Legendre Polynomials - Electricity and Magnetism - Past Paper, Exams of Electromagnetism and Electromagnetic Fields Theory

Download Legendre Polynomials - Electricity and Magnetism - Past Paper and more Exams Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! NIU Ph.D. qualifier examination 2003 Spring (112512003) Electricity and Magnetism Solve 3 out of 5 problems. I. Find the potential @ and field E for an unchanged conducting sphere placed in an initially uniform electric field, using an expansion in Legendre polynomials. Choose the z axis to be the initially uniform field direction. a. Write the most general solution to Laplace's equation in terms of the radial functions U(r\" \'/ : A,rt + B,lr'*t and the Legendre polynomials f (cos9). Since the problem hasr azimuthal symmetry, no spherical harmonics are required. b. Use boundary conditionat r = co to determine alltheAl. c. Use boundary conditi on at r = a te determine all the By except Bo. What determines Bo? d. With all coefficients determined, wite explicit forms for O and E for the space outside the sphere. e. Determine the induced charge density on an initially uncharged sphere. f. Determine the total induced charge on an initially uncharged sphere. II. A spherical shell of radius R carries a uniform surface charge density o. Calculate the vector potential A and the magnetic field B, which are created when the sphere is rotating with an angular speed o. III. Classical model of Zeemaneffect. a. Consideer an electron that executes 3-dimensional simple harmonic motion (SHM), i.e., it is subject to the potential per unit mztss, V(x,y,z) =I*o'(*' + y' + z'). Now turn on an external magnetic field B = Bi,where B is a constant. Show that the frequency of vibration is modified such that (in mks units) 0r: wherein e : 1.6 x 10-1e, e is the electron charge, and m: 9. 1 x 10-31 kg is the electron rest mass.