Electrodynamics — DU B.Sc. (Hons) Physics Previous Year Question Papers

B.Sc. Hons.-(Physics)_NEP: UGCF-2022 VIII- Semester Electrodynamics @SE Paper) Unless specified, the paper uses SI units. The metric used is g„v = (+ ,-,-,- ). Students are advised to mention the units and metric they are using. Some useful formulae are provided at the end of question paper. a) The electric and magnetic fields in an inertial frame have values E = (0, 3, 0)V/in and B = (0, 0, 3 X 10-8)T respectively. Determine whether there exists a frame in which the magnetic field vanishes. b) c) Obtain the condition satisfied by the scalar function % in order to preserve the Lorenz gauge condition under the gauge transformation Aft i A# + aft%. Show that the homogeneous Maxwell's equation a„ f#V = 0, f#V being the dual electromagnetic tensor, can be also written in the form aAFqu + a„FVA + avFdy = 0. d) In a velocity selector, the fields are E = 2.5 X 104/^V/in, 8 = 0.08 £ T. Find the speed of particles that pass undeflected. e) I) Show that dipole radiation vanishes if the charge distribution is symmetric under spatial inversion, i.e. p(-?) = p(7) . A non-relativistic charge oscillates as x(C) = xo cosct)i, where %o is the amplitude and a) is the angular frequency. Find the instantaneous radiated power P(Z) using Larmor's formula. g) A system is described a Lagrangian density I(¢,a„¢). If under an infinitesimal transformation of field ¢ i ¢ + Ea¢, the Lagrangian density changes by 6£ = €a„Kft, show that the Euler-Lagrange equation satisfied the system does not change. a) b) Suppose that the four-potentials A# = (= ,A+) do not satisfy Lorenz gauge condition. Show that it is always possible to find a scalar field % such that the gauge-transfomed potential A'4` = Aft - aftx satisfies the Lorenz condition. Derive the differential equation satisfied by the scalar field %. Using Maxwell's equations, show that once the Lorenz condition is imposed, a and A+ satisfy non-homogeneous wave equations. c) Evaluate the Lorentz invariant quantity I = det(Fvff ) in terns of the electric and magnetic fields. (5, 5, 8) a) A particle of mass " and charge g is moving with velocity 3 in the presence of electric field E and magnetic field 3. Write down the covariant form of Lorentz-force equation for this particle and use it to derive the relativistic equation of motion for a charged particle in an electromagnetic field. Show that in the non-relativistic limit, this equation reduces to md==q(-E+3xB). b) A proton gyrates in magnetic field 8 = Bo(1 + cr y) fi, where Bo = 0.3 T and c¥ = 0.5 in-1. The component of the proton velocity perpendicular the magnetic field is ui = 3 x 105 in/s. Using the guiding-center approximation, calculate (i) the gradient drift velocity, (ii) the drift direction for proton vs. electron, and (iii) the net current density associated with the grad- 8 drift if the proton and electron number densities are both 71 = 1016 in-3.