1. (20 points) Consider an equilateral triangle, of side length 2a, in the x-y plane, centered at the origin, carrying a

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1 20 Points Consider An Equilateral Triangle Of Side Length 2a In The X Y Plane Centered At The Origin Carrying A 1 (974 KiB) Viewed 41 times

1. (20 points) Consider an equilateral triangle, of side length 2a, in the x-y plane, centered at the origin, carrying a current I. a. Use the Biot-Savart Law (not any worked out examples) to find the magnetic field along the z-axis. b. What it the magnetic dipole moment of the loop? Use this to find an approximate expression for the magnetic field along the z-axis. c. Plot both expressions on the same graph assuming a = 1 and -=1, and find when the approximation is good to 1% error. 2. (15 points) An infinitely long cylindrical shell, of negligible thickness and radius b, surrounds an infinitely long solid cylindrical rod, of radius a. Their axes are both along the z-axis and a view down the length of this axis is illustrated. The volume current density in the rod is (in cylindrical coordinates) J= i za 32 where a is a constant with the dimensions of current. The current 27a along the cylindrical shell, flows in the -z direction, is uniformly distributed across the surface and is such that the total current flowing down the shell is exactly opposite to that flowing through the cylinder. Show that the surface current density in the cylindrical shell is K=- , and determine the magnetic field at 2b all locations. 3. (20 Points) Find the mutual inductance for circuits (loops) Ci and C2 shown in the figure. Ci is a rectangle in the x-y plane of size b times c. C2 is a larger rectangle in the x-v plane, infinite in the y direction and at a distance a from the parallel sides of C, in the x direction. a- t - b--H