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Valter edited Aug 20, 2026, 7:13 PM

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		+	### Statement
		+
		+	$13.3.14.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	### Statement
		+
		+	$13.3.14.$ It is known that if a small hole is carefully pierced in a piece of dark paper, small objects can be viewed through this hole with a magnification of several times. Explain this phenomenon.
		+
		+	### Solution
		+
		+	This phenomenon is explained not by the optical properties of the hole itself (it does not act as a lens), but by a sharp increase in the "depth of field" of the eye due to aperture restriction (stopping down).
		+
		+	1. Limitation of eye accommodation: Under normal conditions, a normal eye cannot focus on objects located closer than the distance of most distinct vision ($d_0 \approx 25 \text{ cm}$). If an object is brought closer, the rays from each of its points will converge behind the retina, forming a wide luminous spot on the retina itself — the image will be blurred.
		+	2. The role of the hole (aperture): A small hole in the paper acts as a strict diaphragm. It cuts off the wide marginal ray bundles, leaving only a narrow central (paraxial) bundle. Because of this, the size of the circle of confusion on the retina becomes negligibly small, and the image is perceived as sharp even for objects located very close to the eye (down to $2-3 \text{ cm}$).
		+	3. Angular magnification: Since we can now view the object from a very close distance $d \ll d_0$, the angle $\phi$ subtended by the object ($\phi \approx \frac{h}{d}$) becomes several times larger than the angle $\phi_0$ subtended by the object at the distance of most distinct vision ($\phi_0 \approx \frac{h}{d_0}$). As a result, the eye receives a sharp image on the retina, the size of which is $\Gamma = \frac{d_0}{d}$ times larger than usual. The brain interprets this as a significant magnification of the object.
		+	4. The role of the dark paper: Very little light passes through the tiny hole, so the image turns out dim. Dark (opaque and non-reflective) paper blocks background glare and prevents the pupil from reflexively constricting, providing the necessary contrast to observe the faint image.
		+
		+	<b>Conclusion:</b> The hole itself does not magnify the object; it merely allows bringing it very close to the eye without losing sharpness, which leads to an increase in the angular size of the object.
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

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			### Statement

			$13.3.14.$ [Insert the problem statement]

			### Solution

			### Statement

			$13.3.14.$ It is known that if a small hole is carefully pierced in a piece of dark paper, small objects can be viewed through this hole with a magnification of several times. Explain this phenomenon.

			### Solution

			This phenomenon is explained not by the optical properties of the hole itself (it does not act as a lens), but by a sharp increase in the "depth of field" of the eye due to aperture restriction (stopping down).

			1. Limitation of eye accommodation: Under normal conditions, a normal eye cannot focus on objects located closer than the distance of most distinct vision ($d_0 \approx 25 \text{ cm}$). If an object is brought closer, the rays from each of its points will converge behind the retina, forming a wide luminous spot on the retina itself — the image will be blurred.
			2. The role of the hole (aperture): A small hole in the paper acts as a strict diaphragm. It cuts off the wide marginal ray bundles, leaving only a narrow central (paraxial) bundle. Because of this, the size of the circle of confusion on the retina becomes negligibly small, and the image is perceived as sharp even for objects located very close to the eye (down to $2-3 \text{ cm}$).
			3. Angular magnification: Since we can now view the object from a very close distance $d \ll d_0$, the angle $\phi$ subtended by the object ($\phi \approx \frac{h}{d}$) becomes several times larger than the angle $\phi_0$ subtended by the object at the distance of most distinct vision ($\phi_0 \approx \frac{h}{d_0}$). As a result, the eye receives a sharp image on the retina, the size of which is $\Gamma = \frac{d_0}{d}$ times larger than usual. The brain interprets this as a significant magnification of the object.
			4. The role of the dark paper: Very little light passes through the tiny hole, so the image turns out dim. Dark (opaque and non-reflective) paper blocks background glare and prevents the pupil from reflexively constricting, providing the necessary contrast to observe the faint image.

			<b>Conclusion:</b> The hole itself does not magnify the object; it merely allows bringing it very close to the eye without losing sharpness, which leads to an increase in the angular size of the object.

			#### Answer

			[Insert a concise answer or boxed result]