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.

•• The charge independence of nuclear forces implies that in the absence of electrostatic forces, the energy levels of 7Li and 7Be would be the same. The main effect of the electrostatic forces is simply to raise all the levels of 7Be compared to those of 7Li. Approximating both nuclei as uniform spheres of charge Q = Ze and the same radius R, estimate the difference in the binding energies for any level of and the corresponding level of . (The electrostatic energy of a uniform charged sphere is 3kQ2/5R — see Problem 1. The observed radius R is about 2.5 fm.) Compare your rough estimate with the observed difference of about 1.7 MeV. (Note that the true charge distribution is not a uniform sphere with a well-defined radius R, but is spread out somewhat beyond R. Therefore, the observed electrostatic energies would be expected to be somewhat smaller than your estimates.)

Problem 1

••• In this problem you will calculate the electrostatic energy of a uniform sphere of charge Q with radius R. The electric potential V(r) at any radius r < R is given by (16.49) in Problem 2, and the potential energy of a charge q at radius r is qV(r). (a) Write down the total charge contained between radius r and r + dr, and hence find the potential energy of that charge. (b) If you integrate your answer to (a) from r = 0 to r = R, you will get twice the total potential energy of the whole sphere since you will have counted the energy of any two charge elements twice. Show that the total Coulomb energy of the sphere is