672
14 Electromagnetism II
I =
P
4πr 2 =
c ε 0 E 2
0
2
∴ E 0 =
P
2π r 2 cε 0
1/2
=
40
2π(1.0) 2 (3 × 10 8 × 8.85 × 10 −12 )
1/2
= 49 V/m
14.48 (a) I 0 =
P
A
=
1.2 × 10 −3
4 × 10 −6 = 300 W/m 2
(b) I 0 =
cε 0 E 2
0
2
→ E 0 =
2I 0
cε 0
1/2
=
2 × 300
3 × 10 8 × 8.85 × 10 −12
1/2
= 475 V/m
(c) B 0 =
E 0
c
=
475
3 × 10 8 = 1.58 × 10 −6 T = 1.58 μT
14.49 I =
P
A
=
P
πr 2 =
314 × 10 −3
3.14 × (0.5 × 10 −3 ) 2 = 4 × 10
5 W/m
2
14.50 The average value of the Poynting vector < s > over a period of oscillation
of the electromagnetic wave is known as the radiant flux density, and if the
energy is incident on a surface it is called irradiance.
E = E 0 cos(kx − ωt), B = B 0 cos(kx − ωt)
∴ S = c
2
ε 0 E × B = c
2
ε 0 E 0 × B 0 cos
2
(kx − ωt)
∴ S = c
2
ε 0 |E 0 × B 0 |
cos
2
(kx − ωt)
But
cos
2
(kx − ωt)
=
1
T
T
0
cos
2
(kx − ωt)dt =
1
2
Further E⊥B and E 0 = cB 0 .
∴ < s >= I =
1
2
cε 0 E
2
0 =
1
2
E 2
0
μ 0 c
14.51 E z = 50 sin
4π × 10
14
t −
x
3 × 10 8
I =
cε 0 E 2
0
2
=
1
2
(3 × 10
8
)(8.85 × 10
−12
)(50
2
) = 0.066 W/m
2
14.52 |E × H| =
|E × B|
μ 0
=
E
2
μ 0 c
(∵ B = E/c)
E × H will be in the direction of S = (E × B)/μ 0 , which is the direction of
propagation.