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ECE 3318 Applied Electricity and Magnetism Spring 2016 Homework #4 Date Assigned: Feb. 11, 2016 Due Date: Feb. 18, 2016 1) Make a flux plot for the case shown below, which has two infinite line charges (l0 on the left and -l0 on the right). The equipotential lines are already there (labeled with voltage values on the curves), so you only need to add the flux lines (the lines with the arrows). Do so by keeping the aspect ratio of the curvilinear squares to be L / W = 1. You may cut and paste the figure below onto your own paper in order to make your plot if you wish, or do your work on this page attach this page with the rest of your homework. F 1 2) On a piece of graph paper, make a flux plot for a line charge along the z axis (the origin on your graph, which shows the xy plane), which shows both the flux lines and the equipotential contours. Assume that there are 28 flux lines that emanate from the line charge. Draw the outer equipotential contour at a radius of 5 [cm] from the origin. Your plot should be drawn to full-scale, so that 1 [cm] on your plot corresponds to an actual distance of 1 [cm]. Draw the plot so that the L / W ratio of the curvilinear squares is unity. 3) Assume that in Prob. 2 the reference point is taken as 5 [cm] from the origin, and that the potential is zero at the reference point. Also assume that that the electric field has a magnitude of 1.0 [V/m] at a distance of 5.0 cm from the line change. Starting with the formula for the electric field of a line charge, show that the electric field in this example must then be 0.05 E ˆ [V/m] , where is in meters. Using this formula, calculate what the value of V is between adjacent equipotential lines in your plot, using the values of that you read off from your plot. Pick at least a couple of different adjacent equipotential curves and verify that you get the same result for V from each of them (or something pretty close to the same result). Then label the voltage for each of the equipotential lines on your plot (with the outer one being at zero volts, since this is where the reference point is). 4) An engineer makes an accurate flux plot for a pair of line charges. One line charge has a fixed line charge density of l = 1 [C/m] and is located on the x axis at x = -h. The other line charge has l = -0.5 [C/m] and is located on the x axis at x = h. The axis of both line charges is parallel to the z axis. Assume that the engineer chooses to have 100 flux lines emanating from the left line charge. a) How many flux lines enter the second line charge? b) How many flux lines end up going to infinity? c) Very close to the left line charge, what is the angle between adjacent flux lines? d) Very far away from the origin, what is the angle between adjacent flux lines? e) How many flux lines cross a vertical line (going from left to right) defined by x = 0? (Hint: Use superposition, treating each line source separately. The total flux crossing the line is the sum of the fluxes that cross the line from each line source, treating each line source as if it were alone.) 2 5) The following spherically-symmetric charge distribution is set up in air (the charge density exists everywhere in space): v e r [C/m3]. Use Gauss’s law to find the electric field vector everywhere. 6) A sphere of uniform volume charge density v0 with radius a is surrounded by a spherical shell of uniform volume charge density v1 that has an inner thickness b and an outer radius c. Determine the electric field vector in all regions: r < a, a < r < b, b < r < c, and c < r. a b c 7) A infinitely long electron beam has a uniform volume charge density of v0 for a . For a the charge density is v v0 e a . Find the electric field vector in the region a and the region a . a z 8) A coaxial cable has an inner conductor (wire) of radius a and an outer conductor of inner radius b. For modeling purposes the inner conductor is modeled as a uniform surface charge density s0 [C/m2] at = a. The outer conductor has a uniform surface charge density -s0 [C/m2] at = b. Assume that the cable is infinitely long in the z direction. b a z Find the electric field vector in all three regions: 0 a , a b , b . 3 Extra Problems (not to be turned in – for extra practice only): Shen and Kong: None Hayt and Buck, 7th Edition: 3.2, 3.3, 3.4, 3.25 (parts c, d), 6.25 (ignore the calculation of capacitance). 4