Ph.D. Candidacy Examination in Electricity and Magnetism - Spring 2012, Exams of Electromagnetism and Electromagnetic Fields Theory

Download Ph.D. Candidacy Examination in Electricity and Magnetism - Spring 2012 and more Exams Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! NIU Ph.D. candidacy examination Spring 2012 (2/18/2012) Electricity and Magnetism Solve 3 out of 4 problems. 1. [40 points] The electric field of a charged sheet. (a) Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge density !. [20 points] (b) Find the electric field, keeping the leading non-zero term, when a " # (infinite plane). [10 points] (c) Find the electric field, keeping the leading non-zero term, when z >> a. [10 points] 2. [40 points] Consider a uniform thin shell of charge spinning about the z axis with the angular velocity $ , with the total charge q and the radius of the sphere R. (a) Express its magnetic dipole moment, ! m . [12 points] (b) Express the magnetic scalar potential (as a function of radius r) outside the spinning shell of charge using $ , q, R. [14 points] (hint: The magnetic scalar potential can be represented by the expansion ! out = a m r m+1 P m cos"( ) m=0 # $ (Outside the sphere) ! in = b m r m P m cos"( ) m=1 # $ (Inside the sphere) % & '' ( ' ' , where P l )( ) is the Legendre polynomials, am and bm are some coefficients. (c) Find the magnetic field outside the spinning shell of charge, in terms of the magnetic dipole moment, ! m . [14 points] 3. [40 points] Induction of a toroidal coil. (a) Find the self-inductance of a toroidal coil with rectangular cross section (inner radius a, outer radius b, height h), which carries a total of N turns. [20 points] (b) Calculate the energy stored in this toroidal coil. [20 points] 4. [40 points] An electromagnetic wave with angular frequency $ moves through a material that obeys Ohm’s Law with conductivity !. The permittivity and permeability of the material are the same as that of vacuum. (a) Derive the separate second-order wave equations for the electric and magnetic fields ! E and ! B . [12 points] (b) Find expressions for the electric and magnetic fields of a wave moving in the ẑ direction and polarized in the x̂ direction. [16 points] (c) Find the distance that the wave travels for which its intensity is decreased by a factor of 10. [12 points] z axis