Relationship between the E and B fields
Case: The fields between two metallic plates
Stationary waves. The Heinrich Hertz experiment


Ei = Em Cos( kx - t ) ŷ
Incident wave
X
Reflected
wave
Y
Incident wave
X
Z



Ei = Em Cos( k x - t ) ŷ

E r = Em Cos( k x + t +R) ŷ
Using the identities
R is included to account for
possible phase shift upon reflection
Cos(a+b) = Cos(a) Cos(b) - Sin(a) Sin(b)
Cos((a-b) = Cos(a) Cos(b) + Sin(a) Sin(b)
gives
Cos(a+b) + Cos((a-b) = 2 Cos(a) Cos(b)
Cos(A) + Cos(B) = 2 Cos[ (A+B)/2 ] Cos[ (A-B)/2 ]



E total (x,t) = Ei + E r = 2 Cos [ kx + R/2 ] Cos [t + R/2 ] ŷ
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We require the total electric field to be zero at the metal plate (i. e. at x=0 ) and at all
time.
At x=0:

E total (0,t) = 2 Cos[R/2] Cos[t + R/2] ŷ = 0
This requires R= 
Thus,

Ei = Em Cos( k x - t ) ŷ


E r = Em Cos( k x + t +) ŷ = -Em Cos( k x + t ) ŷ



E total (x,t) = Ei + E r = 2 Sin( kx ) Sin (t ) ŷ

On the other hand, before we also know that the ME require,
E y
x

Bz
t
From which one obtains,
2kCos( kx) Sin (t )  
Bz = 2
Bz
t
k
Cos( kx) Cos (t )
ω

Btotal (x,t) = 2 Cos( kx) Cos (t )
ẑ
λ
X
XX
Case: The fields in the free and unbounded space
Relationship between the electric field and magnetic field
amplitudes
Let's consider an harmonic plane wave
We will leave for the
HW assignment to
demonstrate that