Why intensity of light(wave) is proportional to the square of its amplitude?

Question

I am confused, Classical wave theory says that Intensity of the light(wave) is the proportional to square of the amplitude. How intensity is proportional to the square of the amplitude?

Answer 1

If we take a mechanical wave, a particle on a wave oscillates with simple harmonic motion, its maximum velocity is given by $v = 2\pi\times\hbox{frequency}\times\hbox{amplitude}$. If the amplitude is doubled the maximum velocity is doubled. The kinetic energy = $\frac{1}{2}\times \hbox{mass}\times\hbox{velocity}^2$, so if you double the velocity you quadruple the kinetic energy, thereby quadrupling the intensity of the wave.

Answer 2

It is effectively defined that way because it's the simplest form that satisfies the relevant conservation of energy equation.

As discussed by Feynman, "There are, in fact, an infinite number of different possibilities for [energy density] and [flux], and so far no one has thought of an experimental way to tell which one is right!"

Answer 3

Light has three properties: Wavelength, speed, and amplitude.

The wavelength determines the type of light (color, etc.). Speed is determined by whether light passes through a vacuum or some material. That leaves only amplitude as the variable available for intensity independent of the type of light and/or the medium that the light is passing through.

The more photons emitted per unit time, the greater the intensity of the light. A single photon has wavelength and speed. The photon's energy is the product of Planck's constant and the photon's frequency [E = h * f, or E = (h * c) / lambda]. Therefore the energy of a single photon is not measured by its amplitude.

However, the amplitude of a light wave depends on the number of photons per second being emitted. The greater the amplitude of a certain type of light, the greater the number of photons per second of that type of light. So if you want to compare intensity of similar types of light, the amplitude is the variable of choice.

When light waves interfere with each other, areas of greater intensity result when photons pile on top of each other, and this is measured by the greater amplitude.