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An electromagnetic wave of wavelength 435 nm is traveling in vacuum in the -z- direction. The electric field has amplitude2.70×10-3V/mand is parallel to the x-axis. What are (a) the frequency and (b) the magnetic-field amplitude? (c) Write the vector equations forE(z,t) and B(z,t).

Short Answer

Expert verified

a) The frequency of electromagnetic wave is 6.89×1014Hz.

b) The amplitude of magnetic field of electromagnetic wave is 9×10-12T.

c) The vector equation forE(z,t)=i[2.70×10-3V/mcos(1.44×107m)z-(4.3×1015rad/s)t] andB(z,t)=-j[9×10-12Tcos(1.44×107m)z-(4.3×1015rad/s)t] .

Step by step solution

01

Step 1:Define frequency and write formulas.

The number of waves that moves a fixed point in unit time is known as frequency.

f=1T

Where,T =Time period in s.

The frequency, wavelength, and speed of light of any wave are related by the wavelength-frequency relationship.

The wavelength-frequency relationship is:

c=λff=cλ

Where,c is the speed of light which is equal to 3×108m/s,λ is the wavelength andf is the frequency.

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

Emax=2lε0cBmax=Emaxc

Where,ε0=8.85×10-12C2/Nm2 and c is the speed of light that is equal to 3.0×108m/s.

The equations of sinusoidal electromagnetic wave for electric and magnetic field are:

E=Emaxjcoskx-ωtB=Bmaxkcoskx-ωt

02

Determine the frequency of wave.

Given that,λ=435×10-9m

The wavelength-frequency relationship is

c=λff=cλ

Substitute the values of variables

f=3×108435×10-9=6.89×1014Hz

Hence, the frequency of a sinusoidal electromagnetic wave is 6.94×1014Hz.

03

Determine the amplitude of magnetic field.

Given that,Emax=2.70×10-3V/m

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

Bmax=Emaxc

Substitute the values of variables

Bmax=2.70×103V/m3×108=9×10-12T

Hence, the amplitude of magnetic field of electromagnetic wave is 9×10-12T.

04

Determine the sinusoidal electromagnetic wave.

The angular frequency of wave is

ω=2πf=2π6.89×1014=4.3×1015rad/s

And the wavenumber of wave is

k=2πλ=2π435×10-9=1.44×107rad/m

Substitute the variable in equations of sinusoidal electromagnetic wave for electric and magnetic field are:

E=Emax=coskx-ωtE=2.70×10-3V/mcos1.44×107mz-4.3×1015rad/stB=Bmaxcoskx-ωtB=9×10-12Tcos1.44×107rad/mz-4.3×1015rad/st

Hence, the vector equation forEz,t=i2.70×10-3V/mcos1.44×107mz-4.3×1015rad/st and Bz,t=-j9×10-12Tcos1.44×107mz-4.3×1015rad/st.

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