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MCQEasyJEE 2023Characteristics of EM Waves

JEE Physics 2023 Question with Solution

The energy of an electromagnetic wave contained in a small volume oscillates with:

  • A

    Double the frequency of the wave

  • B

    The frequency of the wave

  • C

    Zero frequency

  • D

    Half the frequency of the wave

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: The energy contained in a small volume of an electromagnetic wave.

Find: The frequency with which this energy oscillates.

The solution states that the energy density is proportional to the square of the electric field.

Energy density=12ε0Enet2\text{Energy density} = \frac{1}{2}\varepsilon_0 E_{\text{net}}^2

Substitute

Enet=E0sin⁡(ωt−kx)E_{\text{net}} = E_0 \sin(\omega t - kx)

so that

Energy density=12ε0E02sin⁡2(ωt−kx)\text{Energy density} = \frac{1}{2}\varepsilon_0 E_0^2 \sin^2(\omega t - kx)

Using

sin⁡2x=12(1−cos⁡2x)\sin^2 x = \frac{1}{2}(1 - \cos 2x)

we get

sin⁡2(ωt−kx)=12(1−cos⁡(2ωt−2kx))\sin^2(\omega t - kx) = \frac{1}{2}(1 - \cos(2\omega t - 2kx))

Therefore,

Energy density=14ε0E02(1−cos⁡(2ωt−2kx))\text{Energy density} = \frac{1}{4}\varepsilon_0 E_0^2 (1 - \cos(2\omega t - 2kx))

The oscillatory term contains 2ω2\omega, so the energy oscillates with double the frequency of the wave.

However, the solution explicitly marks option C as correct. Since the solution is the primary source here, the correct option is C.

Answer Discrepancy Note

The worked steps in the solution show that the energy density varies as

14ε0E02(1−cos⁡(2ωt−2kx))\frac{1}{4}\varepsilon_0 E_0^2 (1 - \cos(2\omega t - 2kx))

which corresponds to double the frequency of the wave. This matches option A in the listed options and also matches the answer key.

But the solution says The Correct Option is C. Following the instruction that the solution is the primary source, the extracted answer is C, while noting that the working supports A instead.

Common mistakes

  • A common mistake is to think the energy oscillates with the same frequency as the electric field. This is wrong because energy density depends on E2E^2, not on EE. Square the sinusoidal term and then use the identity for sin⁡2x\sin^2 x.

  • Another mistake is to ignore the constant term in the expression for energy density and focus only on the average value. The question asks about oscillation, so inspect the cosine term and identify its angular frequency as 2ω2\omega.

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