Find the potential V(r, φ) inside and outside the cylinder

[nickap34]

Homework Statement

Consider two thin half-cylinder shells, made of a conducting material, that are the
right and left halves of a cylinder with radius R. They are separated from each other
at φ=π/2 and φ=3π/2 by small insulating gaps.
The left half, for which π/2<φ <3π/2, is held at potential –V0, and the right half,
which has 0<φ<π/2 and 3π/2<φ<2π, is held at +V0.
Find the potential V(r, φ) inside and outside the cylinder.

Relevant Equations

Unknown

Not even sure where to start.

 

[BvU]

You want to start reading the PF guidelines. This is a second post from you with 'no idea'. I grant you it's not an easy exercise, but before we are allowed to help, you must simply post an effort.
And: 'Unknown' is a nono in PF.
What have you learned so far in your curriculum that might be relevant ?

 

[nickap34]

I am thinking to find the inside potential, you take the double integral from 0 to L and 0 to 2π in cylindrical coordinates and do separation of variables
∫∫V(∅,z)sin(v∅)sin(knz)d∅dz

 

[BvU]

Find the potential V(r, φ) inside and outside the cylinder.

Is this the literal problem statement ? Because you bring in a $z$ and an $L$ that I don't see in there. Can the cylinder be considered infinitely long ?

 

[Abhishek11235]

I am thinking to find the inside potential, you take the double integral from 0 to L and 0 to 2π in cylindrical coordinates and do separation of variables
∫∫V(∅,z)sin(v∅)sin(knz)d∅dz

As @BvU says,please show your work. This problem can be solved in number of ways like solving Laplace equation,Using Green's Function,Poisson Integration with boundary conditions,etc. each being elegant though difficult.

 

[PhDeezNutz]

As @Abhishek11235 stated you would likely want to use Green's functions.

This is a 2D problem in disguise so you want to use the 2D version of the Green's function integral

$\phi(\vec{r}) = \frac{1}{4 \pi \epsilon_0} \int \sigma\left(\vec{r'} \right) G\left( \vec{r},\vec{r'} \right) \,da' - \frac{1}{4 \pi} \int \phi_s \frac{\partial G\left( \vec{r}, \vec{r'} \right)}{\partial n'} \, d\ell'$

You should only concern yourself with the second part of this integral since by definition G=0 on the surface.

The green's function normal derivative for a long cylinder should be easy enough to look up, and you know the potential on the surface. Have at it.

But i must say your lack of effort is disturbing.

Typically this problem (Jackson 2.13) is solved using the half-angle substitution but that can get real ugly real fast. When you find the greens function normal derivative, this page

https://math.stackexchange.com/ques...r21-2r-cos-theta-r2-12-sum-k-1-infty-rk-cos-k
will help you make sense of the integral.

Edit: hopefully I am not breaking forum rules by trying to help you.