Consider the following two statements regarding a linearly polarized, plane electromagnetic wave:
(A) The electric field and the magnetic field have equal average values.
(B) The electric energy and the magnetic energy have equal average values.
Explanation: Let us assume the linearly polarized plan wave travels
in z direction so the electric and magnetic fields of the wave are
given by
Their average values over one cycle would be the average from time
t=0 to t=T.
Average of electric field
(because and
)
Therefore, the average of electric field is zero.
Similarly average of magnetic field is also zero as
So electric and magnetic field have their average value zero.
Therefore (A) is correct.
Now the electric energy associated with the electric field is given by
…(i)
where ϵ0 is the electric permittivity of free space(vacuum) and is
equal to 8.85 × 10-12 C2 N-1 m-2 and Ude is the energy density of
electric field.
Similarly, the magnetic energy associated with the magnetic field is
given by
…(ii)
where μ0 is the magnetic permeability of free space and its value is
4π × 10-7 T m A-1, Udb is the magnetic field energy density.
Also the amplitudes of electric and magnetic fields are related as
…(iii)
where C is the speed of light in vacuum.
and
Using (iii) in (i)
…. (iv)
The electrical permittivity and magnetic permeability of free space
related as
…. (v)
Using (v) in (iv)
Therefore, the electric and magnetic energy densities are same in
magnitude and so their average values will be same.
Average of electric energy over one cycle will be
So electric and magnetic energy and have non zero average values.
Therefore (B) is also correct.