To obtain three images when an object is placed between two plane mirrors, the mirrors must be positioned at an angle of 90 degrees to each other.
Understanding Image Formation with Inclined Mirrors
When two plane mirrors are placed at an angle to one another, they produce multiple images of an object positioned between them. This phenomenon occurs due to successive reflections of light rays between the mirror surfaces. Each mirror not only reflects the original object but also reflects the images formed by the other mirror, leading to a series of images.
The number of images ($N$) formed by two plane mirrors inclined at an angle $\theta$ is generally determined by the formula:
$N = \frac{360^\circ}{\theta} - 1$
This formula is valid under specific conditions: it works when $\frac{360^\circ}{\theta}$ is an even integer, or when $\frac{360^\circ}{\theta}$ is an odd integer and the object is placed symmetrically on the angle bisector. If $\frac{360^\circ}{\theta}$ is an odd integer and the object is placed asymmetrically, the number of images is simply $\frac{360^\circ}{\theta}$.
However, for the specific requirement of three images, the condition for the angle is straightforward:
Let's use the formula to find the angle for three images:
-
Set the number of images ($N$) to 3:
$3 = \frac{360^\circ}{\theta} - 1$ -
Add 1 to both sides of the equation:
$3 + 1 = \frac{360^\circ}{\theta}$
$4 = \frac{360^\circ}{\theta}$ -
Solve for $\theta$:
$\theta = \frac{360^\circ}{4}$
$\theta = 90^\circ$
Thus, placing the two mirrors at a 90-degree angle will consistently result in the formation of three images of a single object situated between them.
How 90-Degree Mirrors Create Three Images
When mirrors are at 90 degrees, the reflections occur in a predictable pattern:
- First Images: The object produces an image in each of the two mirrors independently. Let's call them Image 1 (from Mirror A) and Image 2 (from Mirror B).
- Second-Order Image: Image 1, formed by Mirror A, acts as a virtual object for Mirror B, producing a third image. Simultaneously, Image 2, formed by Mirror B, acts as a virtual object for Mirror A, producing an image that coincides perfectly with the third image generated from Image 1. This overlapping image is counted as a single, third image.
This configuration effectively places the object and its three images at the vertices of a rectangle, with the mirrors forming two adjacent sides.
Common Mirror Angles and Image Counts
The number of images observed changes significantly with the angle between the mirrors. Here's a quick reference for common angles:
| Angle Between Mirrors ($\theta$) | $\frac{360^\circ}{\theta}$ | Number of Images ($N$) | Notes |
|---|---|---|---|
| $180^\circ$ | 2 | 1 | Mirrors form a straight line. |
| $120^\circ$ | 3 | 2 or 3 | 2 images if the object is symmetrically placed; 3 images if asymmetrical. |
| $90^\circ$ | 4 | 3 | Consistently three images regardless of object placement (for a point object between mirrors). |
| $60^\circ$ | 6 | 5 | |
| $45^\circ$ | 8 | 7 | |
| $0^\circ$ | - | Infinite | Mirrors are parallel, producing an endless series of images (e.g., in a barber shop). |
Practical Applications of Multiple Reflections
The principles governing image formation by inclined mirrors are not just theoretical; they have various real-world applications:
- Kaleidoscopes: These classic toys use multiple mirrors, often at 60-degree angles, to create complex and beautiful symmetrical patterns from small, colorful objects.
- Periscopes: While simple periscopes use two parallel mirrors or prisms for a single view, more advanced optical instruments can leverage multiple reflections for specific sightlines.
- Art and Illusions: Artists and designers use precisely angled mirrors to create optical illusions, expand apparent space, or generate mesmerizing visual effects in installations and architecture.
- Optical Testing: In precision optics, understanding and controlling reflections from angled surfaces is crucial for instrument calibration and alignment.
Experimenting with two plane mirrors and a small object at different angles can provide an insightful hands-on demonstration of these principles. You will clearly observe the formation of three images when the mirrors are set at a right angle.
For a deeper dive into the physics of light and reflection, you might find resources like HyperPhysics' section on mirrors or Wikipedia's article on reflection helpful.
