prove that when a plane mirror is rotated through any angle, the reflected ray is rotated through twice that angle.
To prove that when a plane mirror is rotated through any angle, the reflected ray is rotated through twice that angle, we can follow these steps:
Step 1: Visualize the scenario
Consider a plane mirror placed on a horizontal surface. Let's assume a incident ray of light, also known as the incident beam, is incident on the mirror at any angle. The mirror is then rotated through an angle. We want to prove that the reflected ray produced by the mirror will be rotated through twice that angle of rotation.
Step 2: Understand reflection and angles
In order to understand the relationship between the angle of incidence, angle of reflection, and the angle of rotation, we need to know a few principles related to reflection:
- The angle of incidence (i) is the angle between the incident ray and the normal (perpendicular) to the mirror's surface at the point of incidence.
- The angle of reflection (r) is the angle between the reflected ray and the normal to the mirror's surface at the point of reflection.
- The angle of rotation (θ) is the angle through which the mirror is rotated.
Step 3: Apply the law of reflection
According to the law of reflection, the angle of incidence is equal to the angle of reflection when a light ray strikes a mirror.
Therefore, i = r.
Step 4: Proving the relationship
Now, let's consider the scenario where the mirror is rotated through an angle θ. The incident ray will also be rotated along with the mirror. Let's call the rotated incident ray "ray I" and the reflected ray "ray R."
We can observe that when the mirror is rotated, the angle between ray I and the normal (i) will be equal to the angle between ray R and the normal (r).
Since i = r (as per the law of reflection), it implies that both angles are equal:
i = r.
Next, we need to show that the reflected ray is rotated through twice the angle of rotation (θ).
By rotating the mirror through an angle θ, the incident ray (ray I) will also be rotated through the same angle. This implies that the angle of incidence (i) will still be equal to the angle of rotation (θ). Therefore, i = θ.
Since i = r, we can conclude that r = θ.
Hence, the reflected ray (ray R) is rotated through twice the angle of rotation. This can be expressed as:
r = 2θ.
We have proved that when a plane mirror is rotated through any angle, the reflected ray is rotated through twice that angle (2θ).