Question

For a prism of refractive index $\sqrt{3}$, the angle of minimum deviation is equitation is equal to the angle of prism, then angle of the prism is (A) ${60}^{\circ }$ (B) ${90}^{\circ }$ (C) ${45}^{\circ }$ (D) ${180}^{\circ }$

Short answer

The angle of the prism is ${60}^{\circ }$ (A).

Step-by-step solution

Snell's Law equation for first refraction

Snell's law states that the ratio of the sines of the angles of incidence and refraction is equal to the refractive index. Let's denote the angle of incidence as $i_1$ and the angle of refraction as $r_1$: $$\frac{\sin i_1}{\sin r_1}=\sqrt{3}$$

Snell's Law equation for second refraction

The second refraction occurs when the light ray exits the prism. Let's denote the angle of incidence as $i_2$ and the angle of refraction as $r_2$: $$\frac{\sin i_2}{\sin r_2}=\frac{1}{\sqrt{3}}$$

Combine equations and apply the condition

We now combine the two Snell's law equations. Since the angle of minimum deviation is equal to the angle of the prism, we can use the following relationships: $i_1+i_2=A+\delta$, $r_1+r_2=A$, and since the angle formed by the refracted rays inside the prism at the minimum deviation is supplementary to the angle of the prism, $r_1=i_2$. Substitute $i_2=r_1$ in the equation: $$\frac{\sin i_1}{\sin r_1}=\frac{\sin r_1}{\sin r_2}$$

Solve for the angle of the prism

To find the angle of the prism, we need to solve the equation above for $r_1$: $${\sin }^2{r}_1=\sin i_1\sin r_2$$ Using the fact that ${\sin }^{2}u=\frac{1-\cos 2u}{2}$, we rewrite the equation: $$\frac{1-\cos 2r_1}{2}=\frac{\sin i_1(1-\cos 2r_1)}{2}$$ $$\cos 2r_1=\sin i_1-1$$ Now, for the minimum deviation condition to hold, we have $r_1=A$. Thus, $$\cos 2A=\sin i_1-1$$ Since we know ${\sin }^2{i}_1=3{\sin }^{2}A$, we can replace ${\sin }^2{i}_1$ with $3{\sin }^{2}A$ and re-write the equation: $$\cos 2A=\sqrt{3{\sin }^{2}A}-1$$ $$\cos 2A=\sqrt{3(1-{\cos }^{2}A)}-1$$ Solving this equation, we find the angle of the prism A: $A={60}^{\circ }$ So, the correct answer is: (A) ${60}^{\circ }$

Key concepts

Snell's Law

In the field of optics, Snell's Law is fundamental for understanding how light behaves when it passes from one medium to another. This law, named after Willebrord Snellius, describes how the angle of incidence relates to the angle of refraction. The equation is given by:$$\frac{\sin {\theta }_1}{\sin {\theta }_2}=n$$Where:

• ${\theta }_1$ is the angle of incidence (the angle the incoming ray makes with the normal).
• ${\theta }_2$ is the angle of refraction (the angle the refracted ray makes with the normal).
• $n$ is the refractive index, indicating how much the light slows down in the medium compared to vacuum.

Understanding Snell's Law is essential for predicting how light will change direction when it enters a new medium with a different refractive index. This change in direction is why we see phenomena like bending of light in prisms or the apparent bending of straws in water.

Refractive Index

The refractive index ($n$) is a measure that describes how fast light travels through a material. It is a dimensionless number that helps to quantify the bending of light rays as they pass between different media. For air, the refractive index is close to 1, meaning light travels at nearly its maximum speed as in a vacuum. For other substances like water or glass, the refractive index is greater than 1, indicating light slows down.Key points about refractive index:

• It determines the angle of refraction when light enters a material.
• Higher refractive indices indicate a greater bending effect.
• It varies with wavelength, leading to dispersion, especially in prisms.

In our exercise, the refractive index of the prism is \( \sqrt{3} \), suggesting that light bends more significantly compared to its journey in air.

Angle of Prism

The angle of a prism ($A$) is the angle between its two refracting surfaces. This angle determines the path and behavior of light as it passes through the prism. A standard triangular prism has its own characteristic angles that greatly influence optical phenomena like the minimum angle of deviation.In optics, when describing the behavior of light through a prism, we often calculate when the angle of minimum deviation occurs. This angle is crucial as it corresponds to the situation where light's path through the prism is symmetric.In the problem, the angle of the prism is essential because it equals the angle of minimum deviation, leading us to find:

• $A={60}^{\circ }$, a choice that both satisfies Snell's Law and the conditions of minimum deviation.

Understanding such calculations helps appreciate how prisms can visually demonstrate principles of refraction and light dispersion.

Optics

Optics is the branch of physics that deals with the behavior and properties of light and its interactions with matter. This field covers various phenomena, including reflection, refraction, and dispersion. Key principles in optics include:

• Reflection: Light bounces back when it hits a surface.
• Refraction: Light bends as it enters a different medium, governed by Snell's Law.
• Dispersion: Light separates into different colors, as seen with a prism.

Practical applications of optics are found everywhere, from simple devices like eyeglasses to complex technologies like lasers and optical fibers. Understanding optics allows us to design these and utilize light efficiently across various fields.