Electrodynamics – FS 2019
Exercise sheet 12
Lecture: Prof. Dr. S. Pozzorini Assistants: Dr. J.-N. Lang
H. Zhang M. Ebersold
To be handed in: 29.05.2019 Discussion: 31.05.2019
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Exercise 12.1 Coaxial cable (6 points)
Consider a coaxial cable consisting of two cylindrical conducting sheets with axis
~e3. The inner and outer cylinders have radii a and b, respectively, and constant electric potentials
φ(~x, t)
r=a=Va, φ(~x, t)
r=b =Vb. (1)
Similarly as for hollow wave guides, electromagnetic waves inside a coaxial cable have the form
E(~~ x, t) = E(~~ xT)ei(kx3−ωt), B(~~ x, t) = B(~~ xT)ei(kx3−ωt), (2) where the fields F~ = E, ~~ B are independent of x3 and can be expressed in cylindric coordinates as
F~(~xT) =F~T(r, φ) +F3(r, φ)~e3, with F~T(r, φ) = Fr(r, φ)~er+Fφ(r, φ)~eφ. (3) The goal of the exercise is to show that, at variance with hollow wave guides, coaxial cables can transport purely transverse TEM waves, i.e. waves with E3 =B3 = 0.
(a) Using the Maxwell equations, show that TEM modes correspond to a 2-dimensional electrostatic problem with
E~T(~xT) =−∇Tφ(~xT) and ∆Tφ(~xT) = 0. (4) Show also thatB~T(~x, t)⊥E~T(~x, t), and find the dispersion relation ω(~k).
(1 p.) (b) Using a separation Ansatz for φ(~xT) in cylindrical coordinates, show that the boundary conditions imply a trivial azimuthal dependence and determine the radial dependence of φ as a function of the potential difference ∆V =Va−Vb. (2 p.) Hint: Use ∆T= 1r∂r(r∂r) + 1r∂φ2.
(c) Determine the full solution for the electric and magnetic fields. (1 p.) (d) Derive the surface charge density σ(z, t) and the total current I(z, t) on the
inner conductor using Gauss’s and Ampere’s law, respectively.
(2 p.)
Exercise 12.2 Lifetime of a “classical“ Hydrogen atom (4 points)
Consider a naive classical picture of the Hydrogen atom, where the electron rotates on a circular orbit with Bohr radius a0. According to the laws of classical electrody- namics, due to synchrotron radiation the electron would loose kinetic energy and fall into the nucleus.
(a) Estimate the initial velocity of the electron assuming that the Coulomb and the centrifugal force are in equilibrium. Is it justified to treat this system with a non-relativistic approximation? Use numeric values to justify your answer:
– a0 = 5×10−11m – m= 9.11×10−31kg – q= 1.6×10−19C – 0 = 8.85×10−12mF – µ0 = 4π×10−7Hm
(1 p.) (b) Determine the total power corresponding to the emitted synchrotron radiation
in the non-relativistic limit β = vc 1.
(2 p.) (c) Estimate the lifespan of such a classical Hydrogen atom and discuss the result.
(1 p.) Hint: Derive a differential equation for the total kinetic and potential Coloumb energy of the system, and recast it into a differential equation for the radius of the electron orbitr(t).
Exercise 12.3 Rotating Hertz’s Dipole (5 points)
Consider an electric dipole ~p(t) that is located at ~x = 0 and rotates around the ~e3 axis with angular frequency ~ω =ω~e3. At t= 0 the dipole points in the ~e1 direction, i.e. ~p(0) =p ~e1.
(a) Find an appropriate complex-valued dipole ~q, such that
~
p(t) = Re [~q(t)] for ~q(t) =~q e−iωt. (5) (1 p.) (b) Determine theE(~~ x, t) andB~(~x, t) fields in the far region, i.e. retaining only the
radiation terms of order 1/r.
(2 p.) Hint: Start from the standard formula with a complex-valued dipole and express the complex-valued fields as linear combinations of the basis vectors ~ei using
~
x=r~n=r[cosθ ~e3+ sinθ(cosφ ~e1+ sinφ ~e2)]. Then extract the physical fields by taking the real parts.
(c) Determine the Poynting vector S~ and the radiated power dP/dΩ as a function of θ and φ. Discuss how this angular dependence compares to the case of a standard Hertz dipole that oscillates in a fixed direction. (1 p.) Hint: Avoid computing the electric field E~. Instead express S~ directly in terms of B~ and ~n and simplify the result using E⊥~ B⊥~~ n.
(d) Discuss the polarization of the radiated waves, i.e. the orientation of the E~ or B~ field as a function time, in the~n=~e3 and in the~n=~e1 directions. (1 p.)
