(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the mean-square charge radius of a uniformly charged sphere (with radius R) is

< r^2 > = 3*R^2 / 5

2. Relevant equations

< r^2 > = [tex]\int[/tex] [tex]\varphi[/tex]_{f}* r^2 [tex]\varphi[/tex]_{i}dV

[tex]\varphi[/tex]_{f}* = exp(-iqdotr)

[tex]\varphi[/tex]_{i}= exp(iqdotr)

whereq=k_{f}-k_{i}

andp_{f}= h_bar *k_{f}

p_{i}= h_bar *k_{i}

r > R: V(r) = -Ze^2 / (4*pi*e_{0}) * 1/r^2

r < R: V´(r) = -Ze^2 / (4*pi*e_{0}*R) * (3/2 - 1/2 * (r/R)^2)

E_{i}= 1/2*m*v_{i}

E_{r0}= J_{0}^2/(2*m*r_{0}^2) + V(r_{0}) or V´(r_{0}) according as r > R or r < R.

J_{0}= J_{i}= m*v_{i}*b

where b is the impact parameter and is b = Ze^2/ (4*pi*e_{0}*m*v_{i}^2) * cot(A/2) for hyperbolic orbits, E_{i}> 0

3. The attempt at a solution

I basically want to know if my approach is correct. I think that I have to find < r_{0}^2 > where r_{0}is the distance of closest approach to the centre of the charge distribution either inside it (where the electron feels V´(r)) or outside it (where the electron feels V(r)).

I've done some calculations using the radial energy equation, trying to solve for r_{0}^2, but the calculations are a mess and I get a feeling I'm going about it all wrong. The problem is from Krane's Introductory Nuclear Physics, Chapter 3, problem 3.1.

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# Homework Help: Nuclear Physics - mean-square charge radius of a uniformly charged sphere

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