Question

For an isosceles prism of angle \(A\) and refractive index \(\mu\), it is found that the angle of minimum deviation \(\delta_{m}=A\). Which of the following options is/are correct? [A] For the angle of incidence \(i_{1}=A\), the ray inside the prism is parallel to the base of the prism [B] For this prism, the refractive index \(\mu\) and the angle of prism \(A\) are related as \(A=\frac{1}{2} \cos ^{-1}\left(\frac{\mu}{2}\right)\) [C] At minimum deviation, the incident angle \(i_{1}\) and the refracting angle \(r_{1}\) at the first refracting surface are related by \(r_{1}=\left(i_{1} / 2\right)\) [D] For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is \(i_{1}=\sin ^{-1}\left[\sin A \sqrt{4 \cos ^{2} \frac{A}{2}-1}-\cos A\right]\)

Short answer

Correct options: [A]

Step-by-step solution

Recall the Prism Formula

For an isosceles prism at minimum deviation, the angle of incidence equals the angle of emergence and the refracted ray inside the prism is parallel to the base. At minimum deviation, \( i_1 = e \) and \( r_1 = r_2 = \frac{A}{2} \) where \( i_1 \) is the angle of incidence and \( e \) is the angle of emergence.

Analyze Option [A]

Given that \( \delta_m = A \) and applying the prism formula, for the ray inside the prism to be parallel to the base, \( i_1 \) must be equal to \( A \) when \( \delta_m \) is at its minimum. Hence, the condition \( i_1 = A \) indicates that the ray is parallel to the base which justifies option [A].

Analyze Option [B]

The relationship between \( \mu \) and \( A \) at minimum deviation when \( \delta_m = A \) is not given as \( A = \frac{1}{2} \cos^{-1}\left(\frac{\mu}{2}\right) \) but rather derived from Snell's law \( \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \) which simplifies to \( \mu = \sin(A) \) under the given conditions. Therefore, option [B] is incorrect.

Analyze Option [C]

At minimum deviation, \( r_1 = \frac{A}{2} \) which is not necessarily half of \( i_1 \) unless \( i_1 = A \) as stated in option [A]. Thus, the general relationship claimed in option [C] is incorrect.

Analyze Option [D]

Calculating the specific angle of incidence \( i_1 \) needed for the emergent ray to be tangential to the second surface involves complex trigonometric equations that would not simplify to the expression given in option [D]. Thus, without further specific calculations and a clear derivation, we assume that [D] is not a direct implication of the given condition and not necessarily correct.

Key concepts

Refractive Index

The refractive index, often denoted by \( \mu \), is a fundamental optical property of a transparent medium. It quantifies how much the speed of light is reduced in the medium compared to the speed of light in a vacuum. The refractive index of a material determines how much light bends, or refracts, when it enters the material from another medium. For example, common glass has a refractive index around 1.5, which means light travels 1.5 times slower in glass than in vacuum.

Understanding the refractive index is crucial in studying optics, particularly when dealing with prisms, as it helps in predicting the path of light through different media. Snell's law, which we will discuss later, uses the refractive index to describe the relationship between angles of incidence and refraction.

Isosceles Prism Optics

In optics, an isosceles prism is a type of prism which has two equal sides and angles. The most significant angle in prism optics is the prism's apex angle \( A \), the angle between the two equal sides. When light enters such a prism, its path depends on this apex angle and the material's refractive index.

An important feature of prisms is their ability to deviate light from its original path. The angle of minimum deviation \( \delta_m \) occurs where light's entrance and exit angles are equal, making the light's path parallel to the prism's base. This specific condition not only simplifies many calculations but also is essential for applications like spectroscopy, where precise control over light paths is required.

Angle of Incidence

The angle of incidence is the angle at which a ray of light strikes a surface. It is always measured relative to the normal to the surface—an imaginary line perpendicular to the point where the light hits the surface. In the context of an isosceles prism, when light enters at the angle of incidence \( i_1 \) that results in minimum deviation, the light inside the prism travels parallel to the base of the prism. This incident angle is significant, as it sets the conditions for various phenomena like total internal reflection and polarization.

Snell's Law

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction when a light wave passes from one medium into another. It can be mathematically expressed as \( \mu = \frac{\sin(i)}{\sin(r)} \) where \( \mu \) is the refractive index of the second medium relative to the first, \( i \) is the angle of incidence, and \( r \) is the angle of refraction.

When considering an isosceles prism at minimum deviation, Snell's law becomes particularly simple since the incident and emergent angles are equal, and therefore the refracted ray inside the prism is parallel to the base. This provides us with a direct way to relate the refractive index of the prism to the prism's apex angle and minimum deviation.

Angle of Emergence

The angle of emergence \( e \) is the angle at which light exits a medium, measured from the normal to the surface at the point of exit. In the scenario of an isosceles prism, at the angle of minimum deviation, this angle of emergence equals the angle of incidence \( i_1 \), meaning the path of light entering and leaving the prism is symmetrical. This symmetry simplifies the understanding and calculation of the light's path through the prism, making it a key concept when studying prisms and their optical properties.