Calculating Flux in a Coil with Current

[ashishsinghal]

How can we calculate the flux in a coil which carries a current? I am having trouble determining the field at any point inside the coil other than the center? Any help is appreciated...

 

[Redbelly98]

This is not a straightforward calculation. The Biot-Savart law can be used to calculate the flux, but you have to take the thickness of the wire into account. In the limit the wire becomes infinitesimally thin, the calculated flux would go to infinity due to the 1/r² dependence of B near the wire.

In practice, the flux would be determined experimentally.

 

[silmaril89]

This is a very difficult calculation to do if you want to be exact. This is due to edge effects (i.e. where the coil stops and begins). To get an approximation many people will use the same method used for calculating the magnetic field inside an infinitely long solenoid (you are talking about a solenoid, correct?). Basically what this means is that the magnetic field will be a constant inside the solenoid and zero outside it (much like what ardie said in the previous comment).

So, you can use ampere's law to calculate the magnetic field inside. This comes out to be B = (mu * N * I)/L, where mu is the permeability of free space, I is the current in the coil, N is the number of turns in the coil within the length L.

From that, you should be able to calculate the flux.

 

[ashishsinghal]

What about a single circular loop? I think that may be simpler

 

[silmaril89]

I'm not really sure if a single loop would be simpler. The approximation I used for a solenoid might not work out for a single loop. But, I feel that I have had to do calculations like this before. I'm not at home right now, but when I do get home I can take a look in my E&M book and see if I find anything.

 

[ashishsinghal]

The direction of magnetic field will be constant (inside or outside the plane of loop). Then we can say that surely the flux is non zero. Right?

 

[silmaril89]

The direction of magnetic field will be constant (inside or outside the plane of loop). Then we can say that surely the flux is non zero. Right?

That is certainly correct.

 

[ashishsinghal]

How can we calculate field at a point not at the center?? I am

 

[Redbelly98]

How can we calculate field at a point not at the center?? I am

Use the Biot-Savart law:

http://en.wikipedia.org/wiki/Biot–Savart_law#Introduction

Here dl is an infinitesimal length element on the current loop, and r is the displacement vector from the length element to the location where B is to be evaluated.

This is a complicated problem, as I mentioned before. You would evaluate the above integral to get B anywhere within the plane of the loop.

 

[ashishsinghal]

You would evaluate the above integral to get B anywhere within the plane of the loop.

That's what i am not able to do! Please help me do that!

 

[silmaril89]

I looked in my E&M book last night and found what RedBelly98 said to be true. There is no simpler way to do this problem other than using the Biot-Savart Law. It's probably a bit difficult to go into complete detail on how to use the Biot-Savart Law in this thread, but any advanced undergraduate E&M book will be able to show you some examples of how to use the Biot-Savart Law. Have you taken a look at Introduction to Electrodynamics by Griffiths? That book will no doubt be of some use to you.

 

[Redbelly98]

That's what i am not able to do! Please help me do that!

Sure, I can help out with that. Just want you to understand up front:

1. I'm fairly sure evaluating this will require numerical techniques.

2. I'm pretty certain the flux integral (when we get that far) will diverge if we assume an infinitesimally thin wire. We'll see when we get there.

So, you have two choices of integral to use:

. . or . .

Try to set up either one of those integrals in cylindrical coordinates. Assume the loop is in the xy plane, centered at the origin. To make things somewhat easier, note we really just need the z-component of B, since only that component contributes to the flux. So you only need to bother with the z-component when you take the cross product dlxr

Also, assume the point of interest is somewhere on the +x axis. Draw yourself a figure, and to start out get an expression for dlxr (z-component only).

p.s. Note, I have deleted the unhelpful posts (and replies to them) from this discussion.