A wire is bent into a shape resembling a U. The \"bottom\" of the wire is a semi

Question

A wire is bent into a shape resembling a U. The "bottom" of the wire is a semi-circle of radius R. Each of the "legs" is a straight wire segment with a length 4R. A constant current I enters at a, then exits at b.

a) Derive an expression for the net magnetic field at point P.

b) Evaluate your expression for R = 10 cm and I = 500 mA.

A wire is bent into a shape resembling a U. The bottom of the wire is a semi-circle of radius R. Each of the legs is a straight wire segment with a length 4R. A constant current I enters at a, then exits at b. a) Derive an expression for the net magnetic field at point P. b) Evaluate your expression for R = 10 cm and I = 500 mA.

Explanation / Answer

there are three contributions to the net magnetic fields

one from each straight segments and one due to the semi circle of raidus R

in the top segment , current is flowing directly towards the point P and so sin theta = 0, thus no controbution

the are( semi circle) produces a field, directed into the palne of the paper.

the arc subtends an angle of pi , so B due to arc

B = uoi Pi/4piR)-------------------uoi/4R

B due to botton segment = uoi/4pir)(cos 0 - cos pi/2)

B = uoi/4piR

so BNet = uoi/4R)(1+1/pi)

---------------------------------------------------------------
b.
Bnet = 4pie-7 *0.5/(4*0.1)*(1+1/3.14)

Bnet = 2.07 *10^-6 Tesla

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