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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(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: