# Toroid with Air Gap magnetostatics problem

• Tinaaaaaa
In summary, the conversation discusses the calculation of the magnetic field within an iron ring threaded through a toroidal electromagnet. Using Ampere's law and the boundary conditions on B and H at the interface, the magnitude of the magnetic field H within the iron at a distance r from the center of the ring can be shown to be Hiron=(NI-Md)/2πr. The discontinuity in H is due to the presence of magnetic pole density and surface charge density on the end faces at the air gap.

## Homework Statement

consider a toroidal electromagnet with an iron ring threaded through the turns of wire. The ring is not complete and has a narrow parallel-sided air gap of thickness d. The iron has a constant magnetization of magnitude M in the azimuthal direction. Use Ampere's law in terms of the magnetic field vector H, along with the boundary conditions on B and H at the interface, to show that the magnitude of the field H within the iron at a distance r from the centre of the ring is given by
Hiron=(NI-Md)/2πr[/B]

B=μ0(M+H)
- Ampere's Law

## The Attempt at a Solution

So far I have calculated the B field in the closed toroid which would be, using Ampere's integral law: B=μ0ΝΙ/2πr.

Based on that plus the fact that Bair⊥=Biron⊥, we assume that for the air the B is the same. Thus, I equated
B=μ0(M+H)→(B-μ0M)/μ0=H→H=(NI-M)/2πr

This is clearly not the correct answer so I would appreciate it if you could please show me how to reach the correct answer.

You need another form of Ampere's law which is ## \oint H \cdot dl=NI ##. The ## H ## is not continuous though in this problem, while the ## B ## is assumed to be continuous. ## H ## will take on a different value in the air gap=call that ## H_1(r) ##, and ## H_2(r) ## will be the value in the iron. Expressing the continuity of ## B ## at a distance ## r ## from the center: ##B(r)=\mu_o H_1(r)=\mu_o (H_2(r)+M) ##. ## \\ ## Next, write ## \oint H(r) \cdot dl =NI ## in terms of ## H_1(r) ## and ##H_2(r) ##, and then with just ##H_2(r) ## by substituting in for ##H_1(r) ##. With these hints, you should be able to complete it, by solving for ## H_2(r) ##. ## \\## Note: The standard form of Ampere's law is ## \oint B \cdot dl=\mu_o I_{total} ## where ## I_{total}=I_{conductors}+I_{m} ##, where ## I_{m} ## is from magnetic currents and magnetic surface currents. The ## I_m ## requires extra computation, and it is far simpler to just use ## \oint H \cdot dl=I_{conductors}=NI ##.

Last edited:
You need another form of Ampere's law which is ## \oint H \cdot dl=NI ##. The ## H ## is not continuous though in this problem, while the ## B ## is assumed to be continuous. ## H ## will take on a different value in the air gap=call that ## H_1(r) ##, and ## H_2(r) ## will be the value in the iron. Expressing the continuity of ## B ## at a distance ## r ## from the center: ##B(r)=\mu_o H_1(r)=\mu_o (H_2(r)+M) ##. ## \\ ## Next, write ## \oint H(r) \cdot dl =NI ## in terms of ## H_1(r) ## and ##H_2(r) ##, and then with just ##H_2(r) ## by substituting in for ##H_1(r) ##. With these hints, you should be able to complete it, by solving for ## H_2(r) ##. ## \\## Note: The standard form of Ampere's law is ## \oint B \cdot dl=\mu_o I_{total} ## where ## I_{total}=I_{conductors}+I_{m} ##, where ## I_{m} ## is from magnetic currents and magnetic surface currents. The ## I_m ## requires extra computation, and it is far simpler to just use ## \oint H \cdot dl=I_{conductors}=NI ##.
Thank you!

I should point out, in the pole theory of magnetism, there is a good reason for the discontinuity in ## H ##. Magnetic pole density ## \rho_m ## is given by ## \rho_m= -\nabla \cdot M=\nabla \cdot H ##. This makes a magnetic surface charge density on the end faces at the gap given by ## \sigma_m =\vec{M} \cdot \hat{n} ## which are sources of ## H ## that result in this discontinuity. The calculation can also be done with this surface charge density ## \sigma_m ## and Gauss's law, but using Ampere's law in the modified form, and simply assuming an ## H_1(r) ## and an ## H_2(r) ## allowed us to sidestep this calculation, which would be found to be in complete agreement.

## 1. What is a toroid with air gap magnetostatics problem?

A toroid with air gap magnetostatics problem is a type of electromagnetic problem that involves calculating the magnetic field and flux density within a toroidal shaped object that has an air gap in its core. This type of problem is commonly encountered in the field of electrical engineering and is important for understanding the behavior of magnetic circuits.

## 2. How is a toroid with air gap magnetostatics problem solved?

A toroid with air gap magnetostatics problem is typically solved using the laws of magnetostatics, such as Ampere's law and Faraday's law. These laws are used to calculate the magnetic field and flux density at various points within the toroid, taking into account the effect of the air gap on the magnetic properties of the toroid.

## 3. What factors affect the solution of a toroid with air gap magnetostatics problem?

Several factors can affect the solution of a toroid with air gap magnetostatics problem, including the geometry of the toroid, the material properties of the core and air gap, and the current flowing through the toroid. Other factors, such as temperature and external magnetic fields, may also need to be considered depending on the specific problem.

## 4. What are some common applications of toroid with air gap magnetostatics problems?

Toroid with air gap magnetostatics problems are commonly encountered in the design and analysis of electrical devices such as transformers, inductors, and motors. These problems are also important in the study of magnetic materials and their properties, as well as in the development of new technologies such as magnetic levitation systems.

## 5. What are some techniques for solving toroid with air gap magnetostatics problems?

There are several techniques that can be used to solve toroid with air gap magnetostatics problems, including analytical methods, numerical methods, and computer simulations. Analytical methods involve using mathematical equations and formulas to calculate the solution, while numerical methods use approximations and algorithms to obtain a numerical solution. Computer simulations involve using software programs to model and simulate the behavior of the toroid and its magnetic fields.