Q.Draw the ray diagram of an astronomical telescope showing image formation in the normal adjustment position. Write the expression for its magnifying power.

What is Angular Magnification?

When you look at a tiny object — say a grain of salt — you hold it close to your eye to see it bigger. But there is a limit: bring it too close and it blurs. The closest distance at which your eye can focus comfortably is called the near point, conventionally taken as 25 cm for a normal eye. At that distance, the object subtends a certain angle at your eye. That angle determines how large it appears — not its physical size, but the fraction of your field of view it occupies.

Now imagine using a magnifying glass. The same grain of salt now looks much larger. Why? Because the lens lets you bring the object even closer than 25 cm while still seeing a clear, magnified image. That image is formed at a comfortable viewing distance, but the angle it subtends at your eye is far bigger than the angle the object would subtend at 25 cm without the lens.

Angular magnification is simply the ratio of these two angles:

Angular magnification $M = \frac{\theta_{\text{image}}}{\theta_{\text{object}}}$

where $\theta_{\text{image}}$ is the angle subtended by the image when viewed through the instrument, and $\theta_{\text{object}}$ is the angle subtended by the object when viewed with the naked eye at the near point (25 cm).

Why "Angular" and Not "Linear"?

A common confusion: a microscope or telescope does not give you a physically larger object — it gives you a larger apparent size. The image on your retina is bigger because the rays entering your eye are steeper. That steepness is measured by the angle. So magnification here is about angles, not actual lengths.

A Concrete Example

Take a simple magnifier (a convex lens) of focal length $f = 5$ cm. You place the object just inside the focal point so that a virtual, erect image forms at infinity (or at the near point). For the "image at infinity" case, the angle subtended by the image is $\theta_{\text{image}} \approx h / f$, where $h$ is the object height. The angle subtended by the object at the near point (25 cm) is $\theta_{\text{object}} \approx h / 25$.

Thus:

For $f = 5$ cm, $M = 5$. The image appears 5 times larger than the object seen at 25 cm.

The Big Picture

Angular magnification is the language of all optical instruments:

• Simple magnifier: $M \approx 25/f$ (image at infinity) …

The Core Problem: When Two Points Become One Blur

Imagine you are looking at two tiny dots drawn very close together on a piece of paper. From far away, they look like a single dot. As you bring the paper closer, at some point your eye suddenly sees two separate dots. That moment — the threshold where your eye (or a microscope) can just barely tell that there are two objects instead of one — is the heart of resolving power.

A microscope's job is to show you fine detail. But no matter how good the lenses are, there is a fundamental limit: light itself behaves like a wave. When light passes through the circular opening of a lens, it does not travel in perfect straight lines. It spreads out and forms a pattern called an Airy disk — a bright central spot surrounded by faint rings. Every point in your specimen becomes a tiny blurry disk in the image, not a perfect point.

If two points in the specimen are very close, their Airy disks overlap. When they overlap too much, your eye cannot tell them apart — they merge into one blob. The resolving power of a microscope is its ability to show two closely spaced points as distinct.

The Precise Criterion: Lord Rayleigh's Condition

Lord Rayleigh proposed a practical rule: two points are just resolved when the centre of one Airy disk falls exactly on the first dark ring of the other. At that point, the combined intensity has a small dip between the two peaks — your eye can just detect that there are two sources.

For a microscope, the smallest distance $d$ between two points that can just be resolved is given by:

where:

• $\lambda$ is the wavelength of light used
• $n$ is the refractive index of the medium between the specimen and the objective lens
• $\beta$ is the half-angle of the cone of light entering the objective

The quantity $n \sin \beta$ is called the numerical aperture (NA) of the objective lens. So the formula is often written as:

What This Tells Us: Two Levers for Better Resolution

1. Shorter wavelength $\lambda$ — Blue light resolves better than red light. Ultraviolet light resolves even better, which is why electron microscopes (using much shorter "wavelengths" of electrons) can see atoms.

2. Larger numerical aperture $n \sin \beta$ — You can increase $n$ by using oil between the slide and the objective (oil immersion). Air has $n \approx 1$, but special oils have $n \approx 1.5$. You can increase $\sin \beta$ by using a lens that collects light from a wider cone — a lens with a shorter focal length and larger diameter. …