Helmholtz Coil Problem: Analyzing Magnetic Field at Point Z

[raindrops]

Homework Statement

The diagram illustrates a Helmholtz coil, consisting of two identical circular coils of wire, each one having N loops and carrying the same current I. The radius of each coil is R, and the two coils are R meters apart. (a) What are the direction and strength of the magnetic field at the point Z, midway between the two coils? (b) Express B as a function of the variable z, taking the origin for z to be at the center of the left-hand coil and the positive z direction to be to the left. (c) Find the first and second derivatives of the magnetic field strength B with respect to z, dB/dz and d 2B/dz2. Show that they are both zero at point Z (where z = −R/2). This shows that the magnetic field varies very little at the point Z.

http://i40.tinypic.com/k4b5ag.jpg

that link should have the picture.

Homework Equations

I think the equation for a Helmholtz coil is (4/5)^3/2 (uNI/R)

The Attempt at a Solution

For part a, I think it's to the left. I could really use some direction for parts B and C. Thanks!

 

[chrisk]

Start by finding the differential field at any point on the z axis for one current loop:

$$dB_z=\frac{\mu_0I}{4\pi}\frac{dl}{r^2}cos\theta$$

where theta is the angle made by the radius of the loop and the line from the current element on the loop to the point on the z axis. Intergrate to determine the field contributions from all the dl elements. Then multiply by the factor N to find all the loop contributions. Do the same for the second coil. Express the cosine function in terms of the loop radius and the distance from one of the loop centers to the point in question. Finish by using superposition.