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Physics PHYS 354 Electricity and Magnetism II Problem Set #3 1. Conducting Sphere in a Uniform Electric Field A conducting sphere of radius R carrying a charge +Q is placed in a uniform electric field of intensity E0 . Show through application of Laplace's equation in spherical coordinates that the electrostatic potential V r, at a given point outside the sphere is given by V r, E0r cos E0 R 3 cos . 4 0r r2 Q Could you have predicted this result from the result of Problem 1 on Problem Set #1? Why or why not? From this expression determine on the surface of the sphere. Also show that the total charge on the surface of the sphere is +Q. Finally, determine the dipole moment of the induced charge on the sphere. 2. Cavity in a Dielectric A large block of dielectric material ( D E ) is uniformly polarized with E x E0 . A small spherical cavity of radius a is cut out of the block, as shown in the figure below. E0 a z Calculate the field E inside of the cavity: a) assuming that P x adjusts to the presence of the cavity, so that in r a , D E , in r a , D E , and as r , E E0 . (Hint: Choose suitable forms for in r a and , then match Dr and E at r a .) b) assuming P x is frozen at its original value in r a and P 0 in r a . Find the electric field E by two methods: i) ii) 3. Subtract the field due to an oppositely polarized sphere from E0 . Calculate it by replacing P with P and P nˆ and finding appropriate in r a and r a . Lock in the polarization by using the fact that D 0 E P0 where P0 is the "locked in" polarization, rather than D E , where the polarization depends on E . Laplace's Equation in Plane Polar Coordinates In this problem you will be asked to first explore Laplace's equation in plane polar coordinates, then solve a particular problem by matching the boundary conditions. a) Take Laplace's equation in cylindrical coordinates ( r, ,z ) and convert it into an equation in polar coordinates ( r, ). Assume the solution has the form V r, Rr . Use a separation parameter k 2 to show that the solution is V (r, ) A0 B0 ln r C0 D0 V (r, ) Ak r Bk r k b) k C k cos k Dk sin k when k 0 , and when k 0 . Consider a wedge-shaped region bounded by two grounded conduction surfaces intersecting at the origin with an interior angle together with a line charge of strength per unit length located at the point r0 , within the wedge, as shown below. 2 V=0 r VI r0 V=0 P(r,) VII Show that the potential in the region bounded by the intersecting conducting planes is n n n 1 r sin VI r r0 sin 0 n 1 n r0 and n n n 1 r0 VII r r0 sin sin 0 n 1 n r where it is apparent that the solution must be written in two parts, VI for the region r r0 , and VII for the region r r . The electric field given by the two solutions must follow the correct boundary conditions when r r0 . (Hint: In polar coordinates, the sampling property of the delta function is written, b r0 f d f where 0 , 0 and the line charge density can be written: .) 3