Problem 27 - Master Resource Book in JEE Main Physics

Question

A ray of light makes an angle of \(10^{\circ}\) with the horizontal above it and strikes a plane mirror which is inclined at an angle \(\theta\) to the horizontal. The angle \(\theta\) for which the reflected ray becomes vertical is (a) \(50^{*}\) (b) \(80^{*}\) (c) \(100^{\circ}\) (d) \(4 \overline{0^{*}}\)

Step-by-Step Solution

Answer

The angle \( \theta \) is \( 50^{\circ} \).

Step 1: Understanding the Problem

A ray of light strikes a plane mirror and reflects off it. We need to find the angle \( \theta \), the inclination of the mirror to the horizontal, such that the reflected ray becomes vertical.

Step 2: Analyzing Angles

The given angle of incidence with the horizontal is \( 10^{\circ} \). The mirror's inclination angle to the horizontal is \( \theta \). Therefore, the angle of incidence relative to the mirror's surface is \( \theta - 10^{\circ} \).

Step 3: Using Reflection Law

The law of reflection tells us that the angle of incidence equals the angle of reflection. Thus, the angle of reflection is also \( \theta - 10^{\circ} \). Since the reflected ray is vertical, this angle must complement \( 90^{\circ} \).

Step 4: Setting Up Equation

To make the reflected ray vertical, the sum of the reflected angle and \( 10^{\circ} \) must be \( 90^{\circ} \). Therefore, we set up the equation: \(( \theta - 10^{\circ} ) + \theta = 90^{\circ} \).

Step 5: Solving the Equation

Simplifying the equation: \( 2\theta - 10^{\circ} = 90^{\circ} \). Solve for \( \theta \) by adding \( 10^{\circ} \) to both sides to get \( 2\theta = 100^{\circ} \). Then divide by 2 to find \( \theta = 50^{\circ} \).

Step 6: Conclusion

We found the angle \( \theta \) to be \( 50^{\circ} \). Thus, the correct answer is \( \text{(a) } 50^{*} \).

Key Concepts

Angle of Incidence

The angle of incidence is a fundamental concept in optics, referring to the angle at which a ray of light strikes a surface. In practical terms, it is the angle between the incoming light ray and a line perpendicular to the surface, known as the normal.

Calculating the angle of incidence is crucial in understanding how light interacts with surfaces. In our exercise, the ray of light makes a given angle with the horizontal, but when it strikes a mirror, we are more concerned with its angle relative to the mirror's surface.

• The angle of incidence changes depending on the orientation of the surface.
• In the case of a mirror inclined at angle \( \theta \) to the horizontal, the angle of incidence is \( \theta - 10^{\circ} \), assuming the light ray initially makes a \( 10^{\circ} \) angle with the horizontal.
• This angle determines how the light will reflect off the surface, pivotal in predicting the path of the light after reflection.

Angle of Reflection

The angle of reflection is directly connected to the angle of incidence due to the laws governing light behavior. It is defined as the angle between the reflected ray and a line perpendicular to the surface from which the ray bounces off.

According to the law of reflection, the angle of reflection always equals the angle of incidence.

• In our exercise, if the angle of incidence is \( \theta - 10^{\circ} \), then the angle of reflection is also \( \theta - 10^{\circ} \).
• The reflected ray's direction is determined by this equality, ensuring that the behavior of light can be predicted accurately when interacting with reflective surfaces.
• The importance of this concept cannot be understated, as it forms the basis for further understanding of optical phenomena involving mirrors and refraction.

Law of Reflection

The law of reflection is a simple yet powerful principle in optics that states that the angle of incidence equals the angle of reflection. This law is universally applicable to smooth, flat surfaces like plane mirrors and is vital for determining light paths upon reflection.

• The principle can be expressed mathematically as: \( \angle_i = \angle_r \), where \( \angle_i \) is the angle of incidence and \( \angle_r \) is the angle of reflection.
• It ensures symmetry in the behavior of light, explaining why images are seen in mirrors at predictable angles.
• As applied in our problem, this law helps us set up necessary equations to determine the angle \( \theta \) of the mirror so that the reflected ray is vertical.

Applying the law, we conclude the setup that ensures the trajectory of light meets specific conditions provided by additional factors such as other angles involved.

Plane Mirror

A plane mirror is a flat, smooth reflective surface. It is the simplest form of a mirror and is widely used in various optical devices due to its straightforward reflection properties.

In our exercise, the plane mirror plays a crucial role. It is inclined at an angle \( \theta \) to the horizontal, influencing the angle of incidence directly.

• This setup allows us to explore how changing the orientation of a mirror alters the path of a reflected light ray.
• Plane mirrors obey the fundamental law of reflection, demonstrating predictable results in light behavior.
• They help us explore principles of symmetry and angle manipulation, essential in fields like physics and engineering.

By understanding how plane mirrors work, students learn the basics of more complex systems such as lenses and compound mirrors used in telescopes or cameras.