{r}da$
$da=2\pi RLdr$

2. Oct 14, 2014

ehild

The last equation is not correct. How are the potential function V(r) and the electric field E(r) related ?

You should integrate with respect to the variable the integrand depends on. If you integrate a function depending on r , write it as $\int f(r)dr$ instead of $\int f(r) da$. If you use fixed boundaries in the integral for r the result can not depend on r .
Moreover, a means the inner radius of the shell, do not use it as an integrating variable.

The surface charge densities are all right, but the sign of the surface charge density at a is not correct.
I do not understand what you tried to say with line b). Does the potential change inside a conductor?

ehild

3. Oct 14, 2014

ghostfolk

I should've made it made it more clear that $da$ is the normal vector of the surface area of the cylinder so that $d\vec{a}=2 \pi RLdr\hat{r}$

Yeah I forgot the minus sign. Also the potential does not change in the conductor so I should remove the integral that goes from 0 to $R$

Last edited: Oct 14, 2014
4. Oct 15, 2014

ehild

Note that the potential is line integral of the negative electric field instead of a surface integral.

That is right, but it is zero also somewhere else.

ehild