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12.48: An electromagnetic plane wave of (angular) frequency ωis travelling in the xdirection through the vacuum. It is polarized in the ydirection, and the amplitude of the electric field is Eo.

(a) Write down the electric and magnetic fields, role="math" localid="1658134257504" E(x,y,z,t)and B(x,y,z,t)[Be sure to define any auxiliary quantities you introduce, in terms of ω, Eo, and the constants of nature.]

(b) This same wave is observed from an inertial system Smoving in thexdirection with speed vrelative to the original system S. Find the electric and magnetic fields in S, and express them in terms of the role="math" localid="1658134499928" Scoordinates: E(x,y,z,t)and B(x,y,z,t). [Again, be sure to define any auxiliary quantities you introduce.]

(c) What is the frequency ωof the wave in S? Interpret this result. What is the wavelength λof the wave in S? From ωand λ, determine the speed of the waves in S. Is it what you expected?

(d) What is the ratio of the intensity in to the intensity in? As a youth, Einstein wondered what an electromagnetic wave would like if you could run along beside it at the speed of light. What can you tell him about the amplitude, frequency, and intensity of the wave, as approaches ?

Short Answer

Expert verified

(a) The expression for the electric isE(x,y,z,t)=Eocos(kxωt)yand the expression for the magnetic field is B(fx,y,z,t)=Eoccos(kxωt)z.

(b) The equation of electric filed is E¯(x¯,y¯,z¯,t¯)=E¯ocos(k¯x¯ω¯t¯)yand the equation of magnetic field is B¯(x¯,y¯,z¯,t¯)=Eo¯ccos(k¯x¯ω¯t¯)z.

(c) The expression for the frequency of the wave inS frame is,

ω¯=ω1vc1+vc , wavelength isλα and for the velocity of the wave is c.

(d) The ratio of the intensity is 1vc1+vc.

Step by step solution

01

Electromagnetic wave

The ratio of the maximum value of the electric field to the maximum value of the magnetic field provides the velocity of the electromagnetic waves.

Mathematically, it can be expressed as follows,

c=EoBo

HereEo and Bois the maximum value of the electric and magnetic field,c is velocity of the electromagnetic waves.

02

Find the expression for electric and magnetic field in S plane.

(a)

The plane wave (Electric field wave) is polarized and it’s electric field is expressed as,

E(x,y,z,t)=Eocos(kxωt)y

Herek is equal to k=Eo×Bo,k is called as propagation vector.

As,

c=EoBo

Bo=Eoc

So, the magnetic field wave is,

B(fx,y,z,t)=Eoccos(kxωt)z

03

Find the expression for the electric and magnetic in S→ plane.

(b)

The relativistic equations are given by,

Ex=Ex

Ey=γ(EyvBz)…… (1)

Here, xis the position of the event in frame S,vis the velocity of frameS'relative toS, cis the speed of light in vacuum,γ is the Lorentz factor for a velocity v.

As, Bo=Eoc, the magnetic field equation can be written as,

Bx=Bx

By=γ(By+vc2Ez)

Substitute Eocos(kzωt)for Eyand Eoccos(kzωt)for Bzin the equation (1).

role="math" localid="1658135705784" Ey=γ(Eocos(kzωt)vEoccos(kzωt))=Eocos(kzωt)γ(1vc)

α=γ(1vc) ……. (2)

Then,

Ey=αEocos(kzωt)

Using the special theory of relativity, the expression for the Lorentz factor is given by,

γ=11(vc)2 ……. (3)

Herevis the speed of the object andcis the speed of the light.

From equation (2)

α=γ(1vc)

Substitute 11(vc)2forγin the above equation,

α=11(vc)2(1vc)

Simplify the above equation,

α=1(1vc)(1+vc)(1vc)=1vc1+vc

The expression for the time in frame can be expressed as,

t'=γ(tvc2x)

As, x=γ(x¯+vt) andt'=γ(t¯+vc2x¯)

Then,

kxωt=γ(x¯+vt)ω[γ(t+vc2x¯)]

kxωt=γ(kωvc2)x¯[(ωkv)t¯]=k¯x¯ω¯t

Here,

x¯=γ(kωvc2)=γk(1vc)=αk

And

ω¯=γω(1vc)=αω

Therefore the equation of electric filed is E¯(x¯,y¯,z¯,t¯)=E¯ocos(k¯x¯ω¯t¯)y.

Therefore the equation of magnetic field is B¯(x¯,y¯,z¯,t¯)=Eo¯ccos(k¯x¯ω¯t¯)z.

The expression for the relativistic electric field is E¯o=αEo.

The propagation constant is k¯=αk.

The angular velocity is ω¯=αω.

The constant α=1vc1+vc.

04

Find the expression for the frequency, wavelength and speed of the wave.

(c)

The expression for the frequency of the wave inS frame is,

ω¯=ω1vc1+vc

The expression for the wavelength of the wave in Sframe is,

λ¯=2πk¯=2παk=λα

The expression for velocity of wave inSframe is,

λ¯=ω¯2πλ¯=ω¯2π(λα)=(ω2π)λ=c

Thus the velocity of light is same in all inertial frames of references.

05

Obtain the ratio of the intensity.

(d)

As the intensity is directly proportional to the square of the amplitude,

Then the ratio of the intensity is,

I¯I=E¯o2Eo2=α2=1vc1+vc

Therefore the ratio of the intensity is 1vc1+vc.

Thus amplitude, frequency and intensity of the light wave all reduce to zero as we move faster and faster.

It will get so faint thus we won’t be able to see it, and it becomes red-shifted. Even your light vision goggles won’t help. But the light is moving relative to you.

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