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ECE 3318 Applied Electricity and Magnetism Spring 2017 Homework #7 Date assigned: March 9, 2017 Date due: March 23, 2017 Do Probs. 1, 2, and 6-11. (You are welcome to do the other problems as well for extra practice, but only these should be turned in.) 1) Find the curl of the following vector functions: V xˆ 5 x 2 yz yˆ e y x zˆ 3xyz 2 V ˆ cos 3z sin zˆ z V rˆ sin cos ˆ 3r sin ˆ r 3 cos . 2) A simple “curl meter” is a paddle wheel that consists of a set of vanes attached to a spindle (the axis 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 axis is pointed in direction ˆ . A positive torque T 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 T ˆ curlV , 2 if ˆ curlV 0 , if ˆ curlV 0. 1 ˆ T Part 1) Find the torque T on the curl meter for the vector function V xˆ 2 x 2 y yˆ 3zy 3 zˆ 2 xyz at the following points (in rectangular coordinates) and directions: b) r (1,1,1); ˆ xˆ ˆ yˆ c) r (1,1,1); ˆ zˆ d) r (1,1,1); ˆ zˆ + 2 xˆ + 4 yˆ / 21 a) r (1,1,1); e) r (1, 2,3); ˆ xˆ + yˆ + zˆ / 3 . Part 2) Find a unit vector ˆ that gives the direction to point the curl meter so that the torque will be maximum 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 to be maximum in magnitude but negative at this same point. 3) Consider the vector function V xˆ 2 x 2 y yˆ 3zy 3 zˆ 2 xyz . 2 Calculate an approximate expression for the integral I V dr , C where the path C is a circular path of radius a = 0.0001 that is centered at the point (1, 2, 3) and lies in a plane perpendicular to the vector direction xˆ 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 approximately constant over the surface of the circular region inside the path C, since the radius is so small.) Hine: Use Stokes’s theorem. z C y x 4) Assume a vector function F ˆ (5 sin ) ˆ ( 2 cos ) . a) Evaluate F d r around the contour ABCDA in the direction indicated in the figure. b) Find F . c) Evaluate F nˆ dS over the surface defined by the contour ABCDA and compare S the result with that obtained in part (a), to see if you can verify Stokes’s theorem. 3 y C 3 [m] B 1 [m] x A D 5) Green’s theorem states that if P ( x, y ) and Q ( x, y ) are two differentiable functions in a planar region S of the xy plane bounded by a closed curve C , then Q P x y dS S C Pdx Qdy , where the integral around C 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 xy plane, A zˆ dS A dr , S C in which A x, y P x, y xˆ Q x, y yˆ . 6) Calculate the voltage drop B VAB E d r A along the path C as shown below, for the electric field E xˆ 3x yˆ 4 y 3 zˆ 4 z . 4 You are free to choose a different path than the one specified, if you can prove that it makes no difference (explain 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 C : y x2 e x 1 A x 7) The height of a mountain is described by z 103 1 a 2 x 2 b 2 y 2 for z greater than zero, where a = 0.001 and b = 0.002, and all dimensions are in meters. Note that the peak of the mountain is at x = y = 0. Assume that a mountain climber is on the side of the mountain at x = 500, y = 200. Find a unit vector parallel to the xy plane (i.e., a horizontal unit vector) that points in the direction that the climber located at (x, 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? 8) Consider a circular disk of uniform surface charge density s0 of radius a, lying in the z = 0 plane with the center of the disk at the origin. Calculate the potential at any point on the z axis by using the potential-integral formula. Then use the gradient to find the electric field on the z axis, for both z < 0 and z > 0. 9) Two infinite line charges (running in the z direction) are located at x h as shown below. Calculate the potential at any point (x, y), assuming zero volts on the z axis. 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.) 5 y h h x l 0 l 0 10) Use the gradient to calculate the electric field vector at any point (x, y) in space, for the problem of the two infinite line charges in the previous problem. 11) Consider a sphere of uniform charge density v 0 [C/m3] having a radius a, centered at the origin. Perform the following steps: a) Calculate the electric field both inside and outside the sphere using Gauss’ law. b) Calculate 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