(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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**