CBSE-XI-Physics
40: Electromagnetic Waves
- #7A 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.Ans : 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=\frac{\omega }{2\pi }``, is same.
The electrical energy in the region,
UE = `` \left(\frac{1}{2}{\in }_{0}{E}^{2}\right)\times V``
It can be written as:
`` {U}_{\,\mathrm{\,E\,}}=\left(\frac{1}{2}{\in }_{0}\left({{E}_{0}}^{2}{\,\mathrm{\,sin\,}}^{2}\right(kz-\omega t)\right)\times V``
`` {U}_{E}=\left(\frac{1}{2}{\in }_{0}{{E}_{0}}^{2}\times \frac{\left(1-\,\mathrm{\,cos\,}2(kz-\omega t)\right)}{2}\right)\times V``
`` {U}_{E}=\left(\frac{1}{4}{\in }_{0}{{E}_{0}}^{2}\times \left(1-\,\mathrm{\,cos\,}2(kz-\omega t)\right)\right)\times V``
`` ``
The magnetic energy in the region,
`` {U}_{\,\mathrm{\,B\,}}=\left(\frac{{B}^{2}}{2{\mu }_{0}}\right)\times V``
`` {U}_{\,\mathrm{\,B\,}}=\left(\frac{{{B}_{0}}^{2}{\,\mathrm{\,sin\,}}^{2}(kz-\omega t)}{2{\mu }_{0}}\right)\times V``
`` \Rightarrow {U}_{\,\mathrm{\,B\,}}=\left(\frac{{{B}_{0}}^{2}\left(1-\,\mathrm{\,cos\,}(2kz-2\omega t)\right)}{4{\mu }_{0}}\right)\times V``
The angular frequency of the electric and magnetic energies is same and is equal to 2ω.
Therefore, their frequency, `` f\text{'}=\frac{2\omega }{2\pi }=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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