H and B field at the midpoint inside of a magnet

[Monocles]

Homework Statement

For a magnet of length l = 16 cm, cross-section .25 cm^2, and magnetization M = 7.8x10^5 A/m, estimate H and B at the midpoint.

Homework Equations

Unsure if any of these equations are relevant, but these are the ones that I've tried.

q_m = MA
B = \mu_0 M
\mu_0 H = \mu_0 M

The Attempt at a Solution

The answers given in the book are H = 485 A/m, B = .98T. By just randomly plugging stuff into equations, I figured out that B = \mu_0 M applies. However, I can't explain why (I need to justify all of our answers, I can't just throw around equations). The book isn't very clear as to what situations that this applies in.

I have yet to figure out how they got H = 485 A/m. Does it have anything to do with the hysteresis loop of the magnet? If so, how would I figure out the hysteresis loop specific to this magnet?

I've noticed that I have yet to take into account the length or cross-sectional area of the magnet. However, I can't find any equations where those would be useful. I tried calculating the B-field from the equation for a magnetic dipole and that was incorrect. I thought about taking q_m and -q_m to be distributed evenly across across the north and south poles and then using a triple integral to find the B and H fields, but there haven't been any integrals used in the chapter related to this kind of problem, and besides that I haven't even learned how to do triple integrals yet, so I figured that that would be a dead end.

So, I'm at a loss of what to do now. I kind of wish that I had a textbook that explained things better. Are there any online physics 2 textbooks I could consult?

Edit: Right after I made this post I figured out that B = \mu_0 (H + M) applies, but that still doesn't explain why B = \mu_0 M worked.

Edit2: Another relation: B = 2 pi k_m M^2 A / q_m sort of works. It's the equation for lifting strength, but it's off by a factor of 2. How can I justify that the lifting strength of the center of the magnet is twice as strong in the center of the magnet as it is from one pole? Would saying that both sides of the magnet are being attracted to a soft magnet be correct?

 

[anathema]

Thank you for your post. It seems like you have made some progress in solving this problem. Let me try to provide some guidance and explanation to help you understand the concepts and equations involved.

Firstly, let's review the equations you mentioned:

1. q_m = MA - This equation represents the magnetic dipole moment, where q_m is the dipole moment, M is the magnetization, and A is the cross-sectional area.

2. B = \mu_0 M - This equation is known as the magnetic field intensity equation and relates the magnetic field B to the magnetization M. \mu_0 is the permeability of free space.

3. \mu_0 H = \mu_0 M - This equation is known as the magnetic field strength equation and relates the magnetic field strength H to the magnetization M. Here, \mu_0 is the permeability of free space.

Now, let's apply these equations to the given problem. We are asked to estimate H and B at the midpoint of the magnet. This means we need to find the magnetic field and field strength at the center of the magnet. To do this, we can use the equation B = \mu_0 M, which you mentioned in your post. This equation is valid for a long, thin magnet with uniform magnetization. In this case, the magnet has a length of 16 cm and a cross-sectional area of .25 cm^2, which can be considered long and thin. Therefore, we can use this equation to estimate the magnetic field at the center of the magnet.

To find the magnetic field strength at the center of the magnet, we can use the equation \mu_0 H = \mu_0 M. This equation is valid for any magnet, regardless of its shape or size. Therefore, we can use this equation to estimate the magnetic field strength at the center of the magnet.

Now, let's see how we can justify the equation B = \mu_0 M. This equation is based on the fact that the magnetic field inside a long, thin magnet is uniform and equal to the magnetization M. This means that the magnetic field at any point inside the magnet is equal to the magnetization at that point. The equation B = \mu_0 M simply represents this fact mathematically.

Finally, let's address your question about using the equation B = 2 pi k_m M^2 A / q_m

 

[thomate1]

Dear student,

Thank you for reaching out with your question. I understand your confusion and I will do my best to explain the concepts and equations involved in this problem.

Firstly, let's start with the equation B = μ0M. This equation states that the magnetic field inside a magnet is directly proportional to its magnetization. This means that the stronger the magnetization of the material, the stronger the magnetic field inside the magnet. This equation applies to all points inside the magnet, including the midpoint.

Now, let's look at the equation μ0H = μ0M. This equation is known as the demagnetization curve and it relates the magnetic field strength and the magnetization of a magnet. It is commonly used in the study of hysteresis loops, which you mentioned in your attempt at a solution. In this case, the value of H = 485 A/m is most likely obtained from a demagnetization curve specific to the material and dimensions of the magnet in question.

Next, let's consider the equation B = μ0(H + M). This equation is known as the magnetization curve and it also relates the magnetic field strength and the magnetization of a material. However, this equation is used for external magnetic fields, not for points inside a magnet. This is why it did not work when you tried to use it in your solution.

Finally, let's look at the equation B = 2πkM^2A/qm. This equation is used to calculate the lifting strength of a magnet, as you mentioned. However, it is not relevant to this problem as it is used for external magnetic fields and not for points inside a magnet.

To summarize, the equations B = μ0M and μ0H = μ0M are relevant to this problem and are used to calculate the magnetic field at the midpoint inside the magnet. The value of H = 485 A/m is most likely obtained from a demagnetization curve specific to the material and dimensions of the magnet. As for your question about textbooks, I would recommend checking out online resources such as Khan Academy or HyperPhysics for a more detailed explanation of these concepts.

I hope this helps to clarify the concepts and equations involved in this problem. Keep up the good work in your studies!

Best regards,