# Need help on mutual inductanfce calculation

#### HiEdgar

[SOLVED] need help on mutual inductanfce calculation...

Hi, All:

I am trying to calculate the mutual inductance between the coil (100 turns) and the magnet. Does anybody know any approaches that may lead a satisfactory answer.

So far, I have been using Neumann's formula which is the formula for mutual inductance, M, between two loops. I treated my magnet as another coil and used Neumann's formula to determine M then. The problem with that is that the diameter of the wires of my coil are finite whereas neumann's formula treats them as infinitisemally small. Because of this, my M became a funciton of magnet's wire diameter, which is not right. Any input is appreciated.

Thank you,

Edgar

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#### arcnets

HiEdgar,
I'm afraid I don't exactly understand what your problem is. I assume you do an experiment with a coil and a magnet. Do you want to know what voltage these can produce?
If so, I think this depends crucially on the geometrical setup and relative motion, the magnet's strength, and the dimensions of the coil. Can you provide further information on this?
Plus, you mention a Neumann's formula which I don't recall right now. Can you give some details on your calculation please?

#### HiEdgar

here is more info on mutual inductance

Hi,

Thank for a reply. Here is more information you requested. My objective is to measure the strength (Br) of a given magnet. The experimental setup is as follows: as we move the magnet closer to the coil, there will be emf generated in teh coil. In can be shown that the strength of a magnet is proportional to the
Integral[emf]/delta_[M], where M is the mutual inductance. I can measure Integral[emf]. But I need to know M, which is a function of a position. My first approach was to use Neumann's formula which you can find in Griffiths' or Jackson's E&M textbooks. However Neumann's formula deals with wires whose diameter is infinitesimally small.

From your email, I also got a feeling that you may know some techniques on calculating the emf induced in the coil due to magnet's movement. Formulawise, it is not bad: emf=-N dFlux/dt, Flux=Integral{B.da}, B=mu0 H, H=-Grad[Phi_m], Phi_m=Integral[n.M/r da]. Unfortunately, when I used this approach, the math got so hairy after calculation of B that is was really hard to go further. If you know of any other approaches please let me know,

thank you,

Edgar

#### arcnets

OK. I see.
So, you put your coil and magnet on the same axis. And let the magnet move from distance x1 to distance x2. In doing so, you record the induced voltage V as function of time, and you're able to find the integral V dt. Right?

OK. What do we know about induced voltage?
We have Faraday's law:
V = -d/dt [psi](t) * N.
In words: The induced voltage in each loop equals the change in magnetic flux with time. In your case, N = 100.

Now, let's integrate:
Integral V dt = N[psi]1 - N[psi]2.

Now it would be good if [psi]1 was zero, and [psi]2 was maximum.
So, let's start at such a large distance that [psi]1 is practically zero. And let's choose the final position so that [psi]2 is maximum (meaning, V is zero). That will probably be the case when the magnet passes the center of the coil.

In doing so, we find the maximum flux [psi] created by the magnet:
Integral V dt = N[psi].

Now from flux to induction (= B-field strength).

The unit of flux is
[[psi]]= 1 Weber = 1 V s.
The unit of induction is
= 1 Tesla = 1 V s m-2.

So, I guess
B = N[psi]/A,
where A is the area of cross-section of your coil.

In other words, delta M = A.
Does this help?

Last edited:

#### HiEdgar

One more question...

Arcnets,

Thank you for explanation. I really liked your approach but I still have couple more questions. In particular, it was not obvious to me why you decided that delta M = A. From the formula presented (B=N*Flux/A), I do not immediately see that. Can you please show htat explicitely ie delta_M=A?

Also, in regards to formula 'B=N Flux/A', Should the N be present in there? How did you derive it? As far as I know Flux=Integral [B.da], and there is no 'N' present in there.

Is there any simple way to calculate the induced emf in the coil as a funciton of magnet's position?

Thank you,

Waiting for reply,

Edgar

#### arcnets

Re: One more question...

In particular, it was not obvious to me why you decided that delta M = A. From the formula presented (B=N*Flux/A), I do not immediately see that. Can you please show htat explicitely ie delta_M=A?
Well, we know (following my apprach):
maximum flux = (integral V dt)/N.
Now what is the maximum flux? Imagine the magnet has the same (or greater) length as the coil. Now, at final position, the magnet is centered inside the coil. This means that all the field lines produced by the magnet go thru the coil. That is the maximum flux.

Now from the maximum flux we have to find B. And from analysis of units, B = (maximum flux)/(area).

Aha. Now I see! Of couse, the area in question is not the cross-section of the coil. It's that of the magnet. Because the whole field goes thru the magnet. And we can assume it to be homogenous there.

So: B = (maximum flux)/(magnet cross-sectional area)
Also, in regards to formula 'B=N Flux/A', Should the N be present in there? How did you derive it? As far as I know Flux=Integral [B.da], and there is no 'N' present in there.
I think you are right. Sorry. Should read
B = [psi]/Amagnet = (Int. V dt)/(N*Amagnet)

So, your delta_M is
delta_M = N*Amagnet.
Is there any simple way to calculate the induced emf in the coil as a funciton of magnet's position?
I'm afraid there isn't. Because this would require exact knowledge of the magnet's field geometry, which is in general very complicated.

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