Problem 85 - Master Resource Book in JEE Main Physics

Question

The refractive index of a prism for a monochromatic wave is \(\sqrt{2}\) and its refracting angle is \(60^{\circ}\). For minimum deviation, the angle of incidence will be (a) \(3 \overline{0^{\circ}}\) (b) \(45^{*}\) (c) \(60^{\circ}\) (d) \(75^{\circ}\)

Step-by-Step Solution

Answer

The angle of incidence for minimum deviation is 45°. (Option b)

Step 1: Understanding Minimum Deviation

For a prism, the condition for minimum deviation occurs when the angle of incidence \(i\) is equal to the angle of emergence \(e\). At this condition, the light ray inside the prism travels symmetrically relative to the prism's geometry.

Step 2: Apply Snell's Law

Snell's Law in the context of the prism at minimum deviation gives us: \[ n = \frac{\sin(i)}{\sin(r)}\]where \(n\) is the refractive index of the prism, \(r\) is the angle of refraction. Given \(n = \sqrt{2}\), we get:\[\sqrt{2} = \frac{\sin(i)}{\sin(r)}.\]

Step 3: Angle Relationships in the Prism

For a prism, the relationship between angles at minimum deviation is:\[A = 2r\]where \(A\) is the prism's refracting angle. Given \(A = 60^\circ\), we find:\[60^\circ = 2r\Rightarrow r = 30^\circ.\]

Step 4: Calculate Angle of Incidence

Plugging \(r = 30^{\circ}\) into the modified Snell's Law equation, we solve for \(i\):\[\sqrt{2} = \frac{\sin(i)}{\sin(30^\circ)}\]\[\sqrt{2} = \frac{\sin(i)}{\frac{1}{2}}\]\[\sin(i) = \frac{\sqrt{2}}{2} = \sin(45^{\circ}).\]

Step 5: Conclusion and Answer Selection

Since \(\sin(i) = \sin(45^{\circ})\) implies that \(i = 45^{\circ}\), the correct answer for the angle of incidence at minimum deviation is 45 degrees. Thus, option (b) 45° is the right answer.

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in the study of optics that describes how light propagates through different media. When a light ray passes between two mediums with different refractive indices, like air and glass, Snell's Law helps us predict how the light will change direction.

In its simplest form, Snell's Law is expressed as: \[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the first and second mediums respectively,
• \( \theta_1 \) is the angle of incidence (angle the incoming ray makes with the normal to the surface),
• \( \theta_2 \) is the angle of refraction (angle the refracted ray makes with the normal to the surface).

In the context of the prism, Snell's Law helps us to find the relationship between the angles involved when the light ray passes through the prism.

Refractive Index

The refractive index is a measure of how much a light ray bends, or refracts, as it enters a different medium. It's crucial for understanding how prisms and lenses work, as it affects the angle and direction of incoming light.

The refractive index \( n \) is defined as the ratio of the speed of light in a vacuum to its speed in a given medium:\[ n = \frac{c}{v} \]where:

• \( c \) is the speed of light in a vacuum,
• \( v \) is the speed of light in the medium.

In our problem, the prism's refractive index is \( \sqrt{2} \). This means light travels slower in the prism than in a vacuum, which causes it to bend at the interfaces according to Snell's Law.

Angle of Incidence

The angle of incidence is the angle formed between the incoming light ray and the normal (an imaginary line perpendicular to the surface) at the point of incidence. It plays a critical role in determining how much the light will bend when it enters a new medium.

In the context of our prism problem, the angle of incidence is particularly important for identifying the condition of minimum deviation, which occurs when the path of light through the prism is symmetrical. This means the angle of incidence is equal to the angle of emergence when the deviation of the light ray inside the prism is minimal.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal at the point of exit from the medium. When light enters a new medium, it changes speed which causes it to change direction; the angle of refraction quantifies this change in direction.

In a prism, the relationship between the refracting angle \( A \), and the angle of refraction \( r \), is given by \( A = 2r \) in the condition of minimum deviation. This means the refracted light passes symmetrically, making the analysis simpler and leading to the conclusion that certain angles, like 45° in this case, result in minimum deviation based on calculated values of the angles involved inside the prism.