# Mathematical relationship between energy and wavelength are proportional

### Photon energy - Wikipedia

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the . The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a dispersion relation. or of the physical system, such as for conservation of energy in the wave. The energy of a photon depends on radiation frequency; there are photons of all relationship it is clear that the wavelength of light in inversely proportional to. For light, the relationship between wavelength (λ) and frequency (ƒ) is goes up as the frequency increases, the energy is directly proportional to the frequency.

Predictions based on light as a wave To explain the photoelectric effect, 19th-century physicists theorized that the oscillating electric field of the incoming light wave was heating the electrons and causing them to vibrate, eventually freeing them from the metal surface.

This hypothesis was based on the assumption that light traveled purely as a wave through space. See this article for more information about the basic properties of light.

### Light Website : Wavelength

Scientists also believed that the energy of the light wave was proportional to its brightness, which is related to the wave's amplitude. In order to test their hypotheses, they performed experiments to look at the effect of light amplitude and frequency on the rate of electron ejection, as well as the kinetic energy of the photoelectrons.

Based on the classical description of light as a wave, they made the following predictions: The kinetic energy of emitted photoelectrons should increase with the light amplitude. The rate of electron emission, which is proportional to the measured electric current, should increase as the light frequency is increased.

To help us understand why they made these predictions, we can compare a light wave to a water wave. Imagine some beach balls sitting on a dock that extends out into the ocean. The dock represents a metal surface, the beach balls represent electrons, and the ocean waves represent light waves. If a single large wave were to shake the dock, we would expect the energy from the big wave would send the beach balls flying off the dock with much more kinetic energy compared to a single, small wave.

This is also what physicists believed would happen if the light intensity was increased. Light amplitude was expected to be proportional to the light energy, so higher amplitude light was predicted to result in photoelectrons with more kinetic energy.

Classical physicists also predicted that increasing the frequency of light waves at a constant amplitude would increase the rate of electrons being ejected, and thus increase the measured electric current.

## How does energy relate to wavelength and frequency?

Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them to refract at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a dispersion relation. Nonuniform media[ edit ] Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore [11] Wavelength can be a useful concept even if the wave is not periodic in space.

For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.

The figure at right shows an example. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated with a high loss and the wave dies out. The analysis of differential equations of such systems is often done approximately, using the WKB method also known as the Liouville—Green method.

The method integrates phase through space using a local wavenumberwhich can be interpreted as indicating a "local wavelength" of the solution as a function of time and space. In addition, the method computes a slowly changing amplitude to satisfy other constraints of the equations or of the physical system, such as for conservation of energy in the wave.

Crystals[ edit ] A wave on a line of atoms can be interpreted according to a variety of wavelengths.