Electromagnetic waves Flashcards
Explain what is meant by a linear homogeneous partial differential equation with constant coefficients.
A linear polynomial as unknown function and it’s derivatives. A differential equation is homogenous if there are no terms independent of the unknowns.
. Explain what is meant by a superposition principle in the context of linear differential equations.
In a linear system any linear superposition of solutions is a solution itself.
Explain Fourier’s method by which a linear
homogeneous differential equation with constant coefficients can be reduced to an algebraic equation.
An integral transform, transforming a differential equation from x to k. This leads to simpler spatial derivatives of ik.
State the wave equation for a real function
u(x, t) of one spatial variable x and time t.
See question 19.
Give an expression for a plane wave solution
to a wave equation for a real function u(x, t)
of one spatial variable x and time t. With the
help of a sketch explain what is meant by the
amplitude, the wave length and the phase of
the plane wave.
See question 20.
Give an expression for a plane wave solution
to a wave equation for a real function u(r, t)
of a three-dimensional vector r and time t. Explain what is meant by the frequency, the wave
vector and the dispersion relation. State the
relationship between the wave vector and the
wave length of the wave.
See question 21.
Consider a plane wave solution of Maxwell’s
equations in vacuum. Explain the implications
of the first pair of Maxwell’s equations for such
a solution.
Both the electric and magnetic field inproducts with the k vector are zero so they are both orthogonal to the k vector.
State the geometric relationship between the
wave vector, the electric field and the magnetic
field in an electromagnetic plane wave.
All three orthogonal
Explain what is meant by polarisation of an
electromagnetic plane wave. With the help of
a sketch define the cases of linear, circular and
elliptic polarisation.
Polarisation of a plane electromagnetic wave is a characteristic describing the motion
of the electric field in a fixed plane orthogonal to the wave vector.
See exam 2017 B(i) of self test questions for sketch
With the help of a sketch explain what is
meant by right-handed / left-handed circularly
polarised wave.
Looking at the wave from the source, if the wave rotates anti-clockwise it is righthanded.
Looking at the wave from the source, if the wave rotates clockwise it is lefthanded.
. State the expression for the time averaged
force exerted by a linearly polarised electromagnetic plane wave on an absorbing body.
Give your answer (a) in terms of the amplitude of the electric field in the wave (b) the
energy per unit time absorbed by the body.
See question 26.
State Ohm’s law for a conducting medium.
State the SI dimensions of conductivity and
resistivity.
See question 27.
Explain what is meant by Maxwell’s relaxation
and state the expression for the Maxwell relaxation time in a conducting medium.
Maxwell’s relaxation is a phenomenon by which an inhomogeneous distribution of
charge in a conductor relaxes to a homogeneous one. Mathematically, Maxwell’s
relaxation follows from a combination of Gauss’s law, the continuity equation and
Ohm’s law.
Maxwell’s relaxation time is given by
τ = episolon_0 / sigma
Where sigma is the conductivity of the material.
Considering an electromagnetic wave inside a
conductor, explain what is meant by a skin
layer. Give an expression for the thickness of
the skin layer in the case of a good conductor, also explaining what is meant by a good
conductor in the present context.
Electromagnetic radiation can penetrate an Ohmic conductor only to a certain frequency depth. The electromagnetic radiation can only penetrate the skin layer. The depth of the skin layer is determined by the conductivity of the material. The greater the conductivity the thinner the skin.
For good conductors the penetration depth is tiny and omegatau «1. The depth is then given by d = 1/kappa = csqrt(2*tau/omega)
So the skin depth increases with decreasing frequenct.
State the boundary conditions for the electromagnetic field at the boundary of a perfect
conductor. Explain how these boundary conditions follow from Maxwell’s equations.
The parallel componet of the E field vanishes, this follows from stokes theorem.
The normal component of the B field vanishes, this follows from Gauss’s law.
Explain what are the TE, TM and TEM modes
in a wave guide. Which of the three cannot
propagate in a simple rectangular wave guide?
TE mode has no projection of the electric field on the guide’s axis.
TM mode has no projection of the magnetic field on the guide’s axis TEM mode is
transverse.
That is, it has no projection of either the electric or the magnetic field on the guide’s
axis. Such a mode cannot propagate in a simply-connected wave guide.
Give the expression for the complete set of frequencies of TE and TM modes in a wave guide
consisting of two parallel plane mirrors.
See question 32
Give the expression for the complete set of frequencies of TE and TM modes in a rectangular
waveguide of dimensions a × b. Give the expression for the cutoff frequencies of different
modes and explain its physical meaning.
See question 33.
Maxwell’s relaxation time in terms of the resistivity
tau = epsilon_0/sigma = epsilon_0 * rho
Farad unit in SI units
F = s / OMEGA = C^2 / J
Angular frequency (small omega)
2pic/lambda
