Question

Problem 4 (20 points): An infinite sheet of charge (i.e. infinite in the y- and z- directions) with thickness 2d lies in the 12-plane between x =-d and x = +d. An edge view (ie, cross-section) of this thick, planar sheet of charge is shown in the figure. This particular insulating sheet has a non-uniform volume charge density p (C/m') that varies as a function ofx, given by p(x)-Pn(x1), where Pols a positive constant. a) Can Gauss's Law be used to calculate the electric field at a distance x from the middle of the infinite thick sheet of charge? Please briefly explain why or why not. b) Please indicate the direction of the electric field on both sides of the thick sheet, i.e. in the regions x> d and x<-d. c) Consider three Gaussian pillboxes shown in edge view at right and labeled A, B, and C. These Gaussian pillboxes have the same cross-sectional area (not shown), but are different lengths, and are placed as such: Box A: between x d2 andx +d/2 Box B: between x d andx +d 2d Box C: between x 3d/2 andx +3d/2 Rank the net charge enclosed by each box from largest to smallest. Please explain. In each of the following parts, please be sure to show all steps in your derivations. (Hint: you may want to use one or more of the Gaussian pillboxes suggested above, keeping in mind that the sheet has a non-uniform charge distribution as given above.) d) Calculate the magnitude of the electric field outside the thick sheet of charge, for distances >d from the mid-plane atx -0. e) Calculate the magnitude of the electric field inside the thick sheet of charge, for distances d from the mid-plane at x-0.