Question

To get five images of a single object one should have two plane mirrors at an angle of (A) \(36^{\circ}\) (B) \(72^{\circ}\) (C) \(80^{\circ}\) (D) \(302^{\circ}\)

Short answer

The correct angle between two plane mirrors required to get five images of a single object is \(72^{\circ}\) (option B). This gives a fractional number of images (5.5) and rounds up to the required 5 image count.

Step-by-step solution

Write the formula for the number of images, n

Recall that the formula for the number of images, n, formed by two plane mirrors at an angle θ is: n = \( \frac{360^{\circ}}{\theta} \) - 1. We know that there are five images, so n = 5.

Substitute the known values in the formula and solve for θ

Substitute the known values (n = 5) in the formula: 5 = \( \frac{360^{\circ}}{\theta} \) - 1.

Isolate θ

To isolate θ, we will first add 1 to both sides of the equation: 6 = \( \frac{360^{\circ}}{\theta} \). Now, we will multiply both sides by θ: 6θ = 360^{\circ}. Finally, we will divide both sides by 6: θ = 60^{\circ}. Since 60^{\circ}\( is not an option in the choices given (A: \)36^{\circ},\( B: \)72^{\circ},\( C: \)80^{\circ},\( D: \)302^{\circ}$), we need to recall that the formula n = \( \frac{360^{\circ}}{\theta} \) - 1 is derived from the condition that the number of images is an integer value. If the number of images is fractional, we will round up to the nearest integer (in this case, 5 images). So, in our formula, we must take the smallest integer greater than or equal to n: \( ceil(n) = \frac{360^{\circ}}{\theta} \) - 1. Now we have: ceil(5) = \( \frac{360^{\circ}}{\theta} \) - 1.

Substitute the known values in the updated formula and solve for θ

Substitute the known values (ceil(n) = 5) in the formula: 5 = \( \frac{360^{\circ}}{\theta} \) - 1.

Isolate θ

To isolate θ, we will first add 1 to both sides of the equation: 6 = \( \frac{360^{\circ}}{\theta} \). Now, we will multiply both sides by θ: 6θ = 360^{\circ}. Finally, we will divide both sides by 6: θ = 60^{\circ}. Since 60^{\circ}\( is not an option, the closest option is (B) \)72^{\circ}\(, which gives a fractional number of images (5.5) and rounds up to the required 5 image count. So, the correct answer is (B) \)72^{\circ}$.

Key concepts

Plane Mirrors

A plane mirror is a flat, reflective surface that produces images by reflecting light in a predictable way. When an object is in front of a plane mirror, the image appears behind the mirror. This type of mirror is commonly used because it creates images that are virtual, upright, and of the same size as the object being reflected.

Key characteristics of images in plane mirrors include:

• The image is laterally inverted, meaning left and right are switched.
• There is no magnification or reduction in size. The image is the same size as the actual object.
• The image appears to be as far behind the mirror as the object is in front of it.

Understanding how plane mirrors work is crucial in many applications, such as in periscopes, kaleidoscopes, and everyday grooming mirrors.

Angle Between Mirrors

When two plane mirrors are placed at an angle to each other, they can create multiple images of a single object. The number of images formed depends on this angle. A smaller angle will produce more images, while a larger angle will produce fewer.

The principle behind this is the repeated reflection of light between the two mirrors. As the light bounces back and forth between the mirrors, more images are formed. This concept is used in various optical instruments and toys.

To predict how many images will appear, use the formula that relates the angle between the mirrors to the number of images. This is crucial when designing devices like kaleidoscopes, where multiple reflections are desired.

Number of Images Formula

To find out how many images will be formed when two plane mirrors are placed at a specific angle, you can use a simple formula:\[ n = \frac{360^{\circ}}{\theta} - 1 \]Where:- \( n \) is the number of images.- \( \theta \) is the angle between the mirrors.

For example, if two mirrors are at an angle of \(60^{\circ}\), plug in the value into the formula:\[ n = \frac{360^{\circ}}{60^{\circ}} - 1 = 5 \]So, you'll see five images. This formula helps you understand the impact of adjusting the angle between mirrors.

However, if the solution involves a non-integer \( n \), round up to the nearest whole number. This accounts for practical scenarios where fractional images might appear due to certain positioning of mirrors or viewer perspective, ensuring you can accurately predict the visual outcome.