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ps07sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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PS07 Solutions-1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2008 Problem Set 7 Solutions Problem 1: Quickies… a) Two semicircular arcs have radii R 2 and R 1 , carry current i , and share the same center of curvature C. What is the magnitude of the net magnetic field at C? The inner semi-circle makes a field into the page, the outer one out of the page. The inner is closer and hence stronger, and hence the net field is into the page. In class you calculated the magnetic field from a semi-circle of radius R to be 0 4 B i R µ = . So: 0 1 2 1 1 into the page 4 i R R µ = B G b) A wire with current i is shown at left. Two semi-infinite straight sections, both tangent to the same circle with radius R , are connected by a circular arc that has a central angle θ and runs along the circumference of the circle. The connecting arc and the two straight sections all lie in the same plane. If B = 0 at the center of the circle, what is θ ? The straight portions both make a field out of the page at the center of the circle while the arc makes one into the page. These must be equal so that the fields cancel. Two semi- infinite lines together make an infinite line, and we calculated (using Ampere’s law) that the field from an infinite wire is 0 2 B i R µ = π . The arc is just a fraction of a circle so it creates a fraction of the field that a whole circle does at its center: ( ) ( ) 0 2 2 B i R µ = θ π . For these to be equal we must have 0 0 2 4 2 radians B i R i R µ µ = π = θ π θ = c) The figure at left shows two closed paths wrapped around two conducting loops carrying currents i 1 and i 2 . What is the value of d B s G G i v for (a) path 1 and (b) path 2? To do this you have to use the right hand rule to check whether the currents are positive or negative relative to the path. On path 1 i 1 penetrates in the negative direction while i 2 penetrates in the positive direction, so ( ) 2 1 o d i i = µ B s G G v .
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