Spring 2021 Homework Assignments
A particle with charge q is moving along the z-axis with constant velocity so that it's position is given as z=vt. Time t changes from t=-∞ to t=+∞. An observer is stationary on the x-axis at x=a. Find the electric and magnetic fields at the position of the observer. Make sketches of the components of these fields as functions of time for v comparable to c. Hint: You have two costraints which may be expressed as two delta-functions: δ(t'-[t-R/c]) δ(z'-vt') where R2=(z'2 + a2) . The t' integral is easy. The z' integral involves a little bit of algebra: After the t' integral, you are left with an integral of the form ∫ dz' δ(g(z')) f(z') = f(zo)/|g'(zo)| . The object zo is the solution to g(zo)=0 which is a quadratic equation. (Yes I know, the solution does not result in a "simple" expression, but that is life!) It is probably best for you to obtain solutions on the computer, that will also result in more accurate plots. We will revisit this problem in the relativity chapter. You may want to look at the results of section 11.10 in your textbook.
A point charge Q is placed at the center of a sphere with radius R and permittivity ε, which leads to the polarization of the sphere. The sphere is now spun around an axis (which you can take as the z-direction). Find the magnetic field at all points in space.
The space for z > 0 is filled with a material with dielectric permittivity ε = εR + iεI. The space for z < 0 is vacuum. An electromagnetic wave is incident to the boundary from the vacuum with angle of incidence equal to zero. For both the electric field parallel and perpendicular to the plane of incidence, find (a) the ratio of the strength of the incident and reflected waves (E''/E), (b) the Poynting vector at the boundary, for incident, reflected, and transmitted waves.
Problem 7.22 in Jackson.
Consider a point charge q rotating around the origin, with a small radius R, on the x-y plane. The position of the particle is given by r=R, θ=π/2 φ=ωt . The current density J may then be written as qRω δ(r-R) δ(R cosθ) δ(φ-ωt) times the unit vector in direction φ . Note that this is not a source that has the simple time dependence cos(ωt) . However, since the current is periodic with frequency ω, it can be expanded in terms of cos(nωt) and sin(nωt): j = Σn ( An cos(nωt) + Bn sin(nωt) ) (a) Carry out this expansion and find expressions for radiated fields at each frequency and sum them up. (b) Find the expression for total radiated power.
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
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