14.2 Problems
643
14.40 Using Maxwell’s equations, show that in a conducting medium, the wave
equation can be written as
∇
2 E = μσ
∂E
∂t
+ με
∂ 2 E
∂t 2
and a similar expression for the B-field.
[University of Aberystwyth, Wales 2005]
14.41 Let l be the length of the coaxial cylindrical capacitor, a the radius of the
central wire and b the radius of the tube. The conductors are connected to a
battery of V volts and a current I is passed. Calculate (a) the capacitance per
unit length of the cable, (b) the inductance per unit length.
14.42 Consider a coaxial cable with radius a for the central wire and radius b for
the tube connected to a resistance R and battery of emf ξ . Calculate (a) E;
(b) B; and (c) S for the region a < r < b.
14.43 The general expression for magnetic energy density has the form u B =
1 /2 B.H. Show that in the vacuum the above expression is reduced to u B =
B 2
2μ 0
.
14.44 Show that at any point in the electromagnetic field the energy density stored
in the electric field is equal to that stored in the magnetic field.
14.45 A current I is passed through a coaxial cable with inner radius a and outer
radius b. The cable can function both as a capacitor and as an inductor. If the
stored electric and magnetic energy is equal then show that the resistance R
is approximately given by
R =
377
2π
ln
b
a
14.46 A proton of kinetic energy 20 MeV circulates in a cyclotron with 0.5 m
radius. Calculate its energy loss to radiation per orbit and show that it is
negligible.
14.47 A 40-W point source radiates equally in all directions. Find the amplitude of
the E-field at a distance of 1 m.
14.48 A laser beam has a cross-sectional area of 4.0 mm 2 and a power of 1.2 mW.
Find (a) intensity I , (b) E 0 , (c)B 0 .
14.49 A laser emits a 1-mm diameter highly collimated beam at a power level
314 mW. Calculate the irradiance.
14.50 Beginning with the expression for the Poynting vector show that the timeaveraged power per unit area carried by a plane electromagnetic wave in free
space is given by
643
14.40 Using Maxwell’s equations, show that in a conducting medium, the wave
equation can be written as
∇
2 E = μσ
∂E
∂t
+ με
∂ 2 E
∂t 2
and a similar expression for the B-field.
[University of Aberystwyth, Wales 2005]
14.41 Let l be the length of the coaxial cylindrical capacitor, a the radius of the
central wire and b the radius of the tube. The conductors are connected to a
battery of V volts and a current I is passed. Calculate (a) the capacitance per
unit length of the cable, (b) the inductance per unit length.
14.42 Consider a coaxial cable with radius a for the central wire and radius b for
the tube connected to a resistance R and battery of emf ξ . Calculate (a) E;
(b) B; and (c) S for the region a < r < b.
14.43 The general expression for magnetic energy density has the form u B =
1 /2 B.H. Show that in the vacuum the above expression is reduced to u B =
B 2
2μ 0
.
14.44 Show that at any point in the electromagnetic field the energy density stored
in the electric field is equal to that stored in the magnetic field.
14.45 A current I is passed through a coaxial cable with inner radius a and outer
radius b. The cable can function both as a capacitor and as an inductor. If the
stored electric and magnetic energy is equal then show that the resistance R
is approximately given by
R =
377
2π
ln
b
a
14.46 A proton of kinetic energy 20 MeV circulates in a cyclotron with 0.5 m
radius. Calculate its energy loss to radiation per orbit and show that it is
negligible.
14.47 A 40-W point source radiates equally in all directions. Find the amplitude of
the E-field at a distance of 1 m.
14.48 A laser beam has a cross-sectional area of 4.0 mm 2 and a power of 1.2 mW.
Find (a) intensity I , (b) E 0 , (c)B 0 .
14.49 A laser emits a 1-mm diameter highly collimated beam at a power level
314 mW. Calculate the irradiance.
14.50 Beginning with the expression for the Poynting vector show that the timeaveraged power per unit area carried by a plane electromagnetic wave in free
space is given by