A plane electromagnetic wave is passing through a region. Consider the quantities

(a) electric field,

(b) magnetic field,

(c) electrical energy in a small volume and

(d) magnetic energy in a small volume.

Construct pairs of the quantities that oscillate with equal

frequencies.

Explanation: Let us consider an electromagnetic wave traveling

in the z direction. The electric field E of the wave is assumed to be

where E₀ is the amplitude of the electric field, k is the wave number

of the wave , λ is the wavelength, ω is the angular frequency

of the wave and t is the time. The frequency of the wave

is

Now the magnetic field can be considered as

where B₀ is the amplitude of the magnetic field, k is the wave

number of the wave , λ is the wavelength, ω is the angular

frequency of the wave and t is the time. The frequency of

the wave is

The frequencies of electric and magnetic waves are same.

Hence (a) and (b) oscillate with the same frequency.

Now the electric field energy density of the electromagnetic wave is

given by the relation

where ϵ₀ is the electric permittivity of free space(vacuum) and is equal to 8.85 × 10^-12 C² N^-1 m^-2 and U_de is the energy density of electric field. Now the energy of the wave over small volume V of the region is the product of energy density and the small volume V. The energy of the electric field is

The energy of the electric field is .

The angular frequency of the energy of the wave is 2ω, the

Corresponding frequency will be

Now the magnetic field energy density of the electromagnetic wave

is given by the relation

where μ₀ is the magnetic permeability of free space and its value is

4π × 10^-7 T m A^-1, U_db is the magnetic field energy density. Now the energy of the wave over small volume V of the region is the product of energy density and the small volume V. The energy of the magnetic field is

The energy of the magnetic field is

The angular frequency of the energy of the wave is 2ω, the

Corresponding frequency will be

The electric field energy and magnetic field energy have the same

frequencies of oscillation.

So (c) and (d) form a pair.