Problem 29 - Master Resource Book in JEE Main Physics

Question

It is necessary to illuminate the bottom of a well by reflected solar beam when the light is incident at an angle of \(\alpha=40^{\circ}\) to the vertical. At what angle \(\beta\) to the horizontal should a plane mirror be placed? (a) \(70^{\circ}\) (b) \(20^{*}\) (c) \(50^{\circ}\) (d) \(40^{\circ}\)

Step-by-Step Solution

Answer

The mirror should be placed at an angle of \(50^{\circ}\) to the horizontal (option c).

Step 1: Understanding the Problem

We're given an incident light angle \( \alpha = 40^{\circ} \) with respect to the vertical and we need to find the angle \( \beta \) at which a plane mirror must be inclined to the horizontal to reflect this light down vertically into the well.

Step 2: Reflection Law Application

According to the law of reflection, the angle of incidence \( \theta_i \) is equal to the angle of reflection \( \theta_r \). In this case, \( \theta_i \) with the vertical is \( 40^{\circ} \), which means the angle of incidence with the normal line (perpendicular to the mirror surface) is also \( 40^{\circ} \).

Step 3: Determining Angles Related to Mirror

Since the mirror reflects the light directly downward, the angle between the reflected ray and the horizontal must be \( 90^{\circ} \). Because the incident angle with respect to the vertical is \( 40^{\circ} \), the light must also be reflected such that it complements this angle to point directly down, meaning reflected at \( 40^{\circ} \) with respect to the vertical line or \( 50^{\circ} \) with respect to the horizontal.

Step 4: Finding \(\beta\)

To solve for \( \beta \), note that the mirror forms an angle of \( 50^{\circ} \) with the horizontal plane, since it must reflect the beam at the same angle downward. Therefore, the angle \( \beta \) at which the mirror is to be inclined to the horizontal is \( 50^{\circ} \).

Key Concepts

Law of Reflection

The law of reflection is a fundamental principle in optics, describing how light behaves when it hits a surface. It states that the angle of incidence \( \theta_i \) is always equal to the angle of reflection \( \theta_r \). These angles are measured with respect to the normal line, which is an imaginary line perpendicular to the surface at the point of contact with light.

This law holds true for all types of surfaces, whether it's a shiny glass or a dull texture. When light strikes a surface, the beam that hits the surface is called the incident ray and the light that bounces off is called the reflected ray.

• The angle of incidence is the angle formed between the incident ray and the normal.
• The angle of reflection is the angle formed between the reflected ray and the normal.

This symmetry results in consistent reflection patterns. This principle is essential for designing various optical devices like mirrors, periscopes, and many other gadgets that rely on directing light efficiently.

Plane Mirror Optics

Plane mirrors are flat mirrors that represent one of the simplest forms of optics. They reflect light using the law of reflection, keeping the angle of incidence equal to the angle of reflection.

A plane mirror reflects light in such a way that:

• The image formed is virtual, as it cannot be projected on a screen.
• The image is the same size as the object.
• The distance from the object to the mirror is equal to the distance from the image to the mirror.
• It maintains left-right reversal, meaning what’s on the left appears on the right in the image.

Using these characteristics, plane mirrors are utilized in everyday objects like bathroom mirrors and vehicle rear-view mirrors. In the context of optics problems, like the original exercise, understanding how to manipulate the mirror to achieve the desired light direction is crucial.

Angles of Incidence and Reflection

Angles of incidence and reflection are key concepts in understanding how light interacts with surfaces. These angles determine the path that light will take when it strikes a reflective surface.
The angle of incidence \( \theta_i \) and the angle of reflection \( \theta_r \) are:

• Measured from the normal line at the point of contact, not the surface.
• Equal to each other as per the law of reflection, \( \theta_i = \theta_r \).

In the original problem, an incident angle of \( 40^{\circ} \) relative to the vertical was given. Understanding how to convert this to an angle with respect to the horizontal involves basic trigonometry and visualization:

• If light is incident at \( 40^{\circ} \) to the vertical, it makes a \( 50^{\circ} \) angle with the horizontal plane because the total must equal \( 90^{\circ} \).

Working with these angles helps in tasks like orienting plane mirrors efficiently so that light beams hit their intended targets, such as in illuminating the bottom of a well with a solar beam.