Lec12 - units then in a frequently more convenient system...

units, then in a frequently more convenient system of units.
1
4 π
0
= k = 9 × 10
9
N · m 2
C 2
Therefore
P =
2
3
· 9 × 10
9
Nm
2
C
2
×
(10 × 10 - 9 C) 2 × 5 4 ×
m 4
s 4
1
2
· m
2
(3 × 10
8
m
s
)
3
= 14 , 000 × 10
- 33
W = 1 . 4 × 10
- 29
W
Alternatively,
e 2
4 π
0
= 1 . 44 eV · nm
and the given charge is q/e = 6 . 25 × 10
10
, e being the electron charge. Substituting in P we
get P = 8 . 68 × 10
- 11
eV s
- 1
which equals 1 . 4 × 10
- 29
W. Notice that is these units e
2
/ (4 π
0
)
is at order unity. It is therefore particularly convenient to use in physical systems in which
the energy scale is at the order of the eV and the length scale is at the order of the nm. This
length scale is about atomic size, and the eV is the energy scale for atomic transitions. For
this reason, this alternative system of units is very convenient to use in atomic physics.
Energy density in an electromagnetic wave
In the case of mechanical waves in a string, we found that the radiated power (which is the
energy density per unit time) scales with the square of the waves amplitude. We therefore
expect that in an electromagnetic wave the energy density also scales with the square of E
and B . Indeed, in a general electromagnetic field (i.e., not necessarily that of a wave, with
the electric and magnetic waves are possibly independent fields), the energy density is given
by E =
1
2
0
E
2
+
1
2
1
μ
0
B
2
. The energy density in the electric part is most easily found by
looking at the energy in a capacitor, and the student is referred to such a derivation.
This expression is correct for any electromagnetic field, and in particular for that of an
electromagnetic wave. The di ff erence is that now E and B are no longer independent,
but rather must satisfy E = cB , as we found above. Also, c 2 = 1 / (
0
μ
0
). Put together,
E =
1
2
0
( E
2
+ c
2
B
2
). For an electromagnetic wave E = cB , so that the energy densities
stored in the E and B fields are equal, despite that fact that E appears to be much larger
than B . By how much larger? By a factor of c , and c is very large. Or is it?