An electromagnetic wave of frequency 3 GHz enters a dielectric medium of relative electric permittivity 2.25 from vacuum. The wavelength of this wave in that medium will be _________ × 10⁻² cm.

Solution

Physics sir · 4 steps

Frequency stays fixed at the boundary; only the speed drops, and the wavelength shrinks with it.

1. What stays the same. When a wave crosses into a new medium, its frequency does not change. The speed changes, so the wavelength changes in the same ratio.
2. Speed in the medium. For a non-magnetic dielectric, μᵣ = 1, so the refractive index is n = √εᵣ = √2.25 = 1.5. The speed is v = c/n = 2 × 10⁸ m s⁻¹.

   λ = v/f = c/f√εᵣ = (3 × 10⁸)/(3 × 10⁹ × 1.5) m

3. Put in numbers. In vacuum the wavelength is (3 × 10⁸)/(3 × 10⁹) = 0.1 m = 10 cm. Dividing by 1.5 gives λ = 6.67 cm.
4. So. λ = 6.67 cm = 667 × 10⁻² cm, so the answer is 667.

The common slip: 444. Dividing the vacuum wavelength by εᵣ = 2.25 instead of √2.25 = 1.5 gives 4.44 cm, that is 444, not 667.

The idea to keep. In a medium, frequency is fixed and λ scales as 1/n. For a dielectric with μᵣ = 1, n = √εᵣ, not εᵣ itself.

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