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Physics PHYS 354 Electricity and Magnetism II Problem Set #2 1. Method of Images: Conducting Sphere in a Uniform Electric Field A conducting sphere in a uniform electric field can be thought of as a conducting sphere placed between extremely distant positive and negative point charges as shown below. a) Show that near the origin there is an approximately constant electric field, +Q -Q O z=-R z=+R E0 Q 2 0 R 2 . In the limit as R and Q approach infinity, but with Q R 2 a constant, this approximation becomes exact. b) Using the method of images, it was shown that the potential due to a point charge near a grounded conducting sphere is q 1 1 V x y 4 0 x y x anˆ a where nˆ is a unit vector in the direction of the original source charge. r P +Q z=-R a -Q z=+R By using this relationship twice (once for each charge on either side of the sphere) write down the total potential, expand, then let R go to infinity while Q R 2 remains a constant. Show that to first order, 1 2Q 2Q a 3 r cos cos 2 2 2 4 0 R R r where the omitted terms vanish in the limit R . c) Show then that a3 E0 r 2 cos . r where the first term is just due to the original electric field, and the second term is the dipole strength of the charge distribution induced on the surface. 2 2. Laplace's Equation in Rectangular Coordinates Consider an infinitely long rectangular tube with potentials on each of its four sides as specified in the figure. y V=V0 (0,b) V(x,y) V=0 (0,0) V=0 (a,b) V=0 (a,0) x Show that the potential inside the rectangular tube is given by the following expression: V ( x, y ) 1 nx sinhny a sin . n1,3,5 n a sinhnb a 4V0 3