Multiple Images with Plane Mirrors

Consider two adjacent plane mirrors inclined at an angle. Multiple images are formed when an object is placed between the mirrors. The number of images formed depends on two factors (i) the angle $\theta$ between the mirrors and (ii) whether the object is placed symmetrically or asymmetrically between the mirrors.

Formula for number of images

The numbers of image formed by two inclined plane mirrors is calculated by the following formula:

Step 1: Calculate $360/\theta$ (angle $\theta$ is in degrees). If $360/\theta$ is not an integer then get the greatest integer less than $360/\theta$ i.e., \begin{align} n=\left\lfloor\frac{360}{\theta \text{(in degrees)}}\right\rfloor \end{align} For example, (i) if $\theta=60^\circ$ then $360/60=6$ and $n=6$ (ii) if $\theta=50^\circ$ then $360/50=7.2$ and $n=7$ (iii) if $\theta=75$ then $360/\theta=4.8$ and $n=4$.

Step 2: The number of images is $n$ if $n$ is an odd integer and the object is asymmetrically placed between the mirrors. Otherwise, the number of images is $n-1$.

If mirrors are placed parallel to each other then $\theta=0$ degrees. This type of mirrors arrangement can be found in barber's shop. Infinite number of images are formed in this case.

Solved problems

Problem 1: Calculate the number of images formed when an object is placed between two plane mirrors kept perpendicular to each othet?

Solution: The angle between the mirrors is $\theta=90$ degrees. Calculate $360/90=4$ (an integer). Thus, $n=4$. Since $n=4$ is an even integer, the number of images is $n-1=3$. This is irrespective of whether object is placed symmetrically or asymmetrically.

Problem 2: Calculate the number of images formed when angle between two plane mirrors is 75 degree?

Solution: Calculate $360/75=4.8$ (not an integer). Thus, $n=4$. Since $n=4$ is an even integer, the number of images is $n-1=3$.

Problem 3: Calculate the number of images formed when angle between two plane mirrors is 50 degree?

Solution: Calculate $360/50=7.2$ (not an integer). Thus, $n=7$, an odd integer. The number of images in this case also depend on the object's location. If the object is symmetrically placed between the mirrors then the numbers of image is $n-1=6$. If the object is asymmetrically placed between the mirrors then the number of image is $n=7$.

Exercise: The figure shows two plane mirrors inclined at 40 degrees angle. Two objects A and B are placed between them as shown. Find the number of images of A and B.

Ray diagrams

Let a point object is placed in front of a plane mirror. A person standing in front of the plane mirror see the object's image in the plane mirror. The image size is same as that of the object. The image is formed at the same distance from the mirror as is the perpendicular distance of the object from the mirror. The image can be seen from any place in front of the mirror. Let us understand this through ray diagram.

The point object emits light in all directions. The rays striking the mirror gets reflected according to the laws of reflection. Some of the reflected rays reach the person's eye (eye-A). The person see reflected rays as if they are coming from a point source behind the mirror. This imaginary source is the virtual image of the object.

If the person shifts eye to another location (eye-B), he gets rays reflected from another point on the mirrors. These rays also appears to come from the same imaginary source. Thus, the person can see the image from all points in front of the mirror.

Now, consider two plane mirrors $M_1$ and $M_2$ making an angle $\theta=90^\circ$. Let an object is placed between these mirrors. Three images, $I_1, I_2$ and $I_3$ can be seen from any point between the mirros (Eye). Images $I_1$ and $I_2$ are formed in the mirros $M_1$ and $M_2$ respectively. These two images acts as virtual objects for image $I_3$. The ray diagram is shown in the figure.

We encourage you to draw ray diagrams for various situations.

Experiment

You need two plane mirrors, protractor to measure angle, candle and matchbox for this experiment.

Take two plane mirrors (without frame). Place both the mirrors side by side and fix the junction where they meet with a cello tape. Now you will be able to open and close the mirrors like a book. Place both the mirrors at a small angle apart in the upright position on the floor. Place a lighted candle in the space between the two mirrors. You will observe many images of the candle which makes a beautiful scene.

Now, by gradually decreasing the angle between the mirrors observe the images being formed. You will now observe more and more numbers of images of the candle. Similarly, if the angle between the mirrors is increased the number of images decreases and when this angle is 180 degree, only one image is visible.

When the angle between the two mirrors is 180 degree they together act like a single mirror so that only one image is visible. As the angle between the mirrors gradually decreased, not only the candle but the mirrors themselves get imaged in one another. That is why when the angle between the mirrors is decreased one observe image within image, and image within that image, and so on. In this way one observes a lot many images. If the angle between the mirrors is finally decreased to zero, infinite images are expected to be formed.

1. Place both the mirrors vertically with 180 degree angle between the them.
2. Place a lighted candle in the space between the two mirrors.
3. You will see single image of the candle in mirror. Reduce the angle between mirrors and observe multiple images of the candle which makes a beautiful scene.
4. Now, measure the maximum angle between two mirrors when you get 1, 3, 5, 7, and 9 images. These angles are $$\theta_\text{measured}$$.
5. Calculate $$\theta_\text{calculated}$$ for n=1,3,5,7 and 9 by using the formula given above.
6. Draw the ray diagram when angle between two mirrors is 90 degree.

Discussion

$n$ $\theta_\text{measured}$ $\theta_\text{calculated}$ $\Delta\theta_\text{error}$
1
3
5

Note

Fill a transparent glass tray with water and fix two similar mirrors at its opposite ends. See the image being formed. What you see is a very long water canal. Think how this happens?

This setup is also used to make a funny game. Make a cap with two strips of different colors (say black and white) pasted on the left and the right side. Adjust the angle between the mirrors so that you see two images of yourselves (with cap and strips clearly visible). Ask your friend to wear the cap and see his image in the mirrors. Order him to touch the white or black strip quickly. There is a great chance that your friend gets confused and end up touching wrong color strip. Why this happened?

Reference

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