Problem 27 - Master Resource Book in JEE Main Physics

Question

A small objeet is placed \(10 \mathrm{~cm}\) in front of a plane mirror. If you stand behind the object, \(30 \mathrm{~cm}\) from the mirror and look at its image, for what distance must you focus your eyes? (a) \(20 \mathrm{~cm}\) (b) \(60 \mathrm{~cm}\) (c) \(80 \mathrm{~cm}\) (d) \(40 \mathrm{~cm}\)

Step-by-Step Solution

Answer

(d) 40 cm.

Step 1: Understand the Problem

A small object is placed 10 cm in front of a plane mirror. You stand 30 cm behind the object from the mirror. We need to find the distance at which you must focus your eyes to see the image.

Step 2: Calculate Object to Mirror Distance

The object is positioned 10 cm in front of the mirror, making the mirror to object distance 10 cm. In a plane mirror, the image is created at the same distance behind the mirror as the object is in front.

Step 3: Determine Image Distance from the Mirror

Since the object is 10 cm in front of the mirror, the image will also appear 10 cm behind the mirror because images in plane mirrors are virtual and located symmetrically behind the mirror.

Step 4: Calculate Total Distance for Focusing

The viewer stands 30 cm from the mirror, behind the object. To determine the total distance for focusing your eyes on the image, we add the object distance to the mirror (10 cm) and the viewer's distance from the mirror (30 cm). So, the total focusing distance is: \[\text{Focus Distance} = 30\, \text{cm (to the mirror)} + 10\, \text{cm (image previous location)} = 40\, \text{cm}\]

Step 5: Choose Correct Answer

From the possible answer choices, (d) 40 cm is the correct one at which you must focus your eyes to see the image.

Key Concepts

Plane Mirror

A plane mirror is a flat, reflective surface that creates images with distinct properties. Unlike curved mirrors, a plane mirror does not converge or diverge light waves. Instead, it reflects them in the same direction they arrived from, following the law of reflection. In simple terms, the angle at which light hits the mirror is equal to the angle at which it reflects away.

Key characteristics of images in a plane mirror include:

• The image appears to be the same distance behind the mirror as the object is in front of it. So, if an object is placed 10 cm in front of the mirror, its image is perceived as 10 cm behind it.
• Images are virtual, meaning they cannot be projected onto a screen. You cannot "catch" a plane mirror image, as it's simply where diverging reflected light appears to originate.
• These images are also laterally inverted, meaning they are flipped from left to right. It's why your reflection raises its right hand when you lift your left.

The straightforward nature of plane mirrors makes them a key subject in optics.

Virtual Image

In optics, a virtual image is an image formed where light rays appear to diverge but don't actually meet. This is in contrast to a real image, where light rays converge and can be displayed on a screen. When you observe an image in a plane mirror, you're actually seeing what is called a virtual image.

Here are some notable points about virtual images:

• A virtual image is formed by the apparent intersection of diverging rays. It's not tangible and can't be projected onto a physical surface.
• In a plane mirror, the virtual image appears to be located behind the mirror, at the same distance as the object is from the front.
• This type of image maintains the orientation of the real object, except for the lateral inversion (left-right flip).
• Virtual images are crucial for understanding concepts like angle of incidence and reflection.

By grasping the nature of virtual images, you can better comprehend how reflections work and how images are perceived in mirrors.

Focusing Distance

The focusing distance is the distance at which a viewer must adjust their eyes to clearly see an image. This concept is essential in optics because it helps to understand how and where an image is perceived.

In the context of a plane mirror:

• The focusing distance is the sum of the object-to-mirror distance and the observer-to-mirror distance. This accounts for both the location of the object and how far an observer is from the mirror.
• Since plane mirrors create virtual images at the same distance behind the mirror as the object is in front, these distances directly influence what the viewer perceives.
• For example, if an object is 10 cm in front of the mirror and the observer is 30 cm behind the object, the total focusing distance is 40 cm. This is critical to clearly observing the image in the mirror, as calculated in exercises involving plane mirrors.
• Understanding this concept can be applied in everyday scenarios, like when using a bathroom mirror, ensuring that whatever action you undertake in front of the mirror is aligned with the virtual image you see.

Being aware of focusing distance helps you not just with academic exercises, but also in practical situations! It makes sure that the real and perceived world are in sync.