14.3 Solutions
651
14.100 Calculate the number of states of electromagnetic radiation between 5000
and 6000 Å in wavelength using periodic boundary conditions in a cubical
region 0.5 cm on a side.
[University of Manchester 1972]
14.101 Consider a car entering a tunnel of dimensions 15 m wide and 4 m high.
Assuming the walls are good conductors, can AM radio waves (530–
1600 kHz) propagate in the tunnel?
[The University of Aberystwyth, Wales 2006]
14.102 Calculate the least cut-off frequency for TE mn waves for a rectangular
waveguide of dimensions 5 cm × 4 cm.
14.103 Calculate how the wave and group velocities of the TE 01 wave in a rectangular waveguide with a = 1 cm and b = 2 cm vary with frequency.
[The University of Wales, Aberystwyth 2004]
14.104 Consider a rectangular waveguide of dimensions x = a and y = b, the
TM mn wave travelling in the z-direction which is the axis of the guide.
Given that the z-component E z satisfies the equation
∂ z
∂ x z +
∂ z
∂ y z
E z = (k
2
− ω
2
με)E z
obtain (a) the solution for E z and (b) the cut-off frequency.
14.105 Consider a rectangular waveguide of dimensions x = a and y = b, the
wave travelling along the z-direction, the axis of the guide. Given that the
z-component H z satisfies the equation
∂ 2
∂ x 2 +
∂ 2
∂ y 2
H z = (k
2
− ω
2
με)H z
(a) obtain the solution for H z . (b) Obtain the cut-off frequency. (c) What are
the similarities and differences between TM mn mode and TE mn mode?
14.3 Solutions
14.3.1 The RLC Circuits
14.1 (a) Reactance of capacitor
X c =
1
2π fc
=
1
2π × (1000/2π) × 2.5 × 10 −6 = 400
Impedance of the circuit
Z =
R 2 + X 2
c =
(300) 2 + (400) 2 = 500