# Field outside a solenoid on its axis

#### DD31

1. The problem statement, all variables and given/known data
A tightly wound solenoid of length L and radius a that has n turns of wire per unit length carries a current I. Calculate the the magnitude of the magnetic field at a point P on the axis of the solenoid, outside it and a distance y from one end, as shown in the figure.
Note: other than the solenoid being tightly wound, we are not making any approximations regarding the relative sizes of a, L, and y. You answer should be valid anywhere on the axis, even, with some care with signs, inside the solenoid.

http://img256.imageshack.us/img256/8978/lg4xii.jpg [Broken]

All three of the below questions are looking for formulas:

a): What is the current dI in a narrow section of the solenoid of width dx a distance x from the left end of the solenoid?

b): What is the contribution dB to the magnetic field at P due to this narrow section of the solenoid? Notice that the center of this section is a distance (x+y) from point P. Use your result from a), not dI, in your answer.

c): Find the magnitude of B at P. Note: you may look up the integral.

2. Relevant equations
Not sure if all these are relevant, but here are the ones I've been trying to use:
Solenoid formulas:
B = (mu_0)nI, where n = number of turns per length

Other formulas:
B = $$\frac{(mu_0)IR^2}{2(z^2+R^2)^3/2}$$
where z = distance from the center of a ring of charge and R = the radius of the ring.

3. The attempt at a solution
I really haven't made much of an attempt, because I don't even know where to begin fully.

For a, I first figured that the current would just still be I because the current in any given segment of the wire should be the same throughout the wire...but that's incorrect.

Then, I thought that maybe it wanted the current as a portion of the length, so I found the total length of the wire to be 2pi*a*n*L, or the amount of coil around one loop times the number of loops per length times the length. I divided I by this, but that was incorrect as well.

I feel like the bottom formula I listed in the "relevant equations" section may have something to do with the final integration, as it seems like, at point P, the solenoid may "look like" a simple ring of charge, but if I can't find dI, I don't know how I can even start. I think I'm trying to get something slightly different than what they're asking for right now.

Any help is greatly appreciated...this seems like a very difficult problem so I'll be all the more grateful to anybody who takes the time to help me figure it out.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator:

#### DD31

Sorry for the bump, but I'm still flat-footed on this one. I'm going to look at it more tonight extensively, but I'm not quite sure where to even start to find dI.

#### DD31

Alright...so I've found part a. dI = I*n*dx; I was the current in the wire, so the current from some bundle of wires dx wide is I*n*dx. Makes sense.
Now, for b, I'm trying to find some way to use coils or rings and find the field "above" one of those, but I'm searching for a place to use this dI and I'm not finding.

Thanks in advance for any help.

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