# Formula for Magnetic Field of Cylindrical Magnet | R,L Values

• umby
In summary, the conversation discusses the formula for the magnetic field along the axis of a cylindrical magnet. The formula provided in the conversation is shown to be incorrect, and a corrected version is suggested from a source at MIT. The conversation also addresses the use of different symbols and units in the formula, and concludes with a discussion about the field at different points along the magnet. Finally, a link to a Mathematica expression for the field is provided, but it is noted that it does not converge to the corrected formula at certain limits.
umby
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?

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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.

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:

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umby said:
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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umby said:
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##.
umby said:
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].

umby said:
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?

umby said:
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?

umby said:
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?

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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umby said:
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).

umby said:
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:

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

kuruman said:
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

There is an expression for Hz below the surface (u = 2 in equation 8) and another expression above the surface (u = 0 in equation 8). The expression is double-valued on the surface because Hz is discontinuous across the surface. If you want to compare expressions, you must be on the same side of the surface.

kuruman said:
There is an expression for Hz below the surface (u = 2 in equation 8) and another expression above the surface (u = 0 in equation 8). The expression is double-valued on the surface because Hz is discontinuous across the surface. If you want to compare expressions, you must be on the same side of the surface.

Yes, on the same side of the Surface, u=0. In this way I did the calculation: I took the expression provided for Bz in Wolfram

and I set ρ = 0, z =L/2. I did the integration, to get the result:

Then I took the MIT corrected (with u = 0) or my expression and I get:

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By setting z = L/2 before you do the integral, you lose the "inside - outside" information. Do the integral first, then set z = L/2 and you will see what I mean. At the lower limit the first integral will have a term ##\sqrt{(z-L/2)^2}##. This is ##z-L/2## when ##z>L/2## and ##L/2-z## when ##z<L/2##. If you prematurely set ##z=L/2##, you end up with zero in either case and you lose the discontinuity.

sysprog
kuruman said:
By setting z = L/2 before you do the integral, you lose the "inside - outside" information. Do the integral first, then set z = L/2 and you will see what I mean. At the lower limit the first integral will have a term ##\sqrt{(z-L/2)^2}##. This is ##z-L/2## when ##z>L/2## and ##L/2-z## when ##z<L/2##. If you prematurely set ##z=L/2##, you end up with zero in either case and you lose the discontinuity.
Right!

sysprog
^^^^^ a very nice, cogent, entertaining discussion ^^^^^ thanks

kuruman said:
By setting z = L/2 before you do the integral, you lose the "inside - outside" information. Do the integral first, then set z = L/2 and you will see what I mean. At the lower limit the first integral will have a term ##\sqrt{(z-L/2)^2}##. This is ##z-L/2## when ##z>L/2## and ##L/2-z## when ##z<L/2##. If you prematurely set ##z=L/2##, you end up with zero in either case and you lose the discontinuity.

I have just checked, it gives exactly the corrected MIT expression!

## 1. What is the formula for calculating the magnetic field of a cylindrical magnet?

The formula for calculating the magnetic field of a cylindrical magnet is B = μ₀I/2πr, where B is the magnetic field strength, μ₀ is the permeability of free space, I is the current, and r is the distance from the center of the magnet.

## 2. How do I determine the value of μ₀ for a cylindrical magnet?

The value of μ₀ for a cylindrical magnet can be found in a reference table or can be calculated using the formula μ₀ = 4π x 10⁻⁷ N/A².

## 3. How does the radius of the magnet affect the magnetic field strength?

The magnetic field strength is inversely proportional to the radius of the magnet. This means that as the radius increases, the magnetic field strength decreases.

## 4. What is the significance of the current (I) in the formula for magnetic field of a cylindrical magnet?

The current (I) represents the flow of electric charge through the magnet. It is a crucial factor in determining the strength of the magnetic field.

## 5. Can the formula for magnetic field of a cylindrical magnet be used for all types of magnets?

No, this formula is specifically for cylindrical magnets. Different types of magnets, such as bar magnets or horseshoe magnets, have their own unique formulas for calculating the magnetic field.

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