2317 ECE Applied Electricity and Magnetism Spring 2009 Homework #2
Date Assigned: February 6, 2009; Coordinates, Coulombs Law Due Date: February 12, 2009 1. Derive the identity x = cos sin . 2. Derive the identity = x cos cos + y cos sin z sin . 3. A hemispherical surface defined by a radius r = 5 [m] and z > 0 has a surface charge density of s = 2 sin cos 2 [C/m2]. Find the total charge on the surface. 4. A cylindrical volume is defined by < 0 [m], 0 < z < h [m], and 0 < < 2 . Inside this region there is a volume charge density
v = e cos 2 sin
2
z 3 [C/m ]. h
Find the total charge inside the region. 5. A current density vector is given as
J = z z [A/m 2 ] .
Find the total current crossing the surface of the hemisphere, defined by r = 2 [m] and z > 0. The current should be the current crossing the surface in the upward sense. 6. Point charge 1 having 2 [C] is at the point (1,2,3) [m] in space. Point charge 2 having -3 [C] is at the point (-1,4,2) [m] in space. Find the force that would be exerted on a charge q of 2 [C] if this charge were located at (0,-2,-3) [m]. 7. Two charges, Q and 5Q , are separated by a distance d in air. Assume that the charge Q is at the origin and the charge 5Q is at x = d. A third charge is placed on the x axis so that the entire system is in equilibrium (there is no net force on any of the three charges). Determine the location of the third charge (in terms of d) and its charge q (in terms of Q). (Hint: Consider whether the third charge must be located between or outside the other two. Also, note that if the net force on any two of the charges is zero, then the force on the third one will automatically be zero, by Newtons law.)
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8. thin A line charge of density l [C/m] is in the form of a semicircle of radius a lying on the x-y plane with its center located at the origin. The semicircle starts at = / 2 and ends at = 3 / 2 . Find the electric field vector E at the origin for the cases in which (a) the line charge density l = 0 is a constant, and (b) the line charge density varies along the semicircular ring as l = 0 cos . 9. A uniform (constant) surface charge density s 0 [C/m2] is distributed over a cylindrical surface at = a , extending from z = -h/2 to z = h/2. Find the electric field vector in free space at a point on the z-axis at a height z above the origin, where z > h/2. 10. A uniform (constant) volume charge density v 0 [C/m3] exists inside of a cylinder of radius a and height h. The cylinder is centered at the origin, with the cylinder axis running along the z axis. Calculate the electric field at a point on the z-axis at a height z above the origin, where z > h/2. Hint: Take advantage of your solution to the previous problem to the maximum extent possible. 11. A continuous volume charge density defined as v = ( x 2 + y 2 + z 2 )
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[C/m3] is
distributed in the region 0 x 1 , 0 y 3 , 1 z 1 and is zero elsewhere. Find E y at the origin. 12. An infinite sheet of uniform (constant) charge density s 0 [C/m2] is situated coincident with the x-y plane at z = 0. The sheet has a hole of radius a centered at the origin. Find the electric field vector at points along the z-axis. (Your answer should include the cases where z is positive and z is negative.) You may use superposition in any way that you wish. Hint: If the hole were not there, you would have an infinite sheet of surface charge density.
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