Question

A circular loop has radius R and carries current I2 in a clockwise direction. The center of the loop is a distance D above a long, straight wire.
What is the magnitude of the current I1 in the wire if the magnetic field at the center loop is zero? Express your answer in terms of the variables I2, R, D, and appropriate constants (μ0 and π).

Answer 1

Answer:

$I_{1}$ = (πD$I_{2}$)/R

Explanation:

If we define the magnitude of the field as B, then we have:

Total magnitude of the field $B_{t}$ = magnitude of the field B_loop + magnitude of the field B_wire. The total magnitude is equivalent to zero. Therefore, the field B_loop has an inward direction while the field B_wire has an outward direction.

B_loop = (μ0)*($I_{2}$)/2*R

B_wire = (μ0)*($I_{1}$)/2*π*D

Thus:

B_loop = B_wire at the center of the loop.

(μ0)*($I_{2}$)/2*R = (μ0)*($I_{1}$)/2*π*D

$I_{1}$ = (πD$I_{2}$)/R

Answer 2

The magnitude of current at the center of the loop is $I_1 = \frac{\pi D I_2}{R}$.

The given parameters;

• radius of the loop = R
• current in the loop, I = I₂
• distance of the loop from the wire, = D

The magnetic field at the center of the loop is calculated as follows;

$B_o = \frac{\mu_o I_2}{2R}$

The magnetic field at the distance of the wire is calculated as follows;

$B_o = \frac{\mu_o I_1}{2\pi D}$

The magnitude of current at the center of the loop is calculated as follows;

$\frac{\mu_o I_2 }{2R} = \frac{\mu_o I_1}{2\pi D} \\\\I_1 = \frac{\pi D I_2}{R}$

Thus, the magnitude of current at the center of the loop is $I_1 = \frac{\pi D I_2}{R}$.

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