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An electromagnetic wave has a magnetic field given by B(x,t)=-(8.25×10-9T)j^cos[1.38×104rad/mx+ωt]. (a) In which direction is the wave traveling? (b) What is the frequency of the wave? (c) Write the vector equation for E(x,t).

Short Answer

Expert verified
  1. In - x - direction wave is traveling.
  2. The frequency of the wave is 6.59×1011Hz.

3. The vector equation for E(x,t)is

E=-248V/mcos1.38×104rad/mx+4.14×1012rad/st .

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=1Torω2π

Where, Time period in seconds and is angular frequency in .

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

The wavelength-frequency relationship is:

c=λf

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

The wavelength is inversely proportional to the frequency.

λ=cforvf

Where, v is velocity of wave in .

The relation between wave number and wavelength is:

k=2πλ

Where, k is wavenumber in .

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

Emax=2lε0cBmax=Emaxc

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

E=Emaxcoskx-ωtB=Bmaxcoskx-ωt

02

Determine the direction of wave and frequency of wave.

B(x,t)=-(8.25×10-9T)j^cos[(1.38×104rad/m)x+ωt]Given that,

The values of wavenumberk is1.38×104rad/m and the maximum magnetic field is .

Bmaxis-8.25×10-9T

B=-Bmaxcoskx+ωt

Since, if the phase of wave iskx-ωt then wave travels in positive x- direction. Therefore, the above equation defines the wave is travelling in negative x-direction as the phase of wave is kx-ωt.

The frequency of wave is:

f=cλ=kc2π=1.38×1043.0×1082π=6.59×1011Hz

03

Determine the vector equation.

The angular velocity of wave is:

ω=2πf=2π6.59×1011=4.14×1012rad/s

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

Emax=cBmax=3×108-8.28×109=-2.48V/m

The equation of sinusoidal electromagnetic wave for magnetic field is:

E=Emaxcoskx+ωt=-2.48V/mcos1.38×104rad/mx+4.14×1012rad/st

Hence, the vector equation for is E=x,t.

E=-2.48V/mcos1.38×104rad/mx+4.14×1012rad/st

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