W9_parallel_resonance.eps

344 Week 10: Maxwell’s Equations and Light

• The electromagnetic field also carriesmomentum, solving the dilemma of the “miss-
  ing momentum” left over from our consideration of the magnetic force and the failure
  of Newton’s third law. The field momentum is rather difficult to derive in asim-
  pleway, but it cansomewhatbe understood by assuming that the fieldelectrically
  polarizes atoms that it sweeps over in such a way that it exerts amagneticforce
  along the direction of motion of the electromagnetic wave. We’ll explore this with
  a problem later. The momentum density of the electromagnetic field is:

|pf|=

U

c

(798)

and we can consider the net momentum transported per unit area per unit time by

the electromagnetic field perpendicular to a surfaceAto be:

Pr=

Ithru A

c

(799)

This quantity is called theradiation pressureand it is partially responsible for the

solar wind, created as sunlight pushes gas molecules away from the sun. Light

“sails” have also been proposed as a propulsion for getting around inside the solar

system without rocket fuel. We will explore both of these ideas with homework

problems.

To use radiation pressure properly, one has to compute the forceit exerts on a sur-

face. This force will depend on certain things, such as whether or not the radiation

is perfectly absorbed or perfectly reflected and (eventually) therelative velocity of

source and target (as the incident and reflected waves can be doppler shifted, affect-

ing the momentum transfer). In the simplest cases (perfect absorption or reflection)

the force is best computed by using an expression such as:

FS=

1

c

∫

A

S~·nˆdA (800)

that is, the flux of the Poynting vector yields the power transferred to a (perfectly

absorbing) surface, and 1/cof the power is the effective force exerted along the line

of the original Poynting vector. If the radiation is reflected, one has to construct

a such quantity evaluated (with the same power) with respect to the direction of

the angle of reflection, and vector sum the forces. In the simplestcase of normal

absorption or reflection:

FS=

SA

c

(801)

or

FS=^2 SA

c

(802)

respectively.

• Electromagnetic radiation is produced when electrical chargesaccelerate(this fol-
  lows from construction theinhomogeneous wave equationsfor the electromagnetic
  fields directly from Maxwell’s equations, where moving charge and current terms
  become the sources of the time varying fields). In fact, if one works very hard in a
  graduate Electrodynamics class (as shown in my online book, for example, or in J.
  D. Jackson’sClassical Electrodynamics) one can show that thepower cross-section
  of a single chargeqmoving along the (say)z-axis is:

dP

dΩ

= q

2

16 π^2 ǫ 0

1

c^3

∣∣

∣∣d

(^2) z
dt^2

∣∣

∣∣

2

sin^2 (θ) (803)

The power cross section is the amount of power per unit solid angle (dΩ) radiated

away from theaccelerating charge. The actual power then drops off like 1/r^2 in this

direction.