The index of refraction of water for a given monochromatic light in water is n=4/3. The wavelength of this light in water is 500nm.
a) find the speed of this light in water.
b) find the energy of one photon of this light . Does this energy change if this light is propagated in vacuum ? Justify
a) The speed of light in a medium is given by the equation:
speed of light in medium = (speed of light in vacuum) / (index of refraction of the medium)
From the information provided, we know that the speed of light in vacuum is approximately 3 × 10^8 m/s. The index of refraction of water is given as n = 4/3.
So, to find the speed of light in water, we can substitute the values into the equation:
speed of light in water = (3 × 10^8 m/s) / (4/3)
speed of light in water = (3 × 10^8 m/s) × (3/4)
speed of light in water = 2.25 × 10^8 m/s
Therefore, the speed of light in water is approximately 2.25 × 10^8 m/s.
b) The energy of a photon is given by the equation:
energy of a photon = (Planck's constant) × (speed of light) / (wavelength)
The wavelength of the light in water is given as 500 nm, which can be converted to meters by multiplying by 10^(-9).
Substituting the known values into the equation:
energy of one photon = (Planck's constant) × (speed of light in water) / (wavelength in water)
energy of one photon = (6.626 × 10^(-34) J·s) × (2.25 × 10^8 m/s) / (500 × 10^(-9) m)
energy of one photon = 2.98 × 10^(-19) J
Now, if the light is propagated in vacuum, the energy of one photon does not change because the energy of a photon is determined by its frequency (or wavelength), which remains the same. In vacuum, the speed of light is also constant and equal to the speed of light in vacuum. So, the energy of one photon remains the same.
Justification: According to the equation for the energy of a photon, energy is dependent on the product of Planck's constant, speed of light, and wavelength. As long as the wavelength remains constant, the energy of the photon will also remain constant regardless of the medium in which it is propagated.
To find the speed of light in water, we can use the equation v = c/n, where v is the velocity of light in a medium, c is the speed of light in a vacuum, and n is the refractive index of the medium.
a) Speed of light in water:
In this case, the refractive index of water is given as n = 4/3, and the speed of light in a vacuum is approximately 3 × 10^8 m/s. Plugging these values into the equation, we get:
v = (3 × 10^8 m/s) / (4/3)
v = 2.25 × 10^8 m/s
So, the speed of light in water is approximately 2.25 × 10^8 m/s.
b) Energy of one photon of this light:
To find the energy of one photon, we can use the equation E = h * (c/λ), where E is the energy of a photon, h is Planck's constant (approximately 6.63 × 10^-34 J•s), c is the speed of light in a vacuum, and λ is the wavelength of light.
Since we are given the wavelength of light in water (500 nm), we need to convert it to the wavelength in a vacuum. We can use the equation λ_vacuum = λ_water / n.
λ_vacuum = (500 nm) / (4/3)
λ_vacuum = (500 nm) * (3/4)
λ_vacuum = 375 nm
Now we can calculate the energy of one photon:
E = (6.63 × 10^-34 J•s) * (3 × 10^8 m/s) / (375 × 10^-9 m)
E ≈ 5.304 × 10^-19 J
So, the energy of one photon of this light in water is approximately 5.304 × 10^-19 Joules.
Regarding the second part of the question, the energy of this light does not change when it propagates in a vacuum. The energy of a photon is determined by its frequency, which remains constant regardless of the medium through which the light is passing. However, the speed of light, and therefore the wavelength, do change when light passes from one medium to another due to the different refractive indices, but the energy remains the same.
Therefore, the energy of the light does not change when it is propagated in a vacuum.