ECE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7

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1 EE 3318 Applied Electricity and Magnetism Spring 2018 Homework #7 Date assigned: Tuesday, March 6, 2018 Date due: Tuesday, March 20, 2018 Do Probs. 1, 2, and (You are welcome to do the other problems as well for etra practice, but only these should be turned in.) 1) Find the curl of the following vector functions: ( 2 y 5 ) ( ) ( 3 2 ) V = ˆ yz yˆ e + zˆ yz ( cos ) ( 3 sin ) ˆ ( ) V = ˆ ρ φ φ z φ + z zρφ 3 ( θ φ) ˆ θ( φ) ˆ φ( θ) V = rˆ sin cos 3rsin + r cos. 2) A simple curl meter is a paddle wheel that consists of a set of vanes attached to a spindle (the ais of rotation) as shown below. It is used to determine the curl of a vector function that describes the velocity vector of water. In order to determine the component of the curl of a vector in the direction ˆ, at a given point in space, the curl meter is placed at that point in space and the spindle ais is pointed in direction ˆ. A positive torque component T l indicates that the meter wants to rotate in the direction defined by the right-hand rule, as shown in the figure below. Assume that our curl meter has the property that the magnitude of the torque is equal to the square of the curl component. (This is because when water hits an object, the force is proportional to the square of the velocity.) The torque is therefore given by Tl ( V) 2 = ± ˆ curl, + if ˆ curlv > 0, if ˆ curlv < 0. 1

2 ˆ T l Part 1) Find the torque component T l on the curl meter for the vector function ( 2 2 ) ( 3 3 ) ( 2 ) V = ˆ y + yˆ zy + zˆ yz at the following points (in rectangular coordinates) and directions: a) r = (1,1,1); ˆ = ˆ b) r = (1,1,1); ˆ = yˆ c) r = (1,1,1); ˆ = zˆ d) r = ˆ = ( zˆ + ˆ+ yˆ) e) r = ˆ = ( ˆ + y ˆ + z ˆ ) (1,1,1); 2 4 / 21 (1, 2, 3); / 3. Part 2) Find a unit vector ˆ that gives the direction to point the curl meter so that the torque component T l will be maimum in magnitude and positive, assuming the same vector function V as above and assuming that we are at the point (1,1,1) in rectangular coordinates. Repeat for the case where we want the torque component T l to be maimum in magnitude but negative at this same point. 2

3 3) onsider the vector function ( 2 2 ) ( 3 3 ) ( 2 ) V = ˆ y + yˆ zy + zˆ yz. alculate an approimate epression for the integral I, = V dr where the path is a circular path of radius a = that is centered at the point (1, 2, 3) and lies in a plane perpendicular to the vector direction ( ˆ+ yˆ). The direction of integration around the contour is counter-clockwise, as seen from an observer who is looking towards the loop from far away in the first octant, as shown below. (Note: You may assume that the curl of the vector function V is approimately constant over the surface of the circular region inside the path, since the radius is so small.) Hint: Use Stokes s theorem. z y 4) Assume a vector function ˆ 2 F = ˆ ρ(5ρsin φ) + φ( ρ cos φ). a) Evaluate F dr around the contour ABDA in the direction indicated in the figure. b) Find F. c) Evaluate ( ) ˆ F n ds over the surface defined by the contour ABDA and compare S the result with that obtained in part (a), to see if you can verify Stokes s theorem. 3

4 y 3[ m] B 1[ m] A D 5) Green s theorem states that if Py (, ) and Qyare (, ) two differentiable functions in a planar region S of the y plane bounded by a closed curve, then S Q P ds = Pd + Qdy y, where the integral around is taken in a counterclockwise sense. Show that this is a special case of Stokes s theorem for a planar surface bounded by a closed contour lying in the y plane, S ( ) ˆ A z ds = A d r, in which (, ) (, ) ˆ (, ) Ay = Py+ Qy yˆ. 6) alculate the voltage drop V AB B = E dr A along the path as shown below, for the electric field ( 3 ) ( 4 3 ) ( 4 ) E = ˆ + yˆ y + zˆ z. 4

5 You are free to choose a different path than the one specified, if you can prove that it makes no difference (eplain clearly why). The point A is at the origin, and point B is at (1, e, 0). (Note: This may not be an electrostatic field so the curl may or may not be zero!) y B A y e 2 : = + 1 7) A coil of wire has N = 1000 turns. The area of each loop is 100 [cm 2 ]. The total length of the wire in the coil has a resistance of 100 [Ω]. The loops lie in the z = 0 plane. A 60 [Hz] magnetic field is applied to the coil, having the form ( ) ( ω ) B = zˆ 0.1 cos t [Wb/m 2 ]. a) Find the Thévenin equivalent circuit of this coil system, when it is used as an A generator. b) Find the maimum power that this A generator can supply to a matched load. 8) The height of a mountain is described by ( ) z= 10 1 a by for z greater than zero, where a = and b = 0.002, and all dimensions are in meters. Note that the peak of the mountain is at = y = 0. Assume that a mountain climber is on the side of the mountain at = 500, y = 200. Find a unit vector parallel to the y plane (i.e., a horizontal unit vector) that points in the direction that the climber located at (, y, z) should head towards, in order to ascend the mountain as rapidly as possible from her present location. (This corresponds physically to the compass heading that she must head towards.) Also, find the horizontal unit vector that points from where the climber is towards the top of the mountain. Are the two unit vectors the same? 5

6 9) onsider a circular disk of uniform surface charge density ρ s0 of radius a, lying in the z = 0 plane with the center of the disk at the origin. alculate the potential at any point on the z ais by using the potential-integral formula. Then use the gradient to find the electric field on the z ais, for both z < 0 and z > 0. 10) Two infinite line charges (running in the z direction) are located at = ± h as shown below. alculate the potential at any point (, y), assuming zero volts on the z ais. When calculating the potential, you may start with the potential of a single infinite line charge and use superposition. (The potential of a single infinite line charge was derived in class in Notes 14.) y h h ρ l0 ρ l0 11) Use the gradient to calculate the electric field vector at any point (, y) in space, for the problem of the two infinite line charges in the previous problem. 12) onsider a sphere of uniform charge density ρ v0 [/m 3 ] having a radius a, centered at the origin. Perform the following steps: a) alculate the electric field both inside and outside the sphere using Gauss law. b) alculate the potential inside and outside the sphere by integrating the electric field to get the potential, assuming zero volts at infinity. c) Find the electric field inside and outside the sphere by taking the gradient of the potential. You should be able to verify that your answer is the same as the electric field that you started with from Gauss s law. 6