670
14 Electromagnetism II
Substituting (2) in (1)
u B =
1
2
μ H · H =
1
2
μ H
2
=
B 2
2μ
(3)
In free space μ = μ 0 . Therefore in vacuum
u B =
B 2
2μ 0
14.44 u E =
1
2
ε 0 E
2 (energy density in E-field)
(1)
u B =
1
2
B 2
μ 0
(energy density in B-field)
(2)
The fields for the plane wave are
E = E m sin(kx − ωt)
(3)
B = B m sin(kx − ωt)
(4)
Substituting (3) in (1) and (4) in (2)
u E =
1
2
ε 0 E
2
m sin
2
(kx − ωt)
(5)
u B =
1
2
B 2
m
μ 0
sin
2
(kx − ωt)
(6)
Dividing (5) by (6)
u E
u B
=
ε 0 μ 0 E 2
m
B 2
m
(7)
But ε 0 μ 0 =
1
c 2 and E m = cB m
∴
u E
u B
= 1 or u E = u B
14.45 By prob. (14.41) the magnetic energy stored in a coaxial cable
U B =
μ 0 i 2 l
4π
ln(b/a)
(1)
where i is the current and l is the length of the cable. Further, its capacitance
is given by
C =
2πε 0 l
ln(b/a)
(2)
14 Electromagnetism II
Substituting (2) in (1)
u B =
1
2
μ H · H =
1
2
μ H
2
=
B 2
2μ
(3)
In free space μ = μ 0 . Therefore in vacuum
u B =
B 2
2μ 0
14.44 u E =
1
2
ε 0 E
2 (energy density in E-field)
(1)
u B =
1
2
B 2
μ 0
(energy density in B-field)
(2)
The fields for the plane wave are
E = E m sin(kx − ωt)
(3)
B = B m sin(kx − ωt)
(4)
Substituting (3) in (1) and (4) in (2)
u E =
1
2
ε 0 E
2
m sin
2
(kx − ωt)
(5)
u B =
1
2
B 2
m
μ 0
sin
2
(kx − ωt)
(6)
Dividing (5) by (6)
u E
u B
=
ε 0 μ 0 E 2
m
B 2
m
(7)
But ε 0 μ 0 =
1
c 2 and E m = cB m
∴
u E
u B
= 1 or u E = u B
14.45 By prob. (14.41) the magnetic energy stored in a coaxial cable
U B =
μ 0 i 2 l
4π
ln(b/a)
(1)
where i is the current and l is the length of the cable. Further, its capacitance
is given by
C =
2πε 0 l
ln(b/a)
(2)