Problem 36 - Physics

Question

A ray of light strikes a plane mirror at a \(45^{\circ}\) angle of incidence. The mirror is then rotated by \(15^{\circ}\) into the position shown in red in the drawing, while the incident ray is kept fixed. (a) Through what angle \(\phi\) does the reflected ray rotate? (b) What is the answer to part (a) if the angle of incidence is \(60^{\circ}\) instead of \(45^{\circ} ?\)

Step-by-Step Solution

Answer

(a) The reflected ray rotates by \( 30^{\circ} \). (b) With a \( 60^{\circ} \) angle of incidence, it also rotates by \( 30^{\circ} \).

Step 1: Understanding the Law of Reflection

The law of reflection states that the angle of incidence \( \theta_i \) is equal to the angle of reflection \( \theta_r \). This means that when a ray of light strikes a mirror, the angle it makes with the normal on the way in is the same as the angle it makes on the way out.

Step 2: Analyzing Initial Conditions

Initially, the angle of incidence is \( 45^{\circ} \), so the angle of reflection is also \( 45^{\circ} \). This holds because of the law of reflection.

Step 3: Effect of Rotating the Mirror

When the mirror is rotated by \( 15^{\circ} \), the angle between the incident ray and the normal to the mirror changes by \( 15^{\circ} \). Consequently, the angle of reflection must also change by \( 15^{\circ} \). Therefore, the reflected path rotates by twice this amount.

Step 4: Calculating the Rotation of the Reflected Ray (a)

Since the angle of incidence changes by \( 15^{\circ} \), the reflected ray rotates through an angle \( \phi = 2 \times 15^{\circ} = 30^{\circ} \).

Step 5: Analyzing Second Scenario for Part (b)

Now, consider the angle of incidence \( 60^{\circ} \). The mirror is again rotated by \( 15^{\circ} \). Here too, the change in incidence angle is \( 15^{\circ} \). Due to the law of reflection, the rotation of the reflected ray is still twice the angle the mirror is rotated.

Step 6: Calculating the Rotation of the Reflected Ray (b)

With the angle of incidence at \( 60^{\circ} \), the rotated reflected ray is still \( 2 \times 15^{\circ} = 30^{\circ} \). This result remains consistent regardless of the angle of incidence as long as the mirror rotation remains the same.

Key Concepts

Angle of Incidence

Imagine a ray of light striking a smooth mirror. The angle formed between this ray and an imaginary line perpendicular to the mirror's surface is called the "angle of incidence." This angle is fundamental in understanding how light behaves upon striking surfaces. According to the **Law of Reflection**, the angle of incidence \( \theta_i \) is always equal to the angle of reflection \( \theta_r \).

When you see a light ray approach a mirror at \( 45^{\circ} \), it will reflect off at the same \( 45^{\circ} \) on the other side of the normal line. This similarity in angles ensures that incoming and outgoing paths are predictable.

Knowing the angle of incidence allows us to anticipate how light will reflect and interact with its environment, making it crucial for optical applications and problem-solving in physics.

Plane Mirror

A plane mirror is a flat, smooth reflective surface. Its uniformity means any light ray hitting the mirror will reflect symmetrically. When we say the mirror is "plane," it implies no curvature; hence, the light behaves predictably according to the angles described.

Due to its flatness:

• Images appear right-side up and of equal size as the object.
• There is a one-to-one correspondence between the angle of incidence and the angle of reflection.

The position of the plane mirror can be adjusted; for instance, if the mirror rotates by \(15^{\circ}\), the reflection conditions will change slightly. In our scenario, even a simple adjustment in the mirror's position can lead to varied results in the behavior of the reflected ray. By keeping such a mirror consistent and flat, the learner can focus on the relationships between angles rather than complicated surface interactions.

Reflected Ray Rotation

When you rotate a plane mirror, the reflected ray doesn't just shift slightly; it undergoes a "rotation" effect. This happens because the angles of incidence and reflection change concurrently as you adjust the mirror.

To understand this better:

• If the mirror rotates by an angle \( \theta \), the reflected ray rotates twice that angle, given by \( \phi = 2 \theta \).
• This is why rotating the mirror \( 15^{\circ} \) results in the reflected ray rotating \( 30^{\circ} \).

This doubling effect occurs because both the incoming and outgoing paths are affected. Hence, even minimal changes to the mirror's orientation can yield significant adjustments in the reflection path, a critical concept for precise optical applications. Whether the angle of incidence is \( 45^{\circ} \) or \( 60^{\circ} \), as long as the mirror is rotated \( 15^{\circ} \), the reflection always rotates \( 30^{\circ} \).