Question

If you can help with even one part of this problem it would be a big help. Thanks

Two halves of a spherical metallic shell of radius R and infinite conductivity are separated by a very small, insulating gap. A potential is applied on the two halves of the sphere, such that the upper half is at potential +V, the lower half at potential -V.

1) Find the solutions for the potential in terms of the Legendre polynomials. What's the leading term outside teh sphere in the long wavelength limit?

2) Determine the electric dipole moment of the sphere as a function of V.

3) If the potential were alternating as +/- V cos wt, take the long wavelength limit. Find the radiation fields, angular distribution of radiated power, and total radiated power.

Answer 1

3, SOLUIION Two halves of a sphere that are charged oppositely should immediately remind us of a dipole so that we can assume the dipole term is the dominant term. We just need to find the dipole moment of this configuration and then we can apply the equations in the class notes for the fields radiated by a dipole. fl dontrthat are charged oppositely shot need to find the dipole momete (by a dipole Let us consider the sphere at an instant in time when the voltages are at their peak. We previously found the external potential due to such a sphere in terms of Legendre polynomials. We found the first term to be 2 cos We know that the potential due to a electric dipole pointing in the z direction should look like: cos 0 Setting these equal and solving for p we end up with Now the potentials on the sphere vary according to cos(ω), which is just the real part of the complex exponential signifying harmonic dependence. We already found in the class notes the radiation-zone fields due to an electric dipole varying harmonically in time. We can write them down immediately and plug in the dipole moment we have here. i(kr-ot) 3 VR e'kr-wr) sin θ

I hope you understood the problem, If yes rate me!! or else comment for a better solution.

i(kr-wr) 4πέρ i(kr-wt) sín θ θ 2 The time-averaged angular distribution of radiated power is: 1 3 VkR2 e-kr-1) Ho 2c 2 The total radiated power is: