Simple Nature - Light and Matter

is the magnitude of the momentum vector, and U = U e + U m isthe sum of the energy of the electric and magnetic fields. We cannow demonstrate this without explicitly referring to relativity, andconnect it to the specific structure of a light wave.The energy density of a light wave is related to the magnitudesof the fields in a specific way — it depends on the squares of theirmagnitudes, E 2 and B 2 , which are the same as the dot productsE · E and B · B. We argued on page 582 that since energy is ascalar, the only possible expressions for the energy densities of thefields are dot products like these, multiplied by some constants. Thisis because the dot product is the only mathematically sensible wayof multiplying two vectors to get a scalar result. (Any other wayviolates the symmetry of space itself.)How does this relate to momentum? Well, we know that if wedouble the strengths of the fields in a light beam, it will have fourtimes the energy, because the energy depends on the square of thefields. But we then know that this quadruple-energy light beammust have quadruple the momentum as well. If there wasn’t thiskind of consistency between the momentum and the energy, then wecould violate conservation of momentum by combining light beamsor splitting them up. We therefore know that the momentum densityof a light beam must depend on a field multiplied by a field.Momentum, however, is a vector, and there is only one physicallymeaningful way of multiplying two vectors to get a vector result,which is the cross product (see page 912). The momentum densitycan therefore only depend on the cross products E × E, B × B, andE × B. But the first two of these are zero, since the cross productvanishes when there is a zero angle between the vectors. Thus themomentum per unit volume must equal E × B multiplied by someconstant,dp = (constant)E × B dvThis predicts something specific about the direction of propagationof a light wave: it must be along the line perpendicular to the electricand magnetic fields. We’ve already seen that this is correct, and alsothat the electric and magnetic fields are perpendicular to each other.Therefore this cross product has a magnitude|E × B| = |E||B| sin 90 ◦= |E||B|= |E|2c= c|B| 2 ,where in the last step the relation |E| = c|B| has been used.We now only need to find one physical example in order to fix theconstant of proportionality. Indeed, if we didn’t know relativity, itwould be possible to believe that the constant of proportionality waszero! The simplest example of which I know is as follows. Suppose a704 Chapter 11 Electromagnetism