Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object contins no free chrge. Prove tht the object contins zero bound volume chrge density, ρ b. (Of course, it cn hve nonzero bound surfce chrge, σ b ). 2 HWX Problem 2 ) A sphere of polrized mteril, with uniform polriztion P = P 0 ẑ nd rdius, will hve distribution of bound chrge on its surfce. This bound chrge will crete n electric field, outside nd inside the sphere. ecll tht the resulting electric potentil outside the sphere (r > ) is tht of dipole, with dipole moment p = 4π 3 3 P 0 ẑ. You need not prove this fct. Show tht the electric field inside the sphere is uniform, by mtching the potentil outside (given bove) with the generl form for the potentil in sphericl coordintes (ssume no zimuthl dependence: V = V (r, θ)). Find the mgnitude nd direction of the electric field inside the sphere. b) Consider sphere of rdius, composed of dielectric mteril with uniform reltive permittivity ɛ r. The sphere is plced into uniform, externl electric field E ext = E 0 ẑ. Then, the field inside the sphere is the superposition of the externl electric field, nd the electric field E b produced by bound surfce chrge on the sphere. The polriztion in the dielectric is proportionl to the superposition of these fields: P = χ e ɛ 0 ( Eext + E b ) (1) The proportionlity constnt χ e depends on the prticulr mteril of the dielectric. 1
Find the polriztion P tht yields electric field E b tht stisfies the bove eqution, for given χ e. (Use your results from prt.) Find the net induced dipole moment of the sphere, nd stte the electric field outside. 3 HWX Problem 3 A cpcitor consists of two conducting pltes with lrge res A nd smll seprtion d. The lower plte lies t z = 0 nd is grounded (potentil V = 0). The upper plte lies t z = +d nd is t potentil V 0. Becuse the pltes re lrge, you cn ignore fringing fields ner the edges of the pltes. ) Assume tht the region between the pltes contins vcuum. Find the electric field between pltes (give mgnitude nd direction), the chrge density on the upper plte, the totl chrge on the upper plte, nd the cpcitnce of the cpcitor. b) Now suppose tht the region between the pltes contins dielectric mteril with reltive permittivity ɛ r. Given the potentil of the upper plte V 0, find the electric field between the pltes. Find the polriztion P in the dielectric. Find the bound chrge t the surfces of the dielectric. Find the cpcitnce. d d/2 z V0 A/2 A/2 V0 V=0 c) d) V=0 c) Suppose tht the dielectric mteril extends only hlfwy cross the gp between conducting pltes, from z = 0 to z = d/2. Vcuum fills the reminder of the gp. Wht boundry conditions relte E nd electric displcement D in the two regions, bove nd below the surfce of the dielectric? Find E nd D in these two regions, given tht the potentil difference between top nd bottom pltes is V 0. Find the free surfce chrge, nd totl free chrge, on the top plte. Find the cpcitnce. d) Now suppose tht the dielectric mteril extends ll the wy cross the gp, but covers only hlf of the re of the pltes. Vcuum fills the reminder of the region between the pltes. Wht boundry conditions relte E nd electric displcement D in the two regions, inside nd outside the dielectric? Find E nd D in these two regions, given 2
tht the potentil difference between top nd bottom pltes is the sme: V 0. Find the free surfce chrge on the top plte, in the two regions. Find the cpcitnce. 4 HWX Problem 4 A regulr octhedron hs sides of length, nd is centered t the origin. The 6 vertices, or corners of the octhedron lie on the ±x, ±y, nd ±z xes. A chrge is plced on ech of 5 of the 6 vertices: the chrge on the +z-xis is omitted. ) Find the electric field E t the center of the octhedron. Give mgnitude nd direction. Explin your resoning. Why is Guss s Lw not suitble pproch for determining the electric field in this cse? Would Guss s Lw be suitble if the chrge on the +z-xis were present? Explin your nswers in sentence or 2. b) Find the electric potentil V t the center of the octhedron. Explin your resoning. z y x 3
5 HWX Problem 5 ) A chrge is plced on metl sphere of rdius. A second sphericl metl shell, concentric with the first nd with rdius 3, is given chrge q. Give the electric field between the shells. Find the potentil difference between the shells, V. q 3 q 3 Cross-Section b) Find the cpcitnce of the pir of conducting sphericl shells. Give the energy stored in the cpcitor. You need not derive your nswer. Wter 3 2 q Cross-Section c) Now wter is dded to the spce between the two sphericl shells, from rdius 2 to rdius 3. (Very thin insulting shells lie t 2 nd 3, confining the wter nd keeping chrges ±q on the metl shells.) The wter is dielectric, with electric susceptibility χ e, or reltive dielectric constnt ɛ r = 1 + χ e. Find D in the region between the spheres. Find the electric field E in the region between the spheres. 4
d) Find the polriztion P in the wter. Find the bound volume chrge ρ b within the wter, nd the bound surfce chrge σ b on both surfces of the wter. Wht is the totl bound chrge on the wter? Explin your nswer in sentence or two. 6 HWX Problem 6 The x y plne hs the potentil ( ) 2πx V (x, y, z = 0) = V 0 cos. (2) L No chrge lies in the upper hlf-spce z > 0, or the lower hlf-spce z < 0. Fr from the x y plne, t z, the potentil flls to zero: v 0. Write down the potentil in the upper hlf-spce V (x, y, z > 0), nd tht in the lower hlf-spce V (x, y, z > 0). Explin why this potentil is unique, for the boundry conditions. (You need not derive your nswer.) Find the surfce chrge density on the x y plne, σ(x, y). 5