642
14 Electromagnetism II
voltage on the condenser to fall to 1/e of its initial value; (b) the percentage
error which would have been introduced if thermal effects had been ignored.
[University of Manchester 1958]
14.32 Show that the fractional half-width of the resonance curve of an RLC circuit
is given by
ω
ω
=
√
3
Q
where Q is the quality factor given by Q = ω L/R.
14.2.2 Maxwell’s Equations, Electromagnetic Waves,
Poynting Vector
14.33 A plane em wave E = 100 cos (6×10 8 t + 4x) V/m propagates in a medium.
What is the dielectric constant of the medium?
[Indian Administrative Services]
14.34 An infinite wire with charge density λ and current I is at rest in the Lorentz
frame S. Show that the speed of reference frame S where the electric field
is zero, i.e. that frame in which one observes pure magnetic field, is given by
v =
λc 2
I .
14.35 Show that for a magnetic field B the wave equation has the form ∇ 2 B =
μ 0 ε 0
∂ 2 B
∂t 2
14.36 Use Maxwell’s equation to show that ∇.
j +
1
ε 0
∂E
∂t
= 0.
14.37 The free-space wave equation for a medium without absorption is
∇
2 E − μ 0 ε 0
∂ 2 E
∂t 2 = 0
Show that this equation predicts that electromagnetic waves are propagated
with velocity of light given by c = 1/
√ μ 0 ε 0 .
14.38 An electromagnetic wave of wavelength 530 nm is incident onto a sheet of
aluminium with resistivity ρ = 26.5 × 10 −9 m. Estimate the depth that the
wave penetrates into the aluminium. The expression for the skin depth, δ, is
δ =
√
2/μ 0 σ ω.
[University of Manchester 2008]
14.39 Consider an electromagnetic wave with its E-field in the y-direction. Apply
the relation
∂ E y
∂ x = −
∂ B z
∂t to the harmonic wave
E = E 0 cos (kx − ωt), B = B 0 cos (kx − ωt)
to show that E 0 = cB 0 .
14 Electromagnetism II
voltage on the condenser to fall to 1/e of its initial value; (b) the percentage
error which would have been introduced if thermal effects had been ignored.
[University of Manchester 1958]
14.32 Show that the fractional half-width of the resonance curve of an RLC circuit
is given by
ω
ω
=
√
3
Q
where Q is the quality factor given by Q = ω L/R.
14.2.2 Maxwell’s Equations, Electromagnetic Waves,
Poynting Vector
14.33 A plane em wave E = 100 cos (6×10 8 t + 4x) V/m propagates in a medium.
What is the dielectric constant of the medium?
[Indian Administrative Services]
14.34 An infinite wire with charge density λ and current I is at rest in the Lorentz
frame S. Show that the speed of reference frame S where the electric field
is zero, i.e. that frame in which one observes pure magnetic field, is given by
v =
λc 2
I .
14.35 Show that for a magnetic field B the wave equation has the form ∇ 2 B =
μ 0 ε 0
∂ 2 B
∂t 2
14.36 Use Maxwell’s equation to show that ∇.
j +
1
ε 0
∂E
∂t
= 0.
14.37 The free-space wave equation for a medium without absorption is
∇
2 E − μ 0 ε 0
∂ 2 E
∂t 2 = 0
Show that this equation predicts that electromagnetic waves are propagated
with velocity of light given by c = 1/
√ μ 0 ε 0 .
14.38 An electromagnetic wave of wavelength 530 nm is incident onto a sheet of
aluminium with resistivity ρ = 26.5 × 10 −9 m. Estimate the depth that the
wave penetrates into the aluminium. The expression for the skin depth, δ, is
δ =
√
2/μ 0 σ ω.
[University of Manchester 2008]
14.39 Consider an electromagnetic wave with its E-field in the y-direction. Apply
the relation
∂ E y
∂ x = −
∂ B z
∂t to the harmonic wave
E = E 0 cos (kx − ωt), B = B 0 cos (kx − ωt)
to show that E 0 = cB 0 .