Question

Suppose that the electric field part of an electromagnetic wave in vacuum is E = {(3.1 N/C) cos [(1.8 rad/m) y + (5.4 × 106 rad/s)t]}ˆi . (a) What is the direction of propagation? (b) What is the wavelength l ? (c) What is the frequency n ? (d) What is the amplitude of the magnetic field part of the wave? (e) Write an expression for the magnetic field part of the wave.

Answer 1

a)

The direction of the propagation of the wave can be directly figured out by looking at the given plane wave equation more specifically by the argument of the cosine in the plane wave equation.

In the given equation cos( ky+ωt ), since the kyand ωt have the same sign that is both are positive so the direction of the wave is in –y direction.

If either of the two is −ve then the wave will travel in +y direction.

cos( −θ )=cosθ

Hence, the direction of the propagation of the plane wave is in ( −j )direction so that the amplitude of the wave crest could be maintained.

b)

The given wave equation is,

E=( 3.1 N/C )cos{ ( 1.8 rad/m )y+( 5.4× 10 6  rad/s )t }

Compare this equation with the general equation of the plane wave,

E= E 0 sin( kx−ωt )

We get,

k=1.8 rad/m

ω=5.4× 10 6  rad/s

E 0 =3.1 N/C

The wavelength is given as,

λ= 2π k

By substituting the values in the above equation, we get

λ= 2×3.14 1.8 =3.49 m

Thus, the wavelength of the wave is 3.49 m.

c)

The frequency of the plan wave is given as,

ν= ω 2π

By substituting the values in the above equation, we get

ν= 5.4× 10 8 2π =8.6× 10 7 Hz

Thus, the frequency of the wave is 8.6× 10 7 Hz.

d)

The amplitude of the magnetic field part is given as,

B 0 = E 0 c

Where, E 0 is the amplitude of electric field part and c is the speed of light.

By substituting the values in the above equation, we get

B 0 = 3.1 3× 10 8 =1.03× 10 −7  T

Thus, the amplitude of magnetic field part is 1.03× 10 −7  T.

e)

From the above discussion we can infer that the direction of the magnetic field is in +zdirection as the direction of the electric field is ( i )and wave is propagating in ( −j )and the three components of electromagnetic radiation should be perpendicular to each other.

The expression for the magnetic field part will be,

B= B 0 cos( ky+ωt ) k ^ =[ ( 1.03× 10 −7  T )cos{ ( 1.8 rad/m )y+( 5.4× 10 6  rad/s )t } ] k ^