Fun Magnetic Scalar Potential Problem

In summary, the conversation discussed the use of magnetic scalar potential to solve a problem involving two cylindrical shells and a paramagnetic material. The main challenge was finding the H-field, which required a function for the paramagnetic material's Magnetization density, M. The solution involved using the equation B = μ0 (H + M) = μ0 μ H to determine M.
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hylander4
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"Fun" Magnetic Scalar Potential Problem

Homework Statement



An infinite cylindrical shell of radius b is placed inside a constant field B which points along the upwards z-axis. A second cylindrical shell of radius a<b is placed inside the first cylindrical shell, and the volume from b>r>a is filled with a paramagnetic material of permeability u. Find the magnetic field everywhere.

Homework Equations



H = B/(u_0) + M

H = -grad W

Laplacian W = -grad M

W is continuous over all boundaries.
The change in dW/dr over a boundary is equal to the negative change in Magnetization over the boundary.

Cylindrical laplace equation solution (From my undergraduate E+M notebook)

W(r, phi) = D_0 + A_0*(a+b*phi) +
$\EPSILON$ [r^n + (A_n*r^-n)]*[B_n*cos(n*phi)+C_n*sin(n*phi)]

Another version of this equation can be found here http://www.cord.edu/faculty/gealy/physics315/SepVarsCyl.pdf" [Broken] on page two.

Summed from n = 1 to infinity


The Attempt at a Solution



Since there's no free current in this situation, I tried using magnetic scalar potential to solve this problem. Unfortunately, I end up with too many variable in the proposed Laplace equation solutions that I need to create to use the boundary conditions.

In my main attempt I had four boundary conditions and seven types of variables.

My main problem is that I need to find the H field to find the B-field, but in order to find the H-field, I also need a function for the paramagnetic material's Magnetization density M.

The forum wants me to post my attempts, but I've already filled three pages of notebook paper with failed algebra and I doubt that that would be constructive. I'm not really looking for an exact solution, I really just need general guidance.
 
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  • #2


Usually you have B = μ0 (H + M) = μ0 μ H, which determines M to be (μ-1) H. In vacuum, of course, μ = 1.
 

1. What is a "Fun Magnetic Scalar Potential Problem"?

A "Fun Magnetic Scalar Potential Problem" is a scientific concept that involves the use of magnetic scalar potential to solve problems in physics and engineering. It refers to a type of problem where the magnetic field is described by a scalar potential instead of a vector potential. This approach simplifies the calculations and can lead to more elegant solutions.

2. How is magnetic scalar potential different from magnetic vector potential?

Magnetic scalar potential and magnetic vector potential are both mathematical representations of magnetic fields, but they differ in their mathematical properties. While magnetic vector potential is a vector quantity, magnetic scalar potential is a scalar quantity. This means that magnetic scalar potential only has magnitude, while magnetic vector potential has both magnitude and direction.

3. How is magnetic scalar potential used in solving problems?

Magnetic scalar potential can be used to solve problems in electromagnetism, such as calculating the magnetic field produced by a current-carrying wire or a permanent magnet. It is also used in engineering applications, such as designing magnetic circuits and motors. By using the scalar potential, the calculations can be simplified and more efficient solutions can be obtained.

4. What are the applications of magnetic scalar potential?

Magnetic scalar potential has various applications in physics and engineering. It is used in the design of electric motors, generators, and transformers. It is also used in the study of magnetism in materials and the behavior of magnetic fields in different environments. In addition, magnetic scalar potential is used in medical imaging techniques, such as magnetic resonance imaging (MRI).

5. What are some challenges in solving "Fun Magnetic Scalar Potential Problems"?

One of the main challenges in solving "Fun Magnetic Scalar Potential Problems" is determining the boundary conditions for the problem. These conditions define the behavior of the magnetic field at the boundaries of the system and are crucial in obtaining accurate solutions. Another challenge is choosing the appropriate mathematical techniques and tools to solve the problem, as different methods may be more suitable for different types of problems.

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