# Solving Laplace's equation in polar coordinates for specific boundary conditions

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• Stefan H
In summary, the method for solving the physics problem involves obtaining the Fourier series for the derivative of the magnetic scalar potential with respect to r on the boundary, and using this to determine the coefficients for r^{\pm n}\cos(n\theta) and r^{\pm n}\sin(n\theta). This leads to a set of simultaneous equations that can be solved to determine the constants needed to obtain the solution. However, further understanding of second-order linear ODEs is necessary to fully intuitively understand the process.
Stefan H
TL;DR Summary
Solving Laplace's equation for magnetic potentials inside and outside a ferromagnetic wire with specific boundary conditions. Help needed in understanding how to obtain the solution.
Hello everybody,

Currently I am doing my master's thesis and I've encountered a physics problem which is very difficult for me to solve. The problem I have is finding equations for the magnetic scalar potential inside and outside a ferromagnetic wire for specific boundary conditions.

Explanation of the problem:
A ferromagnetic wire is located in an external magnetic field. This means there is a magnetic potential inside and outside of the wire. I want to find specific solutions on the surface of the wire (r=a).

My solution approach was to first solve Laplace's equation for polar coordinates for both magnetic potentials:

Now I want to find an expression for both potentials on the surface of the wire to obtain this solution (According to Gerber: "Applied Magnetism", 1994):

with the given boundary conditions:

The next goal would be to determine the constants C1, A1, C2 and A2. However, I really don't get how Gerber came to this type of solution. He only mentioned to use Laplace's equation and the boundary conditions with quote: "linear combination of cylindrical harmonics" to get his solution.

My overall goal is to understand his approach for this kind of problem, so I can apply it to a different geometry. It would be very nice if anybody here could help me out :)

Thanks in advance and happy Holidays,
Stefan

Last edited:
Stefan H said:
Summary:: Solving Laplace's equation for magnetic potentials inside and outside a ferromagnetic wire with specific boundary conditions. Help needed in understanding how to obtain the solution.

Hello everybody,

Currently I am doing my master's thesis and I've encountered a physics problem which is very difficult for me to solve. The problem I have is finding equations for the magnetic scalar potential inside and outside a ferromagnetic wire for specific boundary conditions.

Explanation of the problem:
A ferromagnetic wire is located in an external magnetic field. This means there is a magnetic potential inside and outside of the wire. I want to find specific solutions on the surface of the wire (r=a).
View attachment 294600

My solution approach was to first solve Laplace's equation for polar coordinates for both magnetic potentials:View attachment 294601

Now I want to find an expression for both potentials on the surface of the wire to obtain this solution (According to Gerber: "Applied Magnetism", 1994):

View attachment 294602

with the given boundary conditions:

View attachment 294603

The next goal would be to determine the constants C1, A1, C2 and A2. However, I really don't get how Gerber came to this type of solution. He only mentioned to use Laplace's equation and the boundary conditions with quote: "linear combination of cylindrical harmonics" to get his solution.

My overall goal is to understand his approach for this kind of problem, so I can apply it to a different geometry. It would be very nice if anybody here could help me out :)

Thanks in advance and happy Holidays,
Stefan

The method is to obtain the Fourier series for $\partial \Phi/\partial r$ on the boundary, and determine the coefficients of $r^{\pm n}\cos(n\theta)$ and $r^{\pm n}\sin(n\theta$ accordingly. In this case that is straightforward: the Fourier series for $\mu_0M\cos\theta$ is $\mu_0M\cos\theta$. Thus from the boundary conditions you get the following simultaneous equations for $n \geq 1$: $$\begin{split} A_n' - A_n'' &= 0\\ B_n' - B_n'' &= 0\\ \mu_0A_n' + \mu_fA_n'' &= \begin{cases} \mu_0M, & n = 1 \\ 0 & n > 1, \end{cases} \\ \mu_0B_n' + \mu_fB_n'' &= 0 \end{split}$$ These have the unique solution $A_n' = A_n'' = 0$ for $n >1$ while $A_1' = A_1''$ is non-zero, with $B_n' = B_n'' = 0$ for all $n$.

Once you become sufficiently familiar with second-order linear ODEs, it will be intuitively obvious that only the $r^{\pm1}\cos\theta$ terms were required here.

Stefan H

However I still have some questions left:
1. Why is it necessary to obtain the Fourier series? The way I would proceed on this problem is to form the derivative depending on theta and r separately and put it in the boundary conditions. However, this would just lead to more equations which I don't know how to simplify.
2. Maybe I put it wrong, but A´n, A´´n, B´n and B´´n are just coefficients. They don't mean the derivative of something. Is the answer still true then?
3. I sadly don't really understand why the Fourier series of
is required, how it is obtained and how the following simultaneous equations are obtained. If you could elaborate on that, it would be very helpful.

pasmith said:
The method is to obtain the Fourier series for $\partial \Phi/\partial r$ on the boundary, and determine the coefficients of $r^{\pm n}\cos(n\theta)$ and $r^{\pm n}\sin(n\theta$ accordingly. In this case that is straightforward: the Fourier series for $\mu_0M\cos\theta$ is $\mu_0M\cos\theta$. Thus from the boundary conditions you get the following simultaneous equations for $n \geq 1$: $$\begin{split} A_n' - A_n'' &= 0\\ B_n' - B_n'' &= 0\\ \mu_0A_n' + \mu_fA_n'' &= \begin{cases} \mu_0M, & n = 1 \\ 0 & n > 1, \end{cases} \\ \mu_0B_n' + \mu_fB_n'' &= 0 \end{split}$$ These have the unique solution $A_n' = A_n'' = 0$ for $n >1$ while $A_1' = A_1''$ is non-zero, with $B_n' = B_n'' = 0$ for all $n$.

Once you become sufficiently familiar with second-order linear ODEs, it will be intuitively obvious that only the $r^{\pm1}\cos\theta$ terms were required here.
Sorry that I am posting that again, but maybe you didn't see my answer since I did not reply directly to you.

However I still have some questions left:
1. Why is it necessary to obtain the Fourier series? The way I would proceed on this problem is to form the derivative depending on theta and r separately and put it in the boundary conditions. However, this would just lead to more equations which I don't know how to simplify.
2. Maybe I put it wrong, but A´n, A´´n, B´n and B´´n are just coefficients. They don't mean the derivative of something. Is the answer still true then?
3. I sadly don't really understand why the Fourier series of

is required, how it is obtained and how the following simultaneous equations are obtained. If you could elaborate on that, it would be very helpful.

## 1. What is Laplace's equation in polar coordinates?

Laplace's equation in polar coordinates is a partial differential equation that describes the distribution of a scalar field in a two-dimensional polar coordinate system. It is written as ∇²u = 0, where ∇² is the Laplace operator and u is the scalar field.

## 2. How is Laplace's equation solved in polar coordinates?

Laplace's equation in polar coordinates can be solved using separation of variables, which involves breaking down the equation into simpler parts and solving each part separately. This results in a series of solutions, which can be combined to form the general solution.

## 3. What are boundary conditions in Laplace's equation in polar coordinates?

Boundary conditions in Laplace's equation in polar coordinates are constraints that are applied to the solution in order to determine the specific values of the scalar field at the boundary of the system. These conditions can be specified as either Dirichlet or Neumann boundary conditions.

## 4. What are Dirichlet boundary conditions in Laplace's equation in polar coordinates?

Dirichlet boundary conditions in Laplace's equation in polar coordinates specify the value of the scalar field at the boundary of the system. This means that the solution must equal a specific value at the boundary, which is often given as a function of the polar coordinates.

## 5. What are Neumann boundary conditions in Laplace's equation in polar coordinates?

Neumann boundary conditions in Laplace's equation in polar coordinates specify the derivative of the scalar field at the boundary of the system. This means that the solution must have a specific slope or gradient at the boundary, which is often given as a function of the polar coordinates.

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