The Formula
A simple microscope is a single converging (convex) lens used as a magnifier. Its magnifying power is usually given by one of two formulas, depending on where the final image is formed:
When the image is at the near point (maximum magnification):
M = 1 + D / f
When the image is at infinity (relaxed eye):
M = D / f
Here, M is the magnifying power, f is the focal length of the lens, and D is the least distance of distinct vision, conventionally taken as 25 cm (250 mm). D and f must be in the same units.
What Each Term Means
- Focal length (f): The distance from the lens to the point where parallel light comes to a focus. A shorter focal length means a more strongly curved, more powerful lens.
- Least distance of distinct vision (D): The closest distance at which a typical eye can focus comfortably, also called the near point. The standard value of 25 cm is a convention; real near points vary from person to person and with age.
- Magnifying power (M): Strictly, this is an angular magnification. It compares how large the object appears through the lens with how large it would appear to the naked eye if held at 25 cm.
Image at the Near Point vs Image at Infinity
The two formulas describe two ways of using the same lens.
Image at the near point. The object is placed slightly inside the focal length, and the lens forms an upright, enlarged virtual image 25 cm from the eye. This gives the largest possible magnification, M = 1 + D/f, but your eye has to focus at its closest comfortable distance, which can be tiring.
Image at infinity. The object is placed at the focal point. Light leaves the lens as parallel rays, and the eye views the image while relaxed, as if looking at something far away. The magnification is slightly lower, M = D/f, but viewing is more comfortable for long periods.
The difference between the two is exactly 1. For high-power lenses with short focal lengths, that difference becomes small compared with the total.
These formulas assume the lens is held close to the eye. Holding a magnifier far from the eye changes the result.
Worked Examples
Example 1: f = 5 cm
- Near point: M = 1 + 25 / 5 = 1 + 5 = 6x
- Relaxed eye: M = 25 / 5 = 5x
Example 2: f = 2.5 cm
- Near point: M = 1 + 25 / 2.5 = 1 + 10 = 11x
- Relaxed eye: M = 25 / 2.5 = 10x
Example 3: f = 10 mm
Convert to the same units: D = 250 mm.
- Near point: M = 1 + 250 / 10 = 26x
- Relaxed eye: M = 250 / 10 = 25x
Example 4: working backwards
A magnifier is labelled 10x. Using the relaxed-eye convention that many manufacturers follow, f = D / M = 250 mm / 10 = 25 mm.
Why Shorter Focal Length Means More Magnification
A lens with a shorter focal length lets you bring the object closer to your eye while still seeing it sharply. An object brought closer takes up a larger angle in your field of vision, so it looks bigger. Because f sits in the denominator of both formulas, halving the focal length roughly doubles the magnification.
The catch is that short-focal-length lenses are small and strongly curved. They have a small field of view, very short working distance, and more optical distortion unless they are carefully designed.
How This Compares With a Compound Microscope
A compound microscope uses two lens systems. The objective forms an enlarged real image inside the tube, and the eyepiece, acting as a simple magnifier, views that image. In textbook physics, its magnification is often approximated as:
M ≈ (L / f objective) × (D / f eyepiece)
where L is the optical tube length. The second part, D / f eyepiece, is the same simple-magnifier formula as above. That is why a 10x eyepiece has a focal length of about 25 mm.
In everyday use, the magnifications are engraved on the lenses and you simply multiply them. See how to calculate total magnification for that approach, and what magnification is for the difference between linear and angular magnification.
Practical Limits of a Simple Microscope
Hand lenses and loupes commonly range from about 2x to 20x or so. Higher powers are possible with very small lenses: the single-lens microscopes made by Antonie van Leeuwenhoek in the 17th century are believed to have reached around 200x or more, using tiny, carefully made lenses held very close to the eye.
For most modern use, though, a compound microscope is far more practical above roughly 20x to 30x. It gives a larger field of view, more comfortable viewing and, with a good objective, better resolution. Remember that magnification alone does not decide how much detail you see; resolution does. Our guide to magnification vs resolution explains why.