PROBLEM SET #1 ELEC 3909: ELECTROMAGNETIC WAVES
(1) Vector Analysis. Let A = α x̂ + 3ŷ − 2 ẑ and B = 4 x̂ + β ŷ + 8 ẑ.
(a) Find α and β is A and B are parallel.
(b) Determine the relationship between α and β if B is perpendicular to A.
Ans: β = −12, 4α + 3β = 16.
(2) Vector Operation. Let A = 4x2 e−y x̂ − 8xe−y ŷ. Determine ∇ × [∇(∇ · A)].
Ans: Zero.
(3) Gradients. The temperature in the auditorium is given by T = x2 + y 2 − z. A mosquito located at
(1,1,2) in the auditorium desired to fly in such a direction that it will get warm as soon as possible. In
what direction must it fly?
Ans: 2 x̂ + 2 ŷ − ẑ.
(4) Heat Flow. The heat flow vector H = k∇T , where T is the temprature and k is the thermal conductivity. Show that if T = 50 sin(πx/2) cosh(πy/2), then ∇ · H = 0.
(5) Vector Curls. A rigid body spins about a fixed axis with angular velocity ω. if u is the velocity of any
point in the body, show that ω = (∇ × u)/2.
(6) Gauss’s Law. Using Gauss’ law, derive the expression for the electric field intensity vector of an infinite
sheet of charge in free space.
Ans: σ/2ǫ0 .
(7) Application of Gauss’ Law. In a certain region, the electric field is given by E = 4xy x̂ + 2x2 ŷ +
ẑ Volts/m (x, y in meters). The medium is free-space.
(a) Calculate the charge density.
(b) From the result in a), find the total charge enclosed in a cube situated in the first co-ordinate octant
(x, y, z > 0), with one vertex at the co-ordinate origin, and the edges of length 1 m, parallel to the
co-ordinate axis.
(c) Confirm the validity of Gauss’ law in integral and the divergence theorem by evaluating the net
outward flux of E through the surface of the cube defined in b).
Ans: ρ = 4ǫ0 y, Q = 2ǫ0 .
(8) Gauss’ Law A spherical region of radius a has a total charge Q. If the charge is uniformly distributed,
apply Gauss’s law to find D = ǫE both inside and outside the sphere.
Ans: r < a, D = Qr/4πa3 r̂, r > a, D = Q/4πr2 r̂.
(9) Ampere’s Law In a certain region, the magnetic field is given by B = 4(z − 1)2 x̂ + 2x3 ŷ + xy ẑ mT
(x, y in meters). The medium is free-space.
(a) Calculate the current density.
(b) From the result in a), find the total current enclosed by a square contour lying in the x − y plane,
with the centre at the co-ordinate origin, and sides of length 2 m, parallel to the x− and y−axis.
(c) Confirm Ampere’s law in integral form and Stokes’ theorem by evaluating the net circulation of B
along the contour defined in b).
Ans: J = [x x̂ − [y − 8(z − 1)] ŷ + 6x2 ẑ]/µ, I = 6/µ A.
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PROBLEM SET #1 ELEC 3909: ELECTROMAGNETIC WAVES
(10) Faraday’s Law of Induction. A square loop of side a recedes with a uniform velocity u0 ŷ from an
infinitely long filament carrying current I along ẑ. Assuming that the loop is located a distance ρ = ρ0
at t = 0, show that the emf induced in the loop at t > 0 is
Vemf =
u 0 a 2 µ0 I
.
2πρ(ρ + a)
(1)
(11) Displacement Current. In a dielectric, (σ = 10−4 S/m, µr = 1 and ǫr = 4.5), the conduction current
density is given as Jc = 0.4 cos(2π × 108 t) A/m. Determine the displacement current density.
Ans: Jd = −100 sin(2π × 103 t) A/m2 .
(12) Fields in a Capacitor. A parallel-plate capacitor of plate area S is connected to a time-harmonic
generator operating at a low frequency f . The capacitor is filled with a two-layer perfect dielectric. The
thickness of the first layer is d1 with permittivity ǫ1 , while these parameters are d2 and ǫ2 for the second
dielectric. The amplitude of the conduction current intensity is the capacitor terminals is I0 . Neglecting the fringing effects, find (a) the amplitude of the displacement current density vector in each of the
dielectric layers, (b) the amplitude of the electric field intensity vector in each of the layers, and (c) the
amplitude of the voltage across the capacitor. Further, If the plates of the capacitor are circular, a in
radius, and so is the cross-section of each of the two dielectric layers, and the voltage between the plates
is given by v(t) = V0 sin(ωt), compute the magnetic field intensity vector at an arbitrary point in the
dielectric. In particular what is the magnetic field at the interface?
Ans: a) Jd,1 = Jd,2 = I0 /S, b) E01 = I0 /2πǫf S, E02 = I0 /2πǫ2 S, c) V0 = (ǫ2 d1 + ǫ1 d2 )I0 /2πǫ1 ǫ2 f S,
H(r, t) = 0.5ǫ1 ǫ2 ωV0 r/(ǫ2 d1 + ǫ1 d2 ).
(13) E & H-fields. In a certain material, σ = 0, µ = µ0 and ǫ = 81ǫ0 . The magnetic field intensity in this
material is H = 10 cos(2π × 109 t + βx) ẑ A/m. Determine E and β.
Ans: β = 60π rad/m and E = −148 cos(ωt + βx) ŷ V/m.
(14) Plane-wave fields. In a certain region for which σ = 0, µ = 2µ0 and ǫ = 10ǫ0 , Jd = 60 sin(109 t −
βz) x̂ mA/m2 . Find D, H and β.
Ans: D = −60 × 10−12 cos(109 t − βz) x̂ C/m2 , H = −[60β × 10−21 /ǫµ] cos(109 t − βz) ŷ A/m, β =
14.907 rad/m.
(15) Maxwell’s Equations. Starting from two curl Maxwell’s equations for the time-varying electromagnetic fields and the continuity equations, derive the two divergence Maxwell’s equations. Repeat the same
derivation, but using the integral form, i.e. obtain the two flux general Maxwell’s equations combining
the two circulation equations and the continuity equations.
(16) Uniform Plane-Wave. A TEM plane-wave propagating in vacuum has a electric field given in phasor
form by E(y, t) = E0 ej(ωt+ky) ẑ, where k = ω/c, with c, the speed of light in vacuum and E0 is a real
constant.
(a) In what direction is the wave propagating?
(b) Write the expression for the magnetic field H associated with this TEM wave. Lets H0 be the
amplitude of the magnetic field in your expression.
(c) Show that E from eq. (1) and your expression for H from part b) satisfy the Maxwell’s Faraday law,
provided a certain relationship holds between E0 and H0 . Find this relationship.