Problem

Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• It is often a useful approximation to treat a nucleus as a uniformly charged sphere of radius R and charge Q. In this problem you will find the electrostatic potential V(r) inside such a sphere, centered at the origin. (a) Write down the electric field E(r) at any point a distance r from the origin with r > R. (Remember that this is the same as the field of a point charge Q at the origin.) (b) Use the definition

to find the potential difference between points at r1 and r2 (both greater than R). It is usual to define V(r) so that V(r) approaches 0 as r → ∞. By choosing r1 = ∞ and r2 = r, show that

(c) Now find E(r) for r ≤ R. (Remember that Gauss’s law tells us this is kQ′/r2, where Q′ is the/total charge inside the radius r.) Check that your answers for parts (c) and (a) agree when r = R. (d) Use Eq. (16.47) to find V(r2) − V(r1) for any two points inside the sphere. Now choose r2 = r and r1 = R, and use the value of V(R) from part (b) to prove that