A current is set up in a wire loop consisting of a semicircle of radius 4.12 cm,

Question

A current is set up in a wire loop consisting of a semicircle of radius 4.12 cm, a smaller concentric semicircle, and two radial straight lengths, all in the same plane. Figure (a) shows the arrangement but is not drawn to scale. The magnitude of the magnetic field produced at the center of curvature is 47.29 ?T. The smaller semicircle is then flipped over (rotated) until the loop is again entirely in the same plane (Figure (b)). The magnetic field produced at the (same) center of curvature now has magnitude 15.83 ?T, and its direction is reversed. What is the radius of the smaller semicircle?

Explanation / Answer

given,

radius of larger semicircle = 4.12 cm

radius of smaller semicircle = x cm

since in first case magnetic fields are in same direction so they'll add up

magnetic field due to concentric semicircle in first case = (k * I / 4) * ( 1 / x + 1 / 0.0412)

47.29 * 10^-6 = (k * I / 4) * ( 1 / x + 1 / 0.0412) -----(1)

since in second case magnetic fields are in opposite direction so they'll subtract each other

magnetic field due to concentric semicircle in second case = (k * I / 4) * ( 1 / x - 1 / 0.0412)

15.83 * 10^-6 = (k * I / 4) * ( 1 / x - 1 / 0.0412) ------(2)

on solving (1) and (2) we'll get

x = 0.0205347 m

radius of smaller semicircle = 0.0205347 m

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