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## Homework Statement

We have seen that a long solenoid produces a uniform magnetic field directed along the axis of a cylindrical region. However, to produce a uniform magnetic field directed parallel to a diameter of a cylindrical region, one can use saddle coils. The loops are wrapped over a somewhat flattened tube. Assume the straight sections of wire are very long. The overall current distribution is the superposition of two overlapping circular cylinders of uniformly distributed current, one toward you and one away from you. The current density J is the same for each cylinder. The position of the axis of one cylinder is described by a position vector a relative to the other cylinder. Prove that the magnetic field inside the hollow tube is ##\frac{\mu_0 J_0}{2}## downward.

## Homework Equations

Ampere's Law

## The Attempt at a Solution

Given that

$$\int \vec B \cdot \vec dl = \mu_0 I$$, I set ##I = J \ \ A##, where A is the cross sectional area of each cylinder of current.

Then we get $$B*(2 \pi r) = \mu_0 J \pi r^2$$

or

$$B = \frac{\mu_0 J r}{2}$$

Now when I add in the contribution from the other current, which ought to be exactly the same, I get ##B = \mu_0 J r##, which is

*what I want. I'm not sure how to proceed from here.*

**not**Thanks in advance for any help!