206
7 Light in Biology and Medicine
In addition, Maxwell’s equations lead to the following expression for the work
done by an electric field (magnetic fields can do no work) in moving a set of charges
in a small volume, per unit time per unit volume:
J · E = −
∂
∂t
1
8π
E
2
+ B
2
− c ∇ ·
1
4π
E × B
.
(7.3)
The first term on the right is the loss of field energy per unit time per unit volume,
and the second term is the flux of field energy flowing into the volume per unit
volume. That flux of energy through the closed boundary of the volume turns out
to be the speed of the wave, c, times the momentum flow through the boundary. In
this way, Maxwell knew that the general expression for the energy density, u, of an
electromagnetic field is
u =
1
8π
E
2
+ B
2
(7.4)
and that an electromagnetic field can carry a momentum whose flux density, 1 is
given by the ‘Poynting vector.’
P =
1
4π
E × B .
(7.5)
This momentum flux density points in the direction of E × B, showing the direction
that an electromagnetic wave travels. For all electromagnetic waves, Maxwell’s
equation (6.2) makes B perpendicular to E. (See Fig. 7.1.)
It follows that the intensity of the wave arriving at a surface, i.e. the energy
arriving per unit time per unit area, is given by
I =
c
4π
|E|
2
= cu ,
(7.6)
and the momentum arriving at a surface of area A per unit time will be
dp/dt = I A/c .
(7.7)
If this momentum is transferred to the surface, then this expression also gives the
force of the wave on the surface.
By calculating the speed of electromagnetic waves using the measured values of
the electric permittivity and magnetic permeability of air (which are close to their
vacuum values), Maxwell in 1864 found 3.1074 × 10 8 m/s, a number close to the
1 The momentum flux density is defined as the momentum arriving at a given area per unit time per
unit area. The momentum flux density multiplied by c is the energy flux density.
7 Light in Biology and Medicine
In addition, Maxwell’s equations lead to the following expression for the work
done by an electric field (magnetic fields can do no work) in moving a set of charges
in a small volume, per unit time per unit volume:
J · E = −
∂
∂t
1
8π
E
2
+ B
2
− c ∇ ·
1
4π
E × B
.
(7.3)
The first term on the right is the loss of field energy per unit time per unit volume,
and the second term is the flux of field energy flowing into the volume per unit
volume. That flux of energy through the closed boundary of the volume turns out
to be the speed of the wave, c, times the momentum flow through the boundary. In
this way, Maxwell knew that the general expression for the energy density, u, of an
electromagnetic field is
u =
1
8π
E
2
+ B
2
(7.4)
and that an electromagnetic field can carry a momentum whose flux density, 1 is
given by the ‘Poynting vector.’
P =
1
4π
E × B .
(7.5)
This momentum flux density points in the direction of E × B, showing the direction
that an electromagnetic wave travels. For all electromagnetic waves, Maxwell’s
equation (6.2) makes B perpendicular to E. (See Fig. 7.1.)
It follows that the intensity of the wave arriving at a surface, i.e. the energy
arriving per unit time per unit area, is given by
I =
c
4π
|E|
2
= cu ,
(7.6)
and the momentum arriving at a surface of area A per unit time will be
dp/dt = I A/c .
(7.7)
If this momentum is transferred to the surface, then this expression also gives the
force of the wave on the surface.
By calculating the speed of electromagnetic waves using the measured values of
the electric permittivity and magnetic permeability of air (which are close to their
vacuum values), Maxwell in 1864 found 3.1074 × 10 8 m/s, a number close to the
1 The momentum flux density is defined as the momentum arriving at a given area per unit time per
unit area. The momentum flux density multiplied by c is the energy flux density.