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

• nickap34
In summary, the conversation revolves around solving a problem in physics using different methods such as solving Laplace equation, using Green's function, and Poisson integration. The problem involves finding the potential inside and outside a cylinder using cylindrical coordinates and separation of variables. The conversation also mentions using Green's function normal derivative and the half-angle substitution method to solve the problem. The participants also urge the person seeking help to show their work and make an effort towards solving the problem.
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.

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 ?

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

nickap34 said:
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 ?

nickap34 said:
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.

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

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

Abhishek11235

## 1. What is the formula for finding the potential inside and outside a cylinder?

The formula for finding the potential V(r, φ) inside and outside a cylinder is given by V(r, φ) = -k(r^2 + a^2)cos(φ), where k is a constant, r is the distance from the center of the cylinder, and a is the radius of the cylinder.

## 2. How do you determine if a point is inside or outside the cylinder?

To determine if a point is inside or outside the cylinder, you can use the distance formula to calculate the distance from the point to the center of the cylinder. If the distance is less than the radius of the cylinder, the point is inside. If the distance is greater than the radius, the point is outside.

## 3. Can the potential inside and outside a cylinder be the same?

No, the potential inside and outside a cylinder cannot be the same because the potential formula is dependent on the distance from the center of the cylinder. Since the distance from the center is different inside and outside the cylinder, the potential will also be different.

## 4. How does the potential change as you move away from the center of the cylinder?

The potential decreases as you move away from the center of the cylinder. This is because the potential formula is inversely proportional to the distance from the center of the cylinder. As the distance increases, the potential decreases.

## 5. Can the potential inside and outside a cylinder be negative?

Yes, the potential inside and outside a cylinder can be negative. This is because the potential formula includes a negative sign, and the potential is dependent on the distance from the center of the cylinder. If the distance is large enough, the potential can be negative inside or outside the cylinder.

Replies
7
Views
2K
Replies
2
Views
551
Replies
4
Views
2K
Replies
1
Views
2K
Replies
7
Views
4K
Replies
4
Views
5K
Replies
13
Views
2K
Replies
9
Views
4K