EE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok Name: (Last, First)______________________________________________________ 1. In radio frequency, current practically flow ONLY on the surface of a conductor. Consider a coaxial cable with a inner conductor of radius a, and a grounded metal shield of inner radius b and outer radius c as shown. Total current in the inner conductor is I. Plot B field as a function of r. What is the surface current density at r = a? At r = b? And at r = c? a b c 0)(2)(2)(0)(0)(2)2(0)(0)(==−=====>=>==<<=<=<=⋅∫crJbIbrJaIarJcrBbrBrIBIrBbraarBarJIdBsssooenclosedoπππµµπµlrrabcr BaIBoπµ2=bIBoπµ2=conductor Net I = 0EE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok α R r z 2. (a) A ring of charge with linear charge density λ and radius R, is lying on the xy-plane. Find the electric potential at a distance z above the center of the ring. (b) Take the gradient of this V to find the electric field at that point (0,0,z). (c) Apply the electric potential from part (a) to a 2-dimensional disk. Find the electric potential of a charged disk with surface charge density σ, radius R, at a distance z above the center of the disk. 2222zRRrRkVrRdkrkdqdVo+====ελπλθλ( )zzRRzzVzEoˆ2ˆ2/322+=∂∂−=ελr[ ][ ] [ ]zzRzruduuzrrdrVzrrdrdVrdrrQoRoooRoo−+=+===+=+=⇒=∫∫− 220222/12/1022222224422)(22εσεσεσεσεσεσπσπλEE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok v 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 B 3. A conducting rod with length L moves in a magnetic field B directed into the plane of the figure. The rod moves with a speed v in the direction shown. (a) What is the potential difference between the ends of the rod? (b) Which side has higher potential? (c) When the charges in the rod are in equilibrium, what are the magnitude and direction of the electric field within the rod? vBLxdqFVVqvBBvqFLoL=⋅−=−→=×=∫0)(rrrrrPositive charges move to the right. So the right side would end up with more + charges and hence higher V (potential). VL > Vo )(←=×=vBEBvqEqrrrrEE 140, Spring 2015 Mid-Term 1 Dr. Ray Kwok x y I .... .... r dB θ θ z x 4. What is the B field of an infinite wire carrying a current I at a distance d away? Use this result to find the B field of an infinite current sheet. A simple model is this: Infinitely long straight conductors with square cross sections and each carrying current I are laid side-by-side to form an infinite current sheet as shown in figure. The conductors lie in the xy-plane, are parallel to the y-axis and carry current in the +y direction. There are n conductors per unit length along the x-axis. What are the magnitude and direction of the magnetic field a distance z above the current sheet? dIBIdBIdBooenclosedoπµµπµ2)2(===⋅∫lrr(
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