A current is set up in a wire loop consisting of a semicircle of radius 4.16 cm,

Question

A current is set up in a wire loop consisting of a semicircle of radius 4.16 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 47.25 %u03BCT. 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 15.88 %u03BCT, and its direction is reversed. What is the radius (in cm) of the smaller semicircle?

Explanation / Answer

suppose the current I flows in the whole setup

B due to straight segments at the centre will be zero

B1 - B due to larger semi circular portion

B2 - B due to smaller semi circular portion

then

1st case,

B net = B2 + B1 = u*i / 2r + u*i / 2R = 47.25 E-6 T

so (u*i / 2) [ 1/r + 1/0.0416 ] = 47.25 E-6

so 1/r = 75.23/i - 24.038 -----------(1)

2nd case,

B net = B2 - B1 = u*i / 2r - u*i / 2R = 15.88 E-6 T

so (u*i / 2) [ 1/r - 1/0.0416 ] = 15.88 E-6 T

so 1/r = 25.286/i + 24.038-------------(2)

solve 1 and 2

add the two equations, so i = 50.258 r

put this value of i in (2)

so 1/r = 25.286/50.258r + 24.038

so r = 0.02065 m or 2.065 cm

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