A source of light used for newton's ring experiment consist of two wavelength, wavelength 1 = 7500armstrong and wavelength 2 = 5000armstrong. Given that the mth mark ring of wavelength 1 coincides with the (m + 1)th dark ring of wavelength 2. Determine the ring diameter if the radius of the curvature of the lens is 0.5m?
To determine the ring diameter in Newton's ring experiment, we can start by understanding the conditions for the rings to appear.
In the experiment, a thin film of air is formed between a plano-convex lens and a glass plate. When light from a source is incident on this setup, interference patterns are observed due to the phase difference created by the air film.
The condition for the formation of Newton's rings is that the path difference between the reflected rays of light at the upper and lower surfaces of the air film is an integral multiple of the wavelength.
Let's consider the mth and (m + 1)th rings for wavelength 1 (λ1) and wavelength 2 (λ2), respectively.
For the mth ring of wavelength 1, the path difference can be given by:
Δx₁ = (2m + 1) * λ₁ / 2
For the (m + 1)th dark ring of wavelength 2, the path difference can be given by:
Δx₂ = m * λ₂
As per the given condition, the two path differences are equal. Hence, we have the equation:
(2m + 1) * λ₁ / 2 = m * λ₂
Now, let's substitute the given values:
λ₁ = 7500 Å (Angstrom) = 7500 * 10^(-10) m
λ₂ = 5000 Å (Angstrom) = 5000 * 10^(-10) m
After substituting these values, we get:
(2m + 1) * 7500 * 10^(-10) / 2 = m * 5000 * 10^(-10)
Simplifying the equation, we have:
(2m + 1) * 7500 = 2m * 5000
Now, solve for the value of m:
15000m + 7500 = 10000m
5000m = 7500
m = 1.5
Since m represents the number of rings, it cannot be a fraction. Hence, we can take the closest integer value, which is 2.
Now, we can calculate the radius of the mth ring using the formula:
R = (m * λ * R²) / (2 * (R - t))
Here, R is the radius of curvature of the lens, λ is the wavelength, and t is the thickness of the thin air film.
Given that R = 0.5 m, λ = λ₁ = 7500 * 10^(-10) m, and t is extremely small compared to R (considered negligible), we can simplify the formula as follows:
R = (m * λ * R²) / (2 * R)
0.5 = (2 * λ * R) / 2
Now, with all the values, we can find the ring diameter (d):
d = 2 * R = 2 * (0.5) = 1 meter
Therefore, the ring diameter in this experiment is 1 meter.