Homework Set # 10 - Due November 11

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1. 15.4.10 (Warning! The RHS of the equation should be multiplied by (-1)!)

2. 15.4.13

3. 15.5.2

4. a) Express the elements of the quadrupole moment tensor x_ix_j as a linear combination of the spherical harmonics Y_2m and Y₀₀.

   b) The quadrupole moment tensor is usually defined as

Qij=∫(3x_ix_j-r²δ_ij)ρ(r)dτ,

with ρ(r) the charge density. Express the components of (3x_ix_j-r²δ_ij) in terms of r²Y_2m.

c) What is the significance of the -r²δ_ij term?

Solution (pdf file)

5. A charged conducting ring of radius a (Example 15.2.3) may be described by

ρ(r)=qδ(r-a)δ(cosθ)/(2πa²).

Using the known Green's function for this system, obtained in class, find the electrostatic potential.

Solution (pdf file)

6. Expand the Green function found in class for the volume inside a sphere of radius R in terms of spherical harmonics. Your Green
    function has to be a function of (r,θ,φ;r',θ',φ') and it has to be symmetric under exchange of the primed and unprimed variables.