Problem Set -7
Physics of Waves - PH11003 (Autumn 2024-25)
Maxwell equation, EM Waves
October 21, 2024
1. Curl and divergence
(a) Let F1 = x2 k̂ and F2 = (x2 y 2 + x)î (2xy + y)ĵ Calculate the divergence and curl of
F1 and F2 .
(b) Which one can be written as the gradient of a scalar? Find a scalar potential that does
the job.
(c) Which one can be written as the curl of a vector? Find a suitable vector potential.
2. Charge and current density
Electric and magnetic field in a region is given by
h ⇣ z ⌘i
~ = E0 cos ! t
E î
h ⇣ C ⌘i
~ = B0 cos ! t z
B ĵ
C
Calculate divergence and curl of E and B in the region and predict the charge and current
density in the region.
3. Poynting vector, radiation pressure
~ t) = E0 cos (kz
The electric field of an electromagnetic wave is given by E(z, !t)(î + ĵ)
(a) What is the maximum amplitude of the electric field?
~
(b) Compute the corresponding magnetic field B.
~
(c) Find the Poynting vector S.
(d) What is the radiation pressure if the wave is incident normally on a surface and is per-
fectly reflected?
4. Flux density
Suppose that the intensity of the sunlight falling on the ground on a particular day is
140W/m2 . What are the peak values of electric and magnetic fields associated with the
incident radiation?
(b) A linearly polarised harmonic plane wave with a scalar amplitude of 10V /m is propagat-
ing along a line in the x y plane at 45 degrees to the x axis, with the x y plane as the
plane of vibration. Write down a vector expression for the wave, assuming kx and ky are
both positive. Calculate the flux density taking the wave to be in vacuum.
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5. If the electric ~
⇣ ⌘field strength E of an electromagnetic wave in free space is given by E =
z
2 cos ! t U0 ŷ V/m, find the magnetic field B.
6. (a) Show that Maxwell’s equations with magnetic charge are invariant under the duality
transformation,
~0 = E
E ~ cos ↵ + cB
~ sin ↵
~ 0 = cB
cB ~ cos ↵ ~ sin ↵
E
cqe0 = cqe cos ↵ + qm sin ↵
0
qm = qm cos ↵ cqe sin ↵
where ↵ is an arbitrary rotation angle in the ”E/B - space”.
(b) Show that the force law is also invariant under this transformation.
✓ ◆
~ ~ ~ ~ 1 ~
F = qe (E + ~v ⇥ B) + qm B ~v ⇥ E
c2
7. Write down the electric and magnetic fields for a monochromatic plane wave of amplitude
E0 , frequency !, and phase angle zero that is (a) travelling in the negative x-direction and
polarized in z-direction; (b) travelling the direction from origin to the point (1,1,1), with
polarization parallel to xz plane. In each case, sketch the wave and give explicit Cartesian
components of ~k and n̂.
8. Suppose
~ ✓, , t) = A sin ✓ cos(kr
E(r, !t)
1
sin(kr !t) ˆ
r kr
with !/k = c. This is the simplest possible spherical wave. for notational convenience, let
(kr = !t) = u in your calculations.
~ obeys all four of Maxwell’s equations in a vacuum and finds the associated
(a) Show that E
magnetic field.
~ over a full cycle to get the intensity vector
(b) Calculate the Poynting Vector. Average S
~
I.
~ over a spherical surface determine the local power radiated.
(c) Integrate I~ · da
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