The magnetic field in a plane monochromatic electromagnetic wave with wavelength

Question

The magnetic field in a plane monochromatic electromagnetic wave with wavelength lambda = 675 nm, propagating in a vacuum in the z-direction is described by where B_1=- 4.8 Times 10^-6 T, and i-hat and j-hat are the unit vectors in the +x and +y directions, respectively. What is k, the wavenumber of this wave? What is z_max, the distance along the positive X-axis to the position where the magnitude of the magnetic field is a maximum at t = 0? What is E_y, the amplitude of the electric field oscillations? What is E_y. the y component of the electric field at (x = 0, y-0, z = z_max) at t = 0? Which of the following equations describes the spatial and time dependence of the electric field oscillations? What is t_max, the first time after t = 0, when the magnitude of the electric field at the origin (x = y = z = 0) has its max turn um value?

Explanation / Answer

a)

Wave Number

k=2pi/lamvda =2pi/(675*10-9)

K=9.31*106 m-1

b)

Sin(Kz-Wt)=1

at t=0=>SIn(Kz)=1

Kz=sin-1(1) =pi/2

Zmax=pi/2K =pi/2*(9.31*106)

Zmax=1.687*10-7 m =168.7 nm

c)

Bmax=sqrt[(4.8*10-6)2+(4.8*10-6)2]=6.788*10-6 T

Emax=Bmaxc =(6.788*10-6)(3*108)=2036.5 V/m

d)

Ey=Emaxsin(-45) =-2036.5 *sin45

Ey=1440 V/m

e)

E=Emax/sqrt[2] Sin(kx-Wt)(i-j)

f)

tmax=T/4 =lambda/4c =(675*10-9)/4*(3*108)

tmax=5.625*10-16 s

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