# Analytical Solution to a Magnetic Field

• hylander4
In summary, the magnetic field inside a cylindrical shielding bucket is linearly polarizable, but can only be solved for approximately when the field's direction is perpendicular to the shield. The problem could be solved using numerical methods such as finite difference or boundary element, but a more accurate solution could be generated by superpositioning the fields of individual objects.
hylander4
Hello all,

I'm currently working on some magnetic shielding, and my supervisor wants me to try and find an analytical solution to the magnetic field inside the shielding.

The shielding is basically bucket shaped (a hollow cylinder with the top cut out), and its meant to shield its interior from the Earth's magnetic field (constant, in one direction). It has a very high permeability, and I'm assuming that its linearly polarizable.

I tried solving the system using boundary conditions and magnetic scalar potential, but I can only solve the approximate situation of an infinite cylinder with this method. What's worse, I can only solve for the magnetic field when the field's direction is perfectly perpendicular to the infinite cylinder.

Does anybody have any tips or alternative methods for solving this system analytically? My experience is limited to undergraduate E+M courses (griffiths, essentially), but I'm willing to learn new methods.

That's a tricky one. I would expect that the first thing you want to do is work in cylindrical coordinates. With that, you go from three dimensions down to two dimensions right off. But from there I don't have any ideas off-hand. There are of course a myriad of numerical methods you can do but that won't help in terms of finding an analytic solution.

I suppose the answer doesn't have to be analytical, as long as the method can be repeated for a few different locations within the shield (all would be on the axis of symmetry for the cylindrical shield, which should simplify things a lot).

I'm not very well acquainted with numerical methods for approximating these things. Any suggestions on topics I could look into?

I'm currently reviewing electrodynamics for school this fall. I'll give the problem a little thought and see if I can come up with an analytic solution. My gut approach would be to solve the problem for just a disk in a magnetic field (the end of the bucket), then solve the problem for hollow cylinder, then add the two together for a given spatial configuration where the disk is really close to an end ...superpostion of solutions.

This problem is a great review problem! I haven't solved an E+M problem in almost a year, and this completely refreshed the basics of boundary conditions, dielectrics, and magnetization density for me.

I'm sure that the superposition would make for a better approximate solution than what I have right now, but wouldn't the field created by the cylinder affect the field inside the disk, and vice versa?

Also, how are you solving the finite hollow cylinder analytically?

I do not believe you can use superposition of individual isolated objects to get the solution for the sum of the objects. It would give you a very approximate answer but it neglects the interaction between the objects of the induced fields in the material.

Since this is a statics problem, you could have a few ways of solving it numerically that should be fairly simple. The simplest is using finite difference. Most often though, the finite element method is used to solve various PDE problems. There is also the boundary element method which works with integral forms of the PDE. For example, here was one of the first results I pulled up via Google:

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1063879&tag=1

These do not need to be very complicated methods for your problem. I think the boundary element method would be the easiest for you because then you would not have to generate a two-dimensional mesh like you would for FEM or FD. Although, for FD, you can just use a square mesh for your mesh and so there is no real "generation" involved. I recall that Matlab can generate a 2D triangular mesh that would be suitable should you use FEM.

## 1. What is an analytical solution to a magnetic field?

An analytical solution to a magnetic field is a mathematical expression that describes the behavior of a magnetic field in a given system. It is obtained by solving the governing equations that govern the behavior of the magnetic field.

## 2. What are the advantages of using an analytical solution to a magnetic field?

Analytical solutions provide a complete and exact description of the magnetic field, which can be used to predict its behavior in different systems. They also allow for easier and faster calculations compared to numerical methods.

## 3. How is an analytical solution to a magnetic field obtained?

An analytical solution is obtained by solving Maxwell's equations, which are a set of four partial differential equations that describe the behavior of electric and magnetic fields. This solution can be found using various mathematical techniques, such as separation of variables or Green's functions.

## 4. Can an analytical solution to a magnetic field be used for any system?

No, an analytical solution may not be possible for all systems. It depends on the complexity of the system and the equations governing the behavior of the magnetic field. In some cases, numerical methods may need to be used to obtain an approximation of the solution.

## 5. How accurate is an analytical solution to a magnetic field?

Analytical solutions are considered to be the most accurate form of solution since they provide an exact mathematical description of the magnetic field. However, the accuracy also depends on the assumptions and simplifications made in the derivation of the solution.

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