# Cylindrical magnet

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## Main Question or Discussion Point

Hi, I am looking for the formula of the magneti field along the axis of a axially magnetized cylindrical magnet.
Unfortunately, there are quite different answers on Internet.
Is the uploaded formula (where R is the magnet radius and L its length) correct? #### Attachments

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kuruman
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It looks right. If I were you, I would go along the more complete equation (8) given here
http://web.mit.edu/6.013_book/www/chapter9/9.3.html
because it provides the field inside the magnet as well. Note that this equation has the mid point of the magnet as the origin, unlike yours where the origin is at one of the faces.

Thank you very much. However, eq. (8) does not have an anlytical solution, does it? You need to use numerical integration to solve it.

kuruman
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It is an analytical solution. Perhaps you confused the $d~M_0$ up front for a differential. It is not; $d$ is the length of the cylinder as shown in the drawing and $M_0$ is the magnitude of the magnetization.

Ohh, I did!
Thank you so much.
Is a 1/2 the value subtracted to z/d?

Actually I see that the two formulae are different. Indeed, replacing d with L and changing the origing z = z1-d/2 in eq.(8), you get: #### Attachments

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kuruman
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Actually I see that the two formulae are different. Indeed, replacing d with L and changing the origing z = z1-d/2 in eq.(8), you get:
View attachment 226284
Right.

So μ0 is replaced by d in your formula. IS M0 measured in A/m?

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kuruman
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So μ0 is replaced by d in your formula.
No. If you follow the proof in the MIT link, you will see that μ0 drops out in equation (6). The symbol $d$ is the half-length of the cylinder. In other words $2d=L$.
IS M0 measured in A/m?
Yes.

I have a doubt: going back to eq. (8), the term in the square brackets is adimensional, M0 is [A/m] times d[m], you get a Hz in [A].

kuruman
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I have a doubt: going back to eq. (8), the term in the square brackets is adimensional, M0 is [A/m] times d[m], you get a Hz in [A].
You are correct. I re-derived the whole thing and equation (8) should have $\frac{M_0}{2}$ as the overall factor without the $d$. The $d$ in the numerator cancels out when one makes the denominator dimensionless. Whoever posted this probably missed the cancellation, but this goes to show that you should distrust what you see on the web, even if it's from MIT.

Thank you very much. Not distrust, but re-think critically.
Finally I have the field thanks to your help.
Last question: I can swop from B to H with μ0, but if I need the force on a particle in the field of the magnet, I have to use B or H?

kuruman
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Thank you very much. Not distrust, but re-think critically.
Finally I have the field thanks to your help.
Last question: I can swop from B to H with μ0, but if I need the force on a particle in the field of the magnet, I have to use B or H?
I would say use B. What equation do you plan to use to find the force?

The classical derivation from the analogy with the force on an electric dipole: (m*Del) B.

Kuruman, please, do you know a complete expression for the field of a cylindrical magnet, not only along the axis of the magnet?

kuruman
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The classical derivation from the analogy with the force on an electric dipole: (m*Del) B.

Kuruman, please, do you know a complete expression for the field of a cylindrical magnet, not only along the axis of the magnet?
An analytical expression for that involves elliptic integrals so I don't think there is one. If the force is to be calculated very near the z-axis you may do a series expansion of the radial field component. If not, then a numerical calculation may be appropriate.

And and expression to calculate numerically it? Do you know a link in the web?

kuruman
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Sorry, I don't know of any links. I am sure you can research the web as well as me, if not better.

Applying the two formulae, which we have discussed so far, it is easy to calculate the z-component of field at the top of the magnet along its axis: Now, I found a page of Mathematica by Wolfram, which reports an expression, to be computed numerically, for the whole field:
http://demonstrations.wolfram.com/MagneticFieldOfACylindricalBarMagnet
However, this expression does not converge to the above one for ρ -> 0 and z ->L/2.
Do you have the same result?

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kuruman
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Applying the two formulae, which we have discussed so far, it is easy to calculate the z-component of field at the top of the magnet along its axis:
View attachment 226332
Now, I found a page of Mathematica by Wolfram, which reports an expression, to be computed numerically, for the whole field:
http://demonstrations.wolfram.com/MagneticFieldOfACylindricalBarMagnet
However, this expression does not converge to the above one for ρ -> 0 and z ->L/2.
Do you have the same result?
To me the Wolfram expression looks like the same as the integral leading to the corrected MIT expression. I don't understand what you mean by ρ→0. ρ is an integration variable. If you want to see what happens when you have a long thin cylindrical magnet, first you integrate then you look what happens in the limit R/L <<1.

The integration variables are R and Φ. R integrated over [0, a] and Φ over [0, 2 Pi].
After the integration, you have the field in generic ρ and z, namely Bz(ρ,z).
I want the field in (ρ = 0, z =L/2), i.e. Bz(0,L/2).

kuruman
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The integration variables are R and Φ. R integrated over [0, a] and Φ over [0, 2 Pi].
After the integration, you have the field in generic ρ and z, namely Bz(ρ,z).
I want the field in (ρ = 0, z =L/2), i.e. Bz(0,L/2).
Yes you are correct. I was thinking of the other two expressions where R is the radius of the cylinder. I have not done it, but it looks like if you set φ=0 and do the φ integral (i.e. stick in a 2π) in the Wolfram expression, you should get equation (6) in the MIT link. Have you tried that?

This is the problem, you do not get the MIT expression for Bz(0,L/2, "or the surface of the magnet") but: While the MIT expression is: #### Attachments

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kuruman
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Hz is discontinuous at the surface. Look at the plot in the MIT link below equation (5).

Hz is discontinuous at the surface. Look at the plot in the MIT link below equation (5).
I mean the field on the magent Surface coming from the outside of the magnet

kuruman