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

a] By Biot Savart law, B = integral dB

= integral u0 i dL x r^ / 4pi r^2 where r^ is unit vector radially inward.

= integral u0 i dL / 4pi r^2 because r^ is already perpendicular to dL

=  u0 i (pi r/2) / 4pi r^2

= u0 i/ 8r (where r is 2R)

= u0 i/16R out of the page

B] similarly for segment bc , B = u0 i/ 8r (where r is R )

B = u0 i/8R into the page

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