14.2 Problems
647
14.75 Explain the difference between the vectors B and H in the theory of magnetism. Derive the expression B = μ 0 (H + M). Indicate briefly how B
depends on it in the case of (a) paramagnetics and (b) ferromagnetics.
[University of Durham 1962]
14.76 Show that Gauss’ and Ampere’s laws in free space, subject to the Lorentz
condition, can be expressed in the usual notation as
−∇
2
φ +
1
c 2
∂ 2 φ
∂t 2 =
ρ
ε 0
and −∇
2 A +
1
c 2
∂ 2 A
∂t 2 = μ 0 J
respectively.
[The University of Aberystwyth, Wales 2005]
14.77 Using Faraday’s law, ∇ × E = −∂ B/∂t, for the propagation of electromagnetic waves travelling along the z-axis, show that
E y = Z 0 H x
where Z 0 =
√
μ 0 /ε 0 = 376.6 , is the wave impedance of free space.
14.78 (a) The electric field of an electromagnetic wave propagating in free space
is described by the equation
E(z, t) = E 0 [ ˆ
xsin(kz − ωt) + ˆ
ycos(kz − ωt)]
where ˆ
x and ˆ
y are unit vectors in the x- and y-direction, respectively.
What is this wave’s direction of propagation? What is the polarization of
the wave?
(b) State and prove the boundary conditions satisfied by the magnetic intensity H and the magnetic field B at the boundary between two media with
different magnetic properties.
(c) Show that
tan θ 1
tan θ 2
=
μ 1
μ 2
where θ 1 and θ 2 are incident and refraction angles.
14.79 A plane wave is normally incident on a dielectric discontinuity. Use appropriate boundary conditions to calculate R, the reflectance, and T , the transmittance, and show that T + R = 1.
14.80 An electromagnetic wave, propagating and linearly polarized in the xz-plane,
is incident onto an interface between two non-conducting media as shown
in Fig. 14.10. The electric fields and propagation vectors of the incident,
647
14.75 Explain the difference between the vectors B and H in the theory of magnetism. Derive the expression B = μ 0 (H + M). Indicate briefly how B
depends on it in the case of (a) paramagnetics and (b) ferromagnetics.
[University of Durham 1962]
14.76 Show that Gauss’ and Ampere’s laws in free space, subject to the Lorentz
condition, can be expressed in the usual notation as
−∇
2
φ +
1
c 2
∂ 2 φ
∂t 2 =
ρ
ε 0
and −∇
2 A +
1
c 2
∂ 2 A
∂t 2 = μ 0 J
respectively.
[The University of Aberystwyth, Wales 2005]
14.77 Using Faraday’s law, ∇ × E = −∂ B/∂t, for the propagation of electromagnetic waves travelling along the z-axis, show that
E y = Z 0 H x
where Z 0 =
√
μ 0 /ε 0 = 376.6 , is the wave impedance of free space.
14.78 (a) The electric field of an electromagnetic wave propagating in free space
is described by the equation
E(z, t) = E 0 [ ˆ
xsin(kz − ωt) + ˆ
ycos(kz − ωt)]
where ˆ
x and ˆ
y are unit vectors in the x- and y-direction, respectively.
What is this wave’s direction of propagation? What is the polarization of
the wave?
(b) State and prove the boundary conditions satisfied by the magnetic intensity H and the magnetic field B at the boundary between two media with
different magnetic properties.
(c) Show that
tan θ 1
tan θ 2
=
μ 1
μ 2
where θ 1 and θ 2 are incident and refraction angles.
14.79 A plane wave is normally incident on a dielectric discontinuity. Use appropriate boundary conditions to calculate R, the reflectance, and T , the transmittance, and show that T + R = 1.
14.80 An electromagnetic wave, propagating and linearly polarized in the xz-plane,
is incident onto an interface between two non-conducting media as shown
in Fig. 14.10. The electric fields and propagation vectors of the incident,