Optics Concept Page - 4

Example

Image formation by a moving object in a plane mirror

Example: A plane mirror is placed YZ plane facing towards negative X axis. The mirror is moving parallel to Y axis with a speed of 5 cm/s$5cm/s$. A point object P is moving in front of the mirror with a velocity (3i^+4j^+5k^) cm/s$(3\hat{i}+4\hat{j}+5\hat{k})cm/s$ . Find the velocity of image with respect to the mirror.

Solution:
Let the velocities of the object and the image with respect to mirror be V⃗ u${\overrightarrow{V}}_u$ and V⃗ v${\overrightarrow{V}}_v$ respectively.
Velocity of the mirror is 5j^cm/s$5\hat{j}cm/s$
So, V⃗ u=(3i^+4j^+5k^)−(5j^) cm/s=(3i^−1j^+5k^) cm/s${\overrightarrow{V}}_u=(3\hat{i}+4\hat{j}+5\hat{k})-(5\hat{j})cm/s=(3\hat{i}-1\hat{j}+5\hat{k})cm/s$
Components of V⃗ u${\overrightarrow{V}}_u$ and V⃗ v${\overrightarrow{V}}_v$ parallel to the mirror are same.
Therefore V⃗ v∥=(−1j^+5k^) cm/s${\overrightarrow{V}}_v\parallel =(-1\hat{j}+5\hat{k})cm/s$
Components of V⃗ u${\overrightarrow{V}}_u$ and V⃗ v${\overrightarrow{V}}_v$ perpendicular to the mirror are equal in magnitude but opposite in direction. Therefore V⃗ v⊥=(−3i^) cm/s${\overrightarrow{V}}_v\perp =(-3\hat{i})cm/s$
∴$\therefore$ V⃗ v=(−3i^−1j^+5k^)−(5j^) cm/s${\overrightarrow{V}}_v=(-3\hat{i}-1\hat{j}+5\hat{k})-(5\hat{j})cm/s$

Example

Problem on situation when mirror is rotated

Example: For the situations shown in fig, determine the angle by which the mirror should be rotated, so that the light ray will retrace its path after refraction through the prism and reflection from the mirror?

Solution:
The deviation suffered by the light ray is A−1μ=4−11.5=2o=θ${\displaystyle \frac{A-1}{\mu }}={\displaystyle \frac{4-1}{1.5}}=2^o=\theta$, say.
As we can notice from figure, the mirror must be rotated by an angle θ$\theta$ in clockwise direction so that it strikes normal to the mirror and retraces its path.
Hence the mirror must be rotated by an angle 2∘$2^{\circ }$ in clockwise direction.

Definition

Applications of Reflection in Daily Life

The most important application of reflection of light is that it makes things visible.
During rainbow formation light reflects and split into spectrum of colors.Colors would not be visible if there is no reflection.

Definition

Uses of Plane Mirror

Uses of Plane Mirror :
1. They are used as looking glass.
2. They are used in solar cookers.
3. They are also used in constructing periscope which is used in submarines.
4. They are also used to make kaleidoscope, a toy which produces beautiful patterns.
5. They are also used in various scientific instruments.

Diagram

Image formation by parallel plane mirrors

When an object placed is placed between two parallel plane mirrors, infinite images are formed as shown in the ray diagram.

Result

Images formed by mirrors placed parallel to each other

Considering an object placed in between the mirrors and we can actually see the image, in this case, the number of images would range from 1 to X (X being a very large number depending on your position relative to the mirror and object, and X not infinity).

Angle between mirrors = θ=0$\theta =0$

Hence,

Number of images formed = 360θ=∞${\displaystyle \frac{360}{\theta }}=\mathrm{\infty }$

So, case of parallel mirrors we get infinite images.

Example

Image formation by perpendicular plane mirrors

When an object is placed between two perpendicular plane mirrors, three images are formed as shown in the ray diagram.

Formula

Image formation by plane mirrors at an angle

Multiple images of are formed when an object is placed between two mirrors at an angle(θ$\theta$). Number of images formed is given by:
n=360oθ−1$n={\displaystyle \frac{{360}^o}{\theta }}-1$
where θ$\theta$ is in degrees.

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