Question

The energy of a wave will remain constant if which of the following changes are made to it?A The wavelength is cut in half, and the speed is doubled.B The wavelength is doubled, and the amplitude is tripled.c wavelength is cut in half, and the speed is tripled. D The wavelength is doubled, and the speed is also doubled.

Answer 1

The energy of a wave will remain constant if the wavelength is doubled and the speed is also doubled.

Option D

Explanation:

Based on the dual nature of waves, Planck's equation states that the energy of the wave is directly proportional to the frequency of the wave. The Planck constant is termed as the proportionality constant.

So, $E = hv$

It is known that frequency is the ratio of speed of light to wavelength of wave, so the energy equation can be written as

$E = \frac{hc}{lambda}$

Thus, energy is inversely proportional to the wavelength of wave and directly proportional to the speed of wave. So in order to keep the energy constant, both the wavelength and the speed should be doubled as shown below.

Let  c = 2c and λ = 2λ, then the new energy will be

$E'=\frac{hc}{lambda}$

Since, c = 2c and lambda = 2 lambda, E' = 2hc/2lambda = E

So the wavelength is doubled and the speed is also doubled to keep the energy of the wave constant.