EE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok B x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x V L W Name: (Last, First)______________________________________________________ 1. A rectangular coil with N turns and total resistance R, (length L < 1, and width W < 1) as shown in figure. The coil moves into a uniform magnetic field B with constant velocity v. Plot the induced current I2 as shown, as a function of x when the coil is entering the region. Clearly state the magnitude of the current. I2 x 0 1 x (t) I2 1 RNBvWRINBvWdtdNBxWadB==−=Φ−==⋅=Φ∫εε2rrL 1+L RNBvW−RNBvWEE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok 2. A toroid is a doughnut-shaped toroidal solenoid as shown. In practice, it has many more turns of wire closely packed than it is in figure. Use Ampere's Law and the loops shown in Figure (b), find B field as a function of r (for N turns of wire carrying a current I). What is the direction of this B field? [The inner radius of the toroid is a, and the outer radius is b.] B field circulates in the direction shown. 0)(02)()2(0)(0)(=>===<<=<=<=⋅∑∫>brBIrNIBNIrBbraarBarIIdBbrooenclosedoπµµπµlrrEE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok 3. A total charge Q is deposited onto a conducting sphere with radius a, which is shielded by a metal shell of inner radius b and outer radius c as shown. The outer shell is grounded outside. Plot E field as a function of r. Clearly explain your answer. What is the surface charge density at r = a? At r = b? And at r = c? a b c 0)(4)4(0)(0)(22=>===⋅<<=<=<∫brErQEQrEQdaEbraarEarQoooinsideπεεπεa b cr E24 aQEoπε=24 bQEoπε=0)(4)(4)(22==−====crbQbraQarσπσπσEE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok α R r z 4. (a) A ring of charge with linear charge density λ and radius R, is lying on the xy-plane. Find the electric field at a distance z above the center of the ring. (b) Integrate this E field to find the electric potential at that point (0,0,z). (c) Apply the E field from part (a) to a 2-dimensional disk. Find the E field of a charged disk with surface charge density σ, radius R, at a distance z above the center of the disk. (d) What is the E field of a disk as R goes to infinity? ( )zzRRzrRzkErzrRdkrkdqdEozˆ22cos2/322322+==⋅==ελπλθλαs( )22222/12/32/322212242zRRzRRVuRduuRdzzRRzzdEVozzooozoz+=+==−=+−=⋅−=∞=−−∞∞∫∫∫ελελελελελrr( )( )[
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