CALC An insulating hollow sphere has inner radius a and outer radius b. Within the insulating material the volume charge density is given by ?(r) = ?/r, where a is a positive constant. (a) In terms of ? and a, what is the magnitude of the electric field at a distance r from the center of the shell, where a < r < b? (b) A point charge q is placed at the center of the hollow space, at r = 0. In terms of ? and a, what value must q have (sign and magnitude) in order for the electric field to be constant in the region a < r < b, and what then is the value of the constant field in this region?
Solution 53P Step 1 of 3: First apply the Gauss’s law and take a spherical Gaussian surface because of the spherical symmetry of the charge distribution. The net field is the vector sum of the field due to q and the field due to the sphere. r 2 (r) = , rv = 4r dr and Q = ( )dv a Step 2 of 3: r r 1 2 2 a)For a Gaussian sphere of radius r, Q enc = (r )dv = 4 r dr = 4 (r2 a ) a a 2 2 Gauss law suggests that E(4r ) = 2 2(r a ) 0 Now E = = (1 2) 20 r