646
14 Electromagnetism II
k
2
= μ 0 ε 0 ε r ω
2
+ iμ 0 σ N ω
where k is the wave vector and ω is the angular frequency.
[University of Durham 2006]
14.69 Show that the general identity
∇ × (∇ × E) = −∇
2 E + ∇(∇.E)
is true for the specific vector field F = x 2 z 3 ˆ
i.
14.70 Use the Poynting vector to determine the power flow in a coaxial cable by
a DC current I when voltage V is applied. Neglect the resistance of the
conductors. How are the results affected if this assumption is not made?
14.71 A wire with radius a and of conductivity σ E carries a constant, uniformly
distributed current I in the z-direction. Apply Poynting’s theorem to show
that power dissipated in the wire is given by the familiar expression I 2 R for
Joule’s heat.
14.72 A super-conductor is a material which offers no DC resistance and satisfies
the equation
B = −
m e
ne 2 ∇ × J
where n is the number of conduction electrons per unit volume, B is the
magnetic field and J is the current density. Using Maxwell’s equations, show
that the equation for a superconductor leads to the relation
∇
2 B =
μ 0 ne 2
m e
B
[University of Durham 2004]
14.73 An oscillating voltage of high frequency is applied to a load by means of
copper wire of radius 1 mm. Given that the skin depth is 6.6 × 10 −5 m for
this frequency, what is the high-frequency resistance per unit length of the
wire in terms of its direct current resistance per unit length?
[University of Durham 1966]
14.74 Use Stokes’ theorem to derive the expression
Curl E = −
∂B
∂t
[University of New Castle upon Tyne 1964]
14 Electromagnetism II
k
2
= μ 0 ε 0 ε r ω
2
+ iμ 0 σ N ω
where k is the wave vector and ω is the angular frequency.
[University of Durham 2006]
14.69 Show that the general identity
∇ × (∇ × E) = −∇
2 E + ∇(∇.E)
is true for the specific vector field F = x 2 z 3 ˆ
i.
14.70 Use the Poynting vector to determine the power flow in a coaxial cable by
a DC current I when voltage V is applied. Neglect the resistance of the
conductors. How are the results affected if this assumption is not made?
14.71 A wire with radius a and of conductivity σ E carries a constant, uniformly
distributed current I in the z-direction. Apply Poynting’s theorem to show
that power dissipated in the wire is given by the familiar expression I 2 R for
Joule’s heat.
14.72 A super-conductor is a material which offers no DC resistance and satisfies
the equation
B = −
m e
ne 2 ∇ × J
where n is the number of conduction electrons per unit volume, B is the
magnetic field and J is the current density. Using Maxwell’s equations, show
that the equation for a superconductor leads to the relation
∇
2 B =
μ 0 ne 2
m e
B
[University of Durham 2004]
14.73 An oscillating voltage of high frequency is applied to a load by means of
copper wire of radius 1 mm. Given that the skin depth is 6.6 × 10 −5 m for
this frequency, what is the high-frequency resistance per unit length of the
wire in terms of its direct current resistance per unit length?
[University of Durham 1966]
14.74 Use Stokes’ theorem to derive the expression
Curl E = −
∂B
∂t
[University of New Castle upon Tyne 1964]