6 (20 points) A wire is bent into the shape shown in the figure. A current I flo

Question

6 (20 points) A wire is bent into the shape shown in the figure. A current I flows counter-clockwise in the wire (abcda). The outer part of the wire (segment ab) has radius 2R, the inner part of the wire (segment cd) has radius R, both centered at point O. These two segments subtend an angle of 90. Segments da and bc are both straight. For this problem, take +z to be OUT of the page. (a) Use the Biot-Savart law to find the magnetic field at point O due to segment ab. take the z directon to o the page. Hint: this question is NOT asking for the field along an axis, is asking for the field at a point.

Explanation / Answer

for the given wire, current is I and inner radius = R, outer radius = 2R, and angle made by the arc = 90 deg

a. using biot savarts law

magnetic field at the center of an arc of circle of loop radius R carrying current I is given by

dB = mu*Idl/4pi*r^2

hence for quarter arc given in the question

B = mu*I*pi*r/2*4*pi*r^2 = mu*I/8r

but r = 2R

B = mu*I/16R

b. foeld due to segment BC = 0 T ( as segment BC intersects point O when extended making perpendicular distance form the sengment 0, and hence magnetic ifled is 0)

c. form part A, segment cd has field B = -mu*I/8R

d. from b. segment da has field = 0 T

e. net field = muI/16R - muI/8R = -mu/16 R T

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