Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason

Q: Why is the angle subtended by the object and image the same?

Because the ray from the top of the object passing through the optical center goes undeviated and forms the top of the virtual image. They lie on the exact same straight line from the optical center, sharing the same angle.

Q: If the angles are the same, why does the image look bigger?

The image looks bigger because the angle $\theta'$ (with the lens) is much larger than the angle $\theta$ you would get if you looked at the object without the lens at the near point $D$. The lens lets us bring the object closer.

Q: What is the difference between linear and angular magnification?

Linear magnification is the ratio of physical heights ($h'/h$), while angular magnification is the ratio of angles subtended at the eye ($\theta'/\theta$). For visual instruments, angular magnification is what determines how big things appear to our brain.

Q: Can we keep the object at any distance less than D?

Yes, but to see a clear image, the virtual image must form at a distance greater than or equal to $D$. So the object distance $u_0$ must be between the focal length $f$ and some minimum distance that places the image exactly at $D$.

Q: Is the simple microscope just a magnifying glass?

Yes! A simple microscope is simply a converging (convex) lens used to produce a magnified, virtual, and erect image of a small object placed close to it.

Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) For a simple microscope, the angular size of the object equals the angular size of the image.
Reason (R) Magnification is achieved as the small object can be kept much closer to the eye than 25 cm and hence, it subtends a large angle.
In the light of the above statements, choose the most appropriate answer from the options given below.

Options:

(A) A is true but R is false.

(B) Both A and R are true but R is not the correct explanation of A.

(C) Both A and R are true and R is the correct explanation of A.

(D) A is false but R is true.

Sigma Percentile · Accepted Answer

### The Limit of Human Vision

Imagine you are trying to read the tiny print on a medicine bottle. Instinctively, you bring the bottle closer to your eyes to make the letters appear larger. However, there is a physical limit to this. If you bring the object closer than approximately $25	ext{ cm}$, your eye's lens can no longer accommodate, and the image becomes blurry. This minimum distance is known as the **least distance of distinct vision**, denoted by $D$.

At this distance $D$, an object of height $h$ subtends a maximum possible angle $	heta$ at your unaided eye, given approximately by:

$$	heta \approx \frac{h}{D}$$

To see the object even larger and clearly, we need a tool to bypass this biological limit. Enter the **simple microscope**, which is essentially a single convex lens.

### The Role of the Convex Lens (Assertion Analysis)

When we use a convex lens as a simple microscope, we place the object between the optical center and the principal focus of the lens. This setup produces a virtual, erect, and magnified image.

Let's analyze the ray diagram carefully. The ray of light originating from the top of the object and passing through the **optical center** of the lens goes completely undeviated. Because this ray travels in a straight line, it forms the top of the virtual image as well.

Geometrically, this means that the object and the virtual image lie on the exact same line of sight from the optical center. Therefore, the angle subtended by the object at the lens ($	heta'$) is exactly equal to the angle subtended by the virtual image at the lens.

$$	heta' = \frac{h}{u_0} = \frac{h'}{D}$$

This confirms that the **Assertion (A) is absolutely true**: the angular size of the object equals the angular size of the image.

### The Secret of Magnification (Reason Analysis)

This leads to a fascinating paradox: If the angular size of the object and the image are identical, why does the simple microscope make things look bigger? Where does the magnification come from?

The secret lies in *where* the object is placed. Without the lens, the closest you could place the object was at distance $D$. But with the converging power of the lens assisting your eye, you can place the object much closer, at a distance $u_0$ (where $u_0 < D$).

Because the object is now physically closer to your eye, it subtends a much larger angle $	heta'$ compared to the original angle $	heta$ it would have subtended at distance $D$.

The **angular magnification** ($m$) is defined as the ratio of the angle subtended by the image ($	heta'$) to the angle subtended by the object when placed at the near point ($	heta$):

$$m = \frac{	heta'}{	heta} = \frac{h/u_0}{h/D} = \frac{D}{u_0}$$

Since $D$ is greater than $u_0$, the magnification $m$ is greater than 1. The lens doesn't magnify by changing the angle between the object and the image; it magnifies by allowing you to bring the object closer than your eye normally permits, thereby increasing the angle it subtends.

This confirms that **Reason (R) is true** and it perfectly explains the mechanism behind the Assertion.

### Conclusion

Both the Assertion and the Reason are correct, and the Reason provides the exact physical justification for the phenomenon described in the Assertion. The simple microscope is a beautiful example of how simple optics can elegantly overcome the biological limitations of the human eye.

Select Answer:

Visualized Solution

Visualizing the Simple Microscope

Analyzing the Assertion

Analyzing the Reason

Final Conclusion

The Sigma Insight: Optical Instruments

The Limit of Human Vision

The Role of the Convex Lens (Assertion Analysis)

The Secret of Magnification (Reason Analysis)

Conclusion

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