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Frequency and Wavelength of Photon
Equation 
The photon is described as a wave, and the photon frequency ($\nu$) is related to ($$) through the speed of Light ($c$), according to the following formula:
 $ c = \nu \lambda $ |
Given that the photon frequency ($\nu$) is the inverse of the period ($T$):
$\nu=\displaystyle\frac{1}{T}$
this means that the speed of Light ($c$) is equal to the distance traveled in one oscillation, which is ($$), divided by the elapsed time, which corresponds to the period:
$c=\displaystyle\frac{\lambda}{T}$
In other words, the following relationship holds:
| $ c = \nu \lambda $ |
This formula corresponds to the mechanical relationship that states the wave speed is equal to the wavelength (distance traveled) divided by the oscillation period, or inversely proportional to the frequency (the inverse of the period).
ID:(3953, 0)
Refraction Index
Equation
The refractive index, denoted as $n$, is defined as the ratio of the speed of light in a vacuum, denoted as $c$, to the speed of light in the medium, denoted as $c_m$:
| $ n =\displaystyle\frac{ c }{ v }$ |
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Refractive index and wavelength
Equation
If $n$ is the refractive index in a medium and $\lambda$ is the wavelength in a vacuum, then when propagating in the medium, the wavelength $\lambda_m$ will be
| $ n =\displaystyle\frac{ \lambda }{ \lambda_m }$ |
The energy of a wave or particle (photon) of light is given by
| $ \epsilon = h \nu $ |
When this energy propagates from one medium, for example, a vacuum with a speed of light $c$, to another medium with a speed of light $c_m$, it is concluded that the frequency of light remains unchanged. However, this implies that, since the speed of light is equal to the product of frequency and wavelength, as expressed in the equation
| $ c = \nu \lambda $ |
the wavelength must change as it transitions between mediums.
Therefore, if we have a wavelength of light in one medium $\lambda_m$ and in a vacuum $\lambda$, the refractive index can be defined as
| $ n =\displaystyle\frac{ c }{ v }$ |
and can be expressed as
$n=\displaystyle\frac{c}{c_m}=\displaystyle\frac{\lambda\nu}{\lambda_m\nu}=\displaystyle\frac{\lambda}{\lambda_m}$
In other words,
| $ n =\displaystyle\frac{ \lambda }{ \lambda_m }$ |
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