Simple Nature - Light and Matter

n/The magnetic field of the

wave. The electric field, not

shown, is perpendicular to the

page.

second, the zero-point is located atx=−(1 s)v. The distance it

travels in one second is therefore numerically equal tov, and this

is exactly the concept of velocity: how far something goes per unit

time.

The wave has to satisfy Maxwell’s equations for ΓE and ΓB

regardless of what Amp`erian surfaces we pick, and by applying them

to any surface, we could determine the speed of the wave. The

surface shown in figure n turns out to result in an easy calculation:

a narrow strip of width 2`and heighth, coinciding with the position

of the zero-point of the field att= 0.

Now let’s apply the equationc^2 ΓB =∂ΦE/∂tatt= 0. Since

the strip is narrow, we can approximate the magnetic field using

sinx≈x, which is valid for smallx. The magnetic field on the right

edge of the strip, atx=`, is thenB` ̃ , so the right edge of the strip

contributesB`h ̃ to the circulation. The left edge contributes the

same amount, so the left side of Maxwell’s equation is

c^2 ΓB=c^2 · 2 B`h ̃.

The other side of the equation is

∂ΦE

∂t

=

∂

∂t

(EA)

= 2`h

∂E

∂t

,

where we can dispense with the usual sum because the strip is nar-
row and there is no variation in the field as we go up and down
the strip. The derivative equalsvE ̃cos(x+vt), and evaluating the
cosine atx= 0,t= 0 gives

∂ΦE

∂t

= 2vE`h ̃

Maxwell’s equation for ΓBtherefore results in

2 c^2 B`h ̃ = 2E`hv ̃

c^2 B ̃=vE ̃.

An application of ΓE=−∂ΦB/∂tgives a similar result, except

that there is no factor ofc^2

E ̃=vB ̃.

(The minus sign simply represents the right-handed relationship of

the fields relative to their direction of propagation.)

Section 11.6 Maxwell’s equations 729