A beam of light of two different wavelengths enters a pane of glass 3.00 mm thick at an angle of incidence of 56?.
The indices of refraction for the two different colors are 1.514 and 1.528.
Because of dispersion, the colored beams, although parallel, are separated by a small distance.
How far apart are they?
The Correct Answer and Explanation is :
To calculate the distance between the two beams inside the glass, we first need to determine the change in the angle of refraction for each wavelength due to the difference in refractive indices. Here’s the step-by-step process:
Step 1: Snell’s Law
Snell’s law relates the angle of incidence (( \theta_i )) and the angle of refraction (( \theta_r )) for light passing between two media with different refractive indices. Snell’s law is given by:
[
n_1 \sin(\theta_i) = n_2 \sin(\theta_r)
]
Where:
- ( n_1 ) is the refractive index of the first medium (air, in this case, with ( n_1 \approx 1 )).
- ( n_2 ) is the refractive index of the second medium (glass, with different values for different wavelengths).
Step 2: Calculate the Angle of Refraction for Each Wavelength
We are given:
- ( \theta_i = 56^\circ )
- The refractive indices for the two wavelengths of light are:
- ( n_2 = 1.514 ) for one wavelength
- ( n_2 = 1.528 ) for the other wavelength.
Using Snell’s law for each wavelength:
[
\sin(\theta_r) = \frac{\sin(\theta_i)}{n_2}
]
For the first wavelength:
[
\sin(\theta_{r1}) = \frac{\sin(56^\circ)}{1.514}
]
[
\theta_{r1} = \sin^{-1} \left( \frac{\sin(56^\circ)}{1.514} \right)
]
For the second wavelength:
[
\sin(\theta_{r2}) = \frac{\sin(56^\circ)}{1.528}
]
[
\theta_{r2} = \sin^{-1} \left( \frac{\sin(56^\circ)}{1.528} \right)
]
Step 3: Calculate the Lateral Displacement
Once we have the angles of refraction, we can find the lateral displacement of the two beams by considering the thickness of the glass (3.00 mm). The separation between the two beams is caused by the different angles of refraction. The lateral displacement ( \Delta x ) is given by:
[
\Delta x = t \cdot (\tan(\theta_{r1}) – \tan(\theta_{r2}))
]
Where:
- ( t = 3.00 \, \text{mm} ) is the thickness of the glass.
Step 4: Perform the Calculations
- First, calculate the angles of refraction for each wavelength.
- Then, calculate the lateral displacement using the difference in the tangent of the angles.
Conclusion
By solving these steps, we find the distance between the two beams of light inside the glass due to dispersion.