Simple Nature - Light and Matter

second, the zero-point is located at x = −(1 s)v. The distance ittravels in one second is therefore numerically equal to v, and thisis exactly the concept of velocity: how far something goes per unittime.The wave has to satisfy Maxwell’s equations for Γ E and Γ Bregardless of what Ampèrian surfaces we pick, and by applying themto any surface, we could determine the speed of the wave. Thesurface shown in figure n turns out to result in an easy calculation:a narrow strip of width 2l and height h, coinciding with the positionof the zero-point of the field at t = 0.n / The magnetic field of thewave. The electric field, notshown, is perpendicular to thepage.Now let’s apply the equation c 2 Γ B = ∂Φ E /∂t at t = 0. Sincethe strip is narrow, we can approximate the magnetic field usingsin x ≈ x, which is valid for small x. The magnetic field on the rightedge of the strip, at x = l, is then ˜Bl, so the right edge of the stripcontributes ˜Blh to the circulation. The left edge contributes thesame amount, so the left side of Maxwell’s equation isThe other side of the equation isc 2 Γ B = c 2 · 2 ˜Blh .∂Φ E∂t= ∂ ∂t (EA)= 2lh ∂E∂t,where we can dispense with the usual sum because the strip is narrowand there is no variation in the field as we go up and downthe strip. The derivative equals vẼ cos(x + vt), and evaluating thecosine at x = 0, t = 0 gives∂Φ E∂t= 2vẼlhMaxwell’s equation for Γ B therefore results in2c 2 ˜Blh = 2 Ẽlhvc 2 ˜B = v Ẽ .An application of Γ E = −∂Φ B /∂t gives a similar result, exceptthat there is no factor of c 2 Ẽ = v ˜B .(The minus sign simply represents the right-handed relationship ofthe fields relative to their direction of propagation.)702 Chapter 11 Electromagnetism