Two coplanar circular loops share the same center but have different radii

PHYSICS · Multi-part · AP · Physics C: Electricity and Magnetism · Magnetic Fields and Electromagnetism · Magnetic Fields and Electromagnetism

Problem

Two coplanar circular loops share the same center but have different radii. The inner loop has radius

R_{\!i}=5.0\,\text{cm}

and carries a current

I_{\!i}=4.0\,\text{A}

that is counterclockwise when viewed from above. The outer loop has radius

R_{\!o}=8.0\,\text{cm}

and carries a current

I_{\!o}=6.0\,\text{A}

that is clockwise. Address each of the following tasks, showing symbolic reasoning before substituting numeric values. (a) Derive an expression for the net magnetic field magnitude

B_{\text{net}}

at the common center in terms of

R_{\!i}

R_{\!o}

I_{\!i}

, and

I_{\!o}

. Clearly indicate how the directions of the two individual fields combine. (b) Using your expression from part (a), calculate the magnitude and direction of the net field at the center for the given numerical values. (c) Determine the current that the outer loop must carry (magnitude and direction) so that the net magnetic field at the common center is exactly zero.

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