PY522 Homework No. 1
Solutions
1. (Jackson 6.11)
A transverse plane wave is incident normally in vacuum on a perfectly absorbing flat
screen.
(a) From the law of conservation of linear momentum, show that the pressure (called
radiation pressure) exerted on the screen is equal to the field energy per unit volume in
the wave.
Let our wave travel in the x direction, with E in the y direction and
B in the z direction
E = E2 yˆ
B = B3 zˆ
From Jackson 6.121 we have
d (P field )α
d (Pmech )α
3
= ∑ ∫ d r ∂βTαβ −
dt
dt
β V
d
d 3 r (E × B )α
∫
dt V
β V
The left-hand-side is the force on the screen and we want the
pressure so we will take the force per volume and then integrate in
the x direction
d d (Pmech )α
d (E × B )α
=
∂
T
−
ε
∑α dV dt
∑
β αβ
0∑
dt
α ,β
α
= ∑ ∫ d 3 r ∂ βTαβ − ε 0
Now evaluate the stress tensor:
(
)
1


Tαβ = ε 0  Eα Eβ + c 2 Bα Bβ − E 2 + c 2 B 2 δαβ 
2


T has only diagonal components
[
[
]
]
]
ε0
2
2
E 2 + c2 B3
2
ε
2
2
T22 = 0 E2 − c 2 B3
2
ε
2
2
T33 = 0 − E 2 + c 2 B3
2
T11 = −
[
The force is only in the x direction because E and B do not depend
on y or z and the cross product (Poynting vector) is also in the x
direction.
d (Pmech )1
d
= ∫ d 3 r ∂1T11 − ε 0 ∫ d 3 r (E × B )1
dt
dt V
V
d (E 2 B3 )
 ε
2
2
= −∫ d 3 r ∂1 0 E 2 + c 2 B3 + ε 0
dt 
 2
V
[
]
The pressure is
0
∞
d d (Pmech )1
ε0
d (E2 B3 )
2
2
2
P = ∫ dx
=−
E 2 + c B3 + ε 0 ∫ dx
dV
dt
2
dt
−∞
0
Now we must take the time average. The last term contributes zero
because its time average is zero.
ε
ε
1
2
2
2
2
P = 0 E2 + c 2 B3 = 0 E2 +
B3 = u
2
2
2µ 0
[
[
]
]
(b) In the neighborhood of the earth the flux of electromagnetic energy from the sun is
approximately 1.4 kW/m2 . If an interplanetary "sailplane" had a sail of mass 1 g/ m2 of
area and negligible other weight, what would be its maximum acceleration in meters per
second squared due to the solar radiation pressure? How does this answer compare with
the acceleration due to the solar "wind" (corpuscular radiation)?
S 1.4 × 103 W/m 2
−6
2
P =u = =
≈
5
×
10
N/m
c
3 × 108 m/s
P A 5 × 10−6 N/m 2
−3
2
a=
=
−3
2 ≈ 5 × 10 m/s
m
10 kg / m
Compare to solar wind:
At http://sec.noaa.gov/ace/MAG_SWEPAM_7d.html
I found the following plot of the solar wind for the last 7 days
beginning 9/12/01:
From this plot we see that the solar wind (in the vicinity of the
earth) is made of protons traveling at
v ~ (4 − 6 )×105 m/s
with a density of
ρ ~ 106 − 107 /m 3
(
)
Thus, the solar wind flux is
N ≈ 5 ×105 m/s 3 ×107 /m 3 ≈ 2 ×1012 m −2s −1
(
)(
)
The corresponding acceleration on our sail plane is
(2 ×10
12
a≈
−3
)
m −2s −1 m p v
10 kg / m
2
≈
(2 ×10
12
)(
)(
)
m −2s −1 1.6 ×10 −27 kg 7 ×105 m/s
≈ 2 ×10−6 m/s 2
−3
2
10 kg / m
significantly smaller than the acceleration due to radiation.
2. (Jackson 6.16)
(a) Calculate the force in newtons acting on a Dirac monopole of the minimum magnetic
charge located a distance 0.05 nm from and in the median plane of a magnetic dipole
with dipole moment equal to one nuclear magneton (eh/2mp ).
The force on monopole in dipole field
F = gH =
gB
µ0
Dipole field,
B=
mµ0
2πr 3
where
2πh
n
e
eh
m=
2m p
g=
This gives
eh
 2πh  eh  1  h
F =
n
=
n


3
3
 e  2m p  2πr  r e 2m p
Numerically (for n = 1)
1.05 ×10−34 J ⋅ s 5.0 ×10−27 J/T
F=
≈ 2.6 ×10−11 N
3
−11
−19
5 × 10 m 1.6 ×10 C
(
)(
)(
(
)
)
(b) Compare the force in part a with atomic forces such as the direct electrostatic force
between charges (at the same separation), the spin-orbit force, the hyperfine interaction.
Comment on the question of binding of magnetic monopoles to nuclei with magnetic
moments. Assume the monopole mass is at least that of a proton.
Reference: D. Sivers, Phys. Rev. D2, 2048 (1970).
For the direct electrostatic force
ke2 (1.44 eV ⋅ nm ) 10−9 m / nm 1.6 × 10−19 J / eV
F= 2 =
≈ 9 × 10−8 N
2
r
5 × 10−11 m
(
(
)(
)
)
The force on the monopole is about 3000 times smaller than the
direct electrostatic force.
The spin orbit force is α 2 times smaller than the direct electrostatic
force, putting it about the same order-of-magnitude as the
monopole force. The hyperfine interaction (electron spin with
proton spin) is about 6 orders of magnitude smaller than direct
electrostatic or about 100 times smaller than monopole.