Simple Nature - Light and Matter

piece of wire of length l is bathed in electromagnetic waves comingin sideways, and let’s say for convenience that this is a radio wave,with a wavelength that is large compared to l, so that the fieldsdon’t change significantly across the length of the wire. Let’s say theelectric field of the wave happens to be aligned with the wire. Thenthere is an emf between the ends of the wire which equals El, andsince the wire is small compared to the wavelength, we can pretendthat the field is uniform, not curly, in which case voltage is a welldefinedconcept, and this is equivalent to a voltage difference ∆V =El between the ends of the wire. The wire obeys Ohm’s law, anda current flows in response to the wave. 17 Equating the expressionsdU/ dt and I∆V for the power dissipated by ohmic heating, we havedU = IEl dtp / A classical calculation ofthe momentum of a light wave.An antenna of length l is bathedin an electromagnetic wave.The black arrows represent theelectric field, the white circles themagnetic field coming out of thepage. The wave is traveling to theright.for the energy the wave transfers to the wire in a time interval dt.Note that although some electrons have been set in motion in thewire, we haven’t yet seen any momentum transfer, since the protonsare experiencing the same amount of electric force in the oppositedirection. However, the electromagnetic wave also has a magneticfield, and a magnetic field transfers momentum to (exerts a forceon) a current. This is only a force on the electrons, because they’rewhat make the current. The magnitude of this force equals lIB(homework problem 6), and using the definition of force, dp/ dt, wefind for the magnitude of the momentum transferred:dp = lIB dtWe now know both the amount of energy and the amount ofmomentum that the wave has lost by interacting with the wire.Dividing these two equations, we finddpdU = B E= 1 cwhich is what we expected based on relativity. This can now berestated in the form dp = (constant)E × B dv (homework problem40).Note that although the equations p = U/c and dp = (constant)E×B dv are consistent with each other for a sine wave, they are notconsistent with each other in general. The relativistic argumentleading up to p = U/c assumed that we were only talking about17 This current will soon come to a grinding halt, because we don’t have acomplete circuit, but let’s say we’re talking about the first picosecond duringwhich the radio wave encounters the wire. This is why real radio antennas arenot very short compared to a wavelength!,q / A simplified drawing ofthe 1903 experiment by Nicholsand Hull that verified the predictedmomentum of light waves.Two circular mirrors were hungfrom a fine quartz fiber, insidean evacuated bell jar. A 150mW beam of light was shoneon one of the mirrors for 6 s,producing a tiny rotation, whichwas measurable by an opticallever (not shown). The force waswithin 0.6% of the theoreticallypredicted value (problem 11 onp. 439) of 0.001 µN. For comparison,a short clipping of a singlehuman hair weighs ∼ 1 µN.Section 11.6 Maxwell’s Equations 705