A plane electromagnetic wave with wavelength 41 m travels in

A plane electromagnetic wave, with wavelength 4.1 m, travels in vacuum in the positive direction of an x axis. The electric field, of amplitude 290 V/m, oscillates parallel to the y axis. What are the (a) frequency, (b) angular frequency, and (c) angular wave number of the wave? (d) What is the amplitude of the magnetic field component? (e) Parallel to which axis does the magnetic field oscillate? (f) What is the time-averaged rate of energy flow associated with this wave? The wave uniformly illuminates a surface of area 1.8 m^2. If the surface totally absorbs the wave, what are (g) the rate at which momentum is transferred to the surface and (h) the radiation pressure?

Solution

(c ) angular wave number of the wave = 2/

                                                                          = (2 x 3.14)/4.1

                                                                          = 1.5317 rad/m

(d) amplitude of magnetic field component ; B = E/c

                                                                                          = 290/(3x10⁸)

          = 9.666x 10^-7 T

(e) Given electric field is oscillating parallel to y axis and plane electromagnetic wave is travelling in positive x direction, so accordingly magnetic field will be oscillating in the direction parallel to x axis.

(f) energy flow per unit area per unit time is called poynting vector ;S

S= ½(₀cE²)

=1/2(8.85x 10^-12x3x10⁸x290²)

= 111.642 W/m²

Given surface area is 1.8 m²,

So time averaged rate of energy flow assosiated with this wave; U=Sxarea

                                                                                                                                = 111.642x1.8

                                                                                                                                = 200.955 W/m²

(g) condition: surface absorbs wave completely

Then rate at which momentum transferred to the surface; p     = U/c

                                                                                                                                = 200.955/(3x 10⁸)

                                                                                                                                = 6.698x 10^-7

(h)                                                                          radiation pressure ; P    = force/area= F/A

                                                                                                                                = 1/A (dp/dt)

                                                                                                                                =1/A(d/dt(U/c)

                                                                                                                                = 1/Ac(dU/dt)

                                                                                                                                =1/c [(dU/dt)/A]

                                                                                                                                =S/c

                                                                                                                                = 111.642/(3x10⁸)

                                                                                                                             = 3.721x 10^-7 Pa