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## Homework Statement

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

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

## Homework Equations

< r^2 > = [tex]\int[/tex] [tex]\varphi[/tex]

_{f}* r^2 [tex]\varphi[/tex]

_{i}dV

[tex]\varphi[/tex]

_{f}* = exp(-i

**q**dot

**r**)

[tex]\varphi[/tex]

_{i}= exp(i

**q**dot

**r**)

where

**q**=

**k**

_{f}-

**k**

_{i}

and

**p**

_{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

## 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.