# Ampere's Law for a toroid (finding relative permeability of iron)

• phantomvommand
Ampere’s Law can be applied to calculate the magnetic field inside the toroid:B_{internal} = 4 \pi \frac {N \pi D}{r}

#### phantomvommand

Homework Statement
Why is this equation: B(D - d)/mu + Bd = mu0 N I true?
Relevant Equations
Ampere's Law
Why is this equation: B(D - d)/mu + Bd = mu0 N I true?

B = magnetic field in the hole of the toroid
D = Average diameter of the toroid
d = Diameter of hole of toroid
mu = relative permeability of iron, or whatever the toroid is made of
mu0 = 4pi x 10^-7
N = Number of turns on the toroid
I = current through each turn of the toroid

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Hi @phantomvommand.

What do you mean by 'in the hole of the toroid'? Do you mean at the geometrical centre-point (on the central axis, in free space)?

The Homework Statement says: "B(D - r)/mu + Br = mu0 N I"
But you then state: "B(D - d)/mu + Bd = mu0 N I"
So 'r' has changed into 'd'. This suggests you might have accidentally changed a radial value to a diameter value.

Does 'average diameter 'D' mean the overall diameter of the toroid or is it the diameter of the coil of wire?

Your value of 'mu0' has missing units,

We use ##\mu_r## for relative permeability.
##\mu## means absolute perpemeability: ##\mu = \mu_r \mu_0##

You need to indicate what have youu tried so far (e.g. links and insights from searching for 'magnetic field at centre of toroid').

You need a diagram. And it's best to use Latex for equations. You'll get a better response that way.

phantomvommand and Delta2
Steve4Physics said:
Hi @phantomvommand.

What do you mean by 'in the hole of the toroid'? Do you mean at the geometrical centre-point (on the central axis, in free space)?

The Homework Statement says: "B(D - r)/mu + Br = mu0 N I"
But you then state: "B(D - d)/mu + Bd = mu0 N I"
So 'r' has changed into 'd'. This suggests you might have accidentally changed a radial value to a diameter value.

Does 'average diameter 'D' mean the overall diameter of the toroid or is it the diameter of the coil of wire?

Your value of 'mu0' has missing units,

We use ##\mu_r## for relative permeability.
##\mu## means absolute perpemeability: ##\mu = \mu_r \mu_0##

You need to indicate what have youu tried so far (e.g. links and insights from searching for 'magnetic field at centre of toroid').

You need a diagram. And it's best to use Latex for equations. You'll get a better response that way.

Yes, I realized my mistake. the 'd' is correct, and it refers to the diameter of the hole in the toroid. I am assuming 'D' refers to the overall diameter of the toroid, taken at a radius (r1 + r2)/2 away from the centre of the toroid, where r1 and r2 are the inner and outer radii of the toroid. (This is probably what is meant by "average diamater.")

Yes, i did not type in the units for ##\mu_0##. Apologies for this.
The book where I took this question from refers to the relative permeability as ##\mu##. Thank you for highlighting this distinction. Side note: This book has multiple printings errors. If your solution suggests that mu is in fact the absolute permeability, instead of the relative permeability, please do share it with me still, as this could be yet another typo.

I have found the following:
https://physics.stackexchange.com/questions/381232/magnetic-induction-at-the-centre-of-a-toroid

This suggests that there is a downward magnetic field at the centre of the coil, equivalent to ##\mu_0*I*\pi/d##, where d is the diameter of the "hole".

Side note: It is quite uncommon to work with diameters instead of radii, and one should take note of this.

As usual, the magnetic field in the toroid can be found with ampere's law. I am puzzled about the fact the 2 terms add together to give the term on the Right hand side, which resembles Ampere's Law. If you apply Ampere's Law to a loop inside the toroid, one can find an expression for the magnetic field inside the toroid (very standard), and you can express the field inside the toroid in terms of the field in the centre of the toroid. I suppose that because relative permeability is involved, when applying Ampere's Law for a loop inside the toroid, ##\mu_0## should be replaced with ##\mu##.

Apologies for not presenting the equations nicely with Latex. I am still in high school, and have not yet had the time to learn Latex.

Yes, you reply to my previous post was very helpful, thank you very much!

This question was taken from Experimental problem 1, Part 4, of the 2004 Asian Physics Olympiad, where the equation in question was mentioned; I am wondering about its proof.

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Steve4Physics
I’m as puzzled as you.

The number of turns per unit length around the toroid is ##\frac N {\pi D}##.
The internal field inside the coil (the same as for a long solenoid) is therefore
##B_{internal} = \frac {\mu_r \mu_0 N I} {\pi D}##

At the central point on the toroid's axis, the current through the toroid is ‘seen’ as a single simple loop (entering toroid, traveling once around, and exiting). We use the standard equation for the field at the centre of a simple current loop:
##B_{centre} = \frac {\mu_0 I} {D}## (note ##\mu_r## is not used)

I don’t see how your formula “B(D - d)/mu + Bd = mu0 N I” can possibly be obtained.

Are you sure you have the original question accurate and complete?

Sorry. Don’t think I can help - maybe someone else will come in.

phantomvommand
It looks like they are trying to work the problem of a transformer with a gap, but it doesn't apply here. See https://www.feynmanlectures.caltech.edu/II_36.html . Looks like they tried to use (36.27) or something of the sort. Instead, all they need is (36.20).

phantomvommand and Steve4Physics

It states “Put the Hall sensor into the gap on the core” and “Measure … the field B in the gap”.

Sounds like this is a toroid with a gap somewhere. So B is not the field at the centre of a toroid, but the field in the gap. See @Charles Link’s Post #5.

This shows the importance of accurately and completely stating the original question verbatim – without interpretation or simplification. And preferably with a diagram. Or a lot of time/effort can get wasted.

Edit. The derivation of the formula requires a clear diagram showing where the 'gap' is (or sufficient information to draw one).

Sometimes the discussions we had when we solved similar problems in previous threads might be useful.

additional note: Upon looking this one over again, iron saturates at an ## M ## of about 1.6-2.2. You can get currents of this kind and that many turns in the primary and still stay in the linear region only if it balanced by offsetting currents in the secondary. In some ways the scenario here is highly impractical, but may be useful for computational purposes.

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phantomvommand
Steve4Physics said:

It states “Put the Hall sensor into the gap on the core” and “Measure … the field B in the gap”.

Sounds like this is a toroid with a gap somewhere. So B is not the field at the centre of a toroid, but the field in the gap. See @Charles Link’s Post #5.

This shows the importance of accurately and completely stating the original question verbatim – without interpretation or simplification. And preferably with a diagram. Or a lot of time/effort can get wasted.

Edit. The derivation of the formula requires a clear diagram showing where the 'gap' is (or sufficient information to draw one).
They used at the top, (note (4)), that the diameter of the core is ## \rho=25 ## mm. I think the letter ## \rho ## in that formula mentioned in the OP needs to be the circumference of the core ## C=\pi \rho ##, and not the diameter of the core. (Note: From the picture, ## \rho ## is clearly not the diameter of the circular cross section of the toroid, but rather the diameter of the ring.)

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phantomvommand and Steve4Physics
They used at the top, (note (4)), that the diameter of the core is ## \rho=25 ## mm. I think the letter ## \rho ## in that formula mentioned in the OP needs to be the circumference of the core ## C=\pi \rho ##, and not the diameter of the core. (Note: From the picture, ## \rho ## is clearly not the diameter of the circular cross section of the toroid, but rather the diameter of the ring.)
I skimmed over the notes and picture in the link - I should have looked more carefully. The value 'd' is the gap-width, not the innner coil diameter.

@Steve4Physics The OP should have posted the "link" that you found, but in any case the formula in the "link" should read

##\frac{B(C-d)}{\mu_r}+Bd=\mu_o NI ##. (The "link" has it incorrectly).

Note: It needs to be a ## C ##,(for circumference), instead of a ## \rho ##, and also a ## \mu_r ##, (where ## \mu=\mu_r \mu_o ##), instead of a ## \mu ##.

and yes, ##d ## is the gap width.

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phantomvommand, Steve4Physics and Delta2
@Steve4Physics The OP should have posted the "link" that you found, but in any case the formula in the "link" should read

##\frac{B(C-d)}{\mu_r}+Bd=\mu_o NI ##. (The "link" has it incorrectly).

Note: It needs to be a ## C ##,(for circumference), instead of a ## \rho ##, and also a ## \mu_r ##, (where ## \mu=\mu_r \mu_o ##), instead of a ## \mu ##.

and yes, ##d ## is the gap width.
Hi @Charles Link . Thanks for this, and the links you’ve provided.

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Steve4Physics said:

It states “Put the Hall sensor into the gap on the core” and “Measure … the field B in the gap”.

Sounds like this is a toroid with a gap somewhere. So B is not the field at the centre of a toroid, but the field in the gap. See @Charles Link’s Post #5.

This shows the importance of accurately and completely stating the original question verbatim – without interpretation or simplification. And preferably with a diagram. Or a lot of time/effort can get wasted.

Edit. The derivation of the formula requires a clear diagram showing where the 'gap' is (or sufficient information to draw one).
Very sorry for this. I see my wrong interpretation of the problem has lead to much confusion. I did not expect the meaning of gap width to be as such, as I have yet to study/see such setups.