From the equations VE=0, VXE = write down the electric field as an integral in terms...

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From the equations VE=0, VXE = write down the electric field as an integral in terms of time derivative of B. Hint: First write down the Biot-Savart solution for V-B = 0, V x B= HoJ. And then compare the mathematical expressions to write down the solution. b) Similarly, from the equations V-A=0, VX = B, write down the vector potential A as an integral in terms of B. c) By comparing the results of a) and b), Write E in terms of A. Check the result by taking a curl for both sides. Now we would like to compute the electric field for a spherical shell of radius R carries a uniform surface charge o and also spinning around a fixed axis with an angular velocity w(t) that changes slowly with time. There are two contributions here: the Coulomb field due to the charge, and the Faraday field due to the changing B. d) Find the electric field inside and outside the sphere due to the charge. e) Find the electric field inside and outside the sphere due to the change of magnetic field. Combine the results with d). From the equations VE=0, VXE = write down the electric field as an integral in terms of time derivative of B. Hint: First write down the Biot-Savart solution for V-B = 0, V x B= HoJ. And then compare the mathematical expressions to write down the solution. b) Similarly, from the equations V-A=0, VX = B, write down the vector potential A as an integral in terms of B. c) By comparing the results of a) and b), Write E in terms of A. Check the result by taking a curl for both sides. Now we would like to compute the electric field for a spherical shell of radius R carries a uniform surface charge o and also spinning around a fixed axis with an angular velocity w(t) that changes slowly with time. There are two contributions here: the Coulomb field due to the charge, and the Faraday field due to the changing B. d) Find the electric field inside and outside the sphere due to the charge. e) Find the electric field inside and outside the sphere due to the change of magnetic field. Combine the results with d).

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