Light from benzene with a refractive index 1.501 travel into glycerine with refractive index 1.473. The system is held at a constant temperature of 20 degrees Celsius. Calculate the angle of refraction if the incident angle is 150 CÂÂ
The Correct Answer and Explanation is :
To calculate the angle of refraction when light passes from benzene ((n_1 = 1.501)) into glycerine ((n_2 = 1.473)), we use Snell’s Law:
[
n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
]
Step-by-Step Solution
- Given:
- Refractive index of benzene, (n_1 = 1.501)
- Refractive index of glycerine, (n_2 = 1.473)
- Incident angle, (\theta_1 = 150^\circ) Convert the incident angle to radians for accurate calculations if necessary:
[
\theta_1 = 150^\circ
]
- Snell’s Law Equation:
Rearrange Snell’s Law to solve for (\theta_2):
[
\sin(\theta_2) = \frac{n_1}{n_2} \sin(\theta_1)
] - Substitute Values:
[
\sin(\theta_2) = \frac{1.501}{1.473} \sin(150^\circ)
] (\sin(150^\circ) = \sin(180^\circ – 30^\circ) = \sin(30^\circ) = 0.5):
[
\sin(\theta_2) = \frac{1.501}{1.473} \times 0.5 = 0.5094
] - Find (\theta_2):
Take the inverse sine:
[
\theta_2 = \arcsin(0.5094) \approx 30.6^\circ
]
Final Answer:
The angle of refraction is approximately 30.6°.
Explanation (300 Words)
When light transitions between two media with different refractive indices, its speed and direction change due to refraction. The relationship between the incident and refracted angles is governed by Snell’s Law, which ensures the continuity of wavefronts at the boundary.
In this problem, light moves from benzene ((n_1 = 1.501)) into glycerine ((n_2 = 1.473)), both of which have similar refractive indices since they are dense liquids. The incident angle, (150^\circ), is measured from the normal to the surface. Since (\sin(150^\circ) = 0.5), we calculate the ratio of refractive indices to determine how the light bends.
The calculated refracted angle, (30.6^\circ), indicates that the light bends slightly toward the normal since glycerine has a slightly lower refractive index than benzene. The bending is subtle because the refractive indices are close in value.
This principle is crucial in optics, as it explains phenomena like lensing and the behavior of light in various media. The result also emphasizes that even small changes in refractive index can significantly affect the direction of light propagation, making this calculation important for designing optical systems such as lenses, fiber optics, and imaging devices.