A plane electromagnetic wave, has
frequency of 2.0 × 10¹⁰ Hz and its energy density is 1.02 × 10 –8 J/m³ in vacuum. The amplitude of the magnetic field of the wave is close to
( 1/(4π ε ₀) = 9 × 10⁹Nm²/C² and speed of light
= 3 × 10⁸ ms –1 )

Solution

Physics sir · 4 steps

Energy density gives you the field directly; the frequency is only there to distract you.

1. Energy density formula. For a plane EM wave in vacuum, the average energy density is shared equally between the electric and magnetic fields. In terms of the magnetic amplitude B₀ it is:

   u = B₀²/2μ₀ ⇒ B₀ = √(2μ₀ u)

2. Find mu0. Use μ₀ = 1/(ε₀ c²) with 1/ε₀ = 4π × 9 × 10⁹. This gives μ₀ = (4π × 9 × 10⁹)/(9 × 10¹⁶) = 4π × 10⁻⁷ T m A⁻¹.
3. Put in numbers. B₀² = 2 × 4π × 10⁻⁷ × 1.02 × 10⁻⁸ = 2.56 × 10⁻¹⁴ T². So B₀ = 1.6 × 10⁻⁷ T.
4. So. B₀ ≈ 160 nT, which is option (c). The frequency 2.0 × 10¹⁰ Hz is never needed.

The idea to keep. Average energy density of an EM wave is 1/2ε₀ E₀² = B₀²/2μ₀. It depends only on amplitude, never on frequency, so ignore frequency data here.

Answer from the NTA final answer key. Solution by PrepWiser.