# Potential around charged infinite cylinder in E field

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1. Feb 22, 2017

### eckerm

1. The problem statement, all variables and given/known data
The cylinder has a radius a and is perpendicular to the electric field, E(r)=E(x_hat). It also carries charge Q. The potential is of the form V(r,φ)=A0+A0'ln(r)+∑(n=1 to ∞)((Ancos(nφ)+Bnsin(nφ))rn+(An'cos(nφ)+Bn'sin(nφ))r-n)

2. Relevant equations
V=-∫E⋅dl

3. The attempt at a solution
The above equation yields V=-Ex=-Ercosφ

The two boundary conditions are V(r,φ)=-Ercosφ above the surface of the cylinder and V=0 at r=a. Since -Ercosφ=-ErP1(cosφ), the n on the left side of the equation have to be equal to 1.

Starting with the first boundary condition, as r→∞ the r-1 term will be negligible, the A0 will be negligible, and ln(r) term will grow slower than the r term so it will be negligible. Thus, (A1cosφ+B1sinφ)r=-Ercosφ. Solving for A1 gives A1=-E-B1tanφ.

The other boundary condition gives A0+A0'ln(a)+(A1cosφ+B1sinφ)a+(A1'cosφ+B1'sinφ)a-1=0.

Here I'm stuck. The plan was to solve for B1 then have two equations and two unknowns, but I don't know what to do about the prime letters.

2. Feb 22, 2017

It would help to know what the type of material is. Is it a dielectric with a dielectric constant $\epsilon$ ? Also, how is the (free) electric charge distributed on the cylinder? The statement of the problem appears to be rather incomplete.

3. Feb 22, 2017

### eckerm

I assume it's a conducting metal and the charge is evenly dispersed on the surface.

4. Feb 22, 2017

Suggestion then is to do a couple of things: Solve it as if it were a dielectric cylinder in a uniform field without any additional free electric charge and then take the limit as the dielectric constant goes to infinity. In addition add the charge Q=it should be a charge per unit length, but in any case, let the charge be uniformly distributed on the surface and superimpose this solution with the previous one. $\\$ The Legendre method in a uniform field for a cylinder is somewhat lengthy, but there is a shortcut that should give the same answer which I will show you here: Given an applied electric field $E_o$ perpendicular to the axis of the cylinder. There will be some uniform polarization $P_i$ that occurs for this geometry, with $E_p=-(1/2)P_i/\epsilon_o$. (The (1/2) factor is a result of the geometry=it comes out of the detailed Legendre analysis=I'm simply giving a result that I have seen previously stated. The $E_p$ is the electric field that occurs from the surface charge density $\sigma_p$ that results in the case of uniform polarization $P_i$. The Legendre solution will show that this is in fact the case). Meanwhile $E_i=E_o+E_p$ and $P_i=\epsilon_o \chi E_i$ where $\chi$ is the dielectric susceptibility. Then the dielectric constant $\epsilon=\epsilon_o(1+\chi)$. A little algebra gives (assuming my algebra is correct) $E_i=\frac{E_o}{1+(1/2)(\frac{\epsilon}{\epsilon_o}-1)}$. This just gives the electric field inside the cylinder, and the surface polarization charge density is $\sigma_p=P \cdot \hat{n} =P cos(\phi)$ where $P=P_i=(\epsilon-\epsilon_o) E_i$. You should be able to compare this result to the Legendre result and they should agree. For the limit as $\epsilon$ gets large (a conductor), I get $P=2 \epsilon_o E_o$ , and also that $E_i=0$. $\\$ Anyway, I'd be interested in seeing your complete Legendre solution, but I recommend first doing it without any free charge on the cylinder, and just superimposing that part afterwards. $\\$ One additional input is I believe the free charge per unit length is going to give you a $A_o' ln(r)$ term which will be absent until you include the free (surface) charge . $\\$ And an additional input: I believe you need a potential term of the form $-E_o cos(\phi) r$ as $r$ gets large. To satisfy the boundary condition of equal potential everywhere at $r=a$, you need an $A_1'cos(\phi)/r$ term to offset the previous term at $r=a$. This means $A_1'=+E_o a^2$. It remains to calculate the electric fields, along with the resulting surface charge to show that this solution for the potential works everywhere. (It appears to me that the $B$ coefficients might be equal to zero and all you need is $A_1$ and $A_1'$ along with a subsequent $A_o'$. This solution for the potential is of course for $r>a$. For $r<a$ the potential is zero everywhere. $A_o$ can be chosen to make $V=0$ for $r=a$ in the solution for $r \geq a$.) Additional comment: This last method is easier than my previous suggestion above to solve it as a dielectric, etc. Using the fact that the potential must be the same everywhere on the surface makes for a simple solution for $A_1'$.