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Question
A plane electromagnetic wave is passing through a region. Consider (a) electric field (b) magnetic field (c) electrical energy in a small volume and (d) magnetic energy in a small volume. Construct the pairs of the quantities that oscillate with equal frequencies.
Solution
Let the electromagnetic wave be propagating in the z-direction. The vibrations of the electric and magnetic fields are given by:
Ex= E0 sin (kz – ωt)
By= B0 sin (kz – ωt)
Let the volume of the region be V.
The angular frequency of the vibrations of the electric and magnetic fields are same and are equal to ω. Therefore, their frequency, `f = ω/(2pi)` , is same .
The electrical energy in the region,
`U_E = (1/2 ∈_0 E^2) xx V`
It can be written as :
`U_E = (1/2 ∈_0 (E_0^2 sin^2(kz - ωt))xx V`
`U_E = (1/2 ∈_0 E_0^2 xx ((1-cos2(kz-ωt)))/2) xx V`
`U_E = (1/4 ∈_0 E_0^2 xx (1-cos2(kz-ωt))) xx V`
The magnetic energy in the region ,
`U_B = (B^2/(2u_0)) xx V`
`U_B = ((B_0^2 sin^2(kz-ωt))/(2u_0)) xx V`
⇒ `U_B = ((B_0^2(1-cos(2kz-2ωt)))/(4u_0)) xx V`
The angular frequency of the electric and magnetic energies is same and is equal to 2ω.Therefore, their frequency,
`f^' = (2ω)/(2pi) = 2f`
, will be same.
Thus, the electric and magnetic fields have same frequencies and the electrical and magnetic energies will have same frequencies.
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