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4321_HW7FA08 - 5 A point charge q is located a large...

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Physics 4321 Homework Set 7 October 20, 2008 Due: October 27, 2008 1. A sphere of radius R is centered at the origin and carries a charge density ρ ( r , θ ) given by ρ ( r , θ ) = k ( R / r 2 )( R -2 r )sin θ , where k is a constant, and r , θ are the usual spherical coordinates. Show that the monopole and dipole contributions to the multiple expansion are zero. 2. For the charge density given in problem 1, calculate the quadrupole moment and use it to write an approximate expression for the potential on the z axis, far from the sphere. 3. In example 3.9, we calculated the exact potential for a spherical shell of radius R , which carries a surface charge density σ = k cos θ . (a) Calculate the dipole moment of this charge distribution. (b) Calculate the potential due to the dipole term and compare it to the exact result. What can you conclude about the higher order multipoles? 4. Two point charges are located as follows: q 1 = - q at (0,0,0) and q 2 = 3 q at (0,0, a ). Calculate the monopole moment and the dipole moment for this arrangement of charges.
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Unformatted text preview: 5. A point charge q is located a large distance r from a neutral atom of polarizability a . Calculate the force between them. 6. A dipole p is a distance r from a point charge q and oriented so that p makes an angle with the vector r from q to p . (a) Calculate the force on p . (b) Calculate the force on q . (c) Is Newtons third law ok here? 7. A sphere of radius R carries a polarization P ( r ) = k r , where k is a constant and r is the vector from the center. (a) Calculate the bound charge densities b and b . (b) Calculate the electric field inside and outside the sphere. 8. A thick spherical shell (inner radius a and outer radius b) is made of dielectric material with a frozen in polarization P ( r ) = ( k / r ) r ^ , where k is a constant and r is the distance from the center. No free charge exists here. (a) Calculate the electric field in all three regions. (b) Calculate all bound charges....
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