TL;DR:
A convex lens bends parallel rays to meet at the focal point. Its focal length f sets what it does.
The thin lens equation is
1/f = 1/dₒ + 1/dᵢ, and magnification ism = −dᵢ/dₒ.A magnifier works with the object closer than f: the image is virtual, upright and larger.
Hold a magnifying glass over a line of small print and the letters swell. Pull the glass away from the page and at some distance the picture flips upside down. Same glass, two very different images. The distance between the object and the lens decides which one you get.
What a convex lens does
A convex lens is thicker in the middle than at the edges. Light bends at each curved surface by refraction, as covered in refraction, and the net effect is that rays travelling parallel to the axis bend toward a single point, the focal point. The distance from the lens to that point is the focal length f.
A fatter, more curved lens bends light more and has a shorter focal length. A flatter lens has a longer one.
Ray diagrams: three easy rays
To find where an image forms, draw rays from the top of the object. Two are enough, and a third checks them.
A ray parallel to the axis bends through the focal point on the far side.
A ray through the centre of the lens carries straight on, barely bent.
A ray that passes through the near-side focal point comes out parallel to the axis.
Where the rays meet is where the top of the image forms. If the rays really converge, the image is real and can be caught on a screen. If they only seem to come from a point, because you trace them backward, it is virtual and exists only for the eye.
Worked example: a 10 cm lens used as a magnifier
A convex lens has f = 10 cm. The thin lens equation is 1/f = 1/dₒ + 1/dᵢ, where dₒ is the object distance and dᵢ is the image distance. A negative dᵢ means a virtual image on the same side as the object. Put a small object 6 cm from the lens, which is inside the focal length.
Rearrange:
1/dᵢ = 1/f − 1/dₒ = 1/10 − 1/6.Common denominator 30:
3/30 − 5/30 = −2/30.Invert:
dᵢ = −15 cm.Magnification:
m = −dᵢ/dₒ = −(−15)/6 = +2.5.
So the image is 15 cm from the lens on the same side as the object (the minus sign), it is virtual, it is upright (m is positive) and it is 2.5 times as tall as the object. You look through the lens and see it behind the glass.
Check by substituting: 1/6 + 1/(−15) = 0.1667 − 0.0667 = 0.1000 = 1/10. It works.
Now move the object farther away
Put the object at 30 cm, beyond the focal length. Then 1/dᵢ = 1/10 − 1/30 = 3/30 − 1/30 = 2/30, so dᵢ = 15 cm, a positive number. The image is real, on the other side of the lens, and m = −15/30 = −0.5: inverted and half the size. That is the flipped picture you saw when you pulled the glass away.
Why a magnifier must be closer than f
Inside the focal length, the lens does not bend the rays enough to make them meet. They still spread out, but more slowly than before, so they look as though they come from a bigger object farther away. As the object moves toward f, the virtual image gets larger and farther. At exactly f the rays emerge parallel and no image forms at a finite distance.
Eye strain is lowest when you view a comfortable distance away. For a reader whose near point is 25 cm and who sees the image there, the object distance needs 1/dₒ = 1/10 + 1/25 = 7/50, so dₒ ≈ 7.1 cm and m = 25/7.14 = 3.5. That is the usual idea behind a magnifier labelled with a power.
Never focus the sun: A convex lens can focus sunlight to a point hot enough to burn skin, paper, wood or a dry leaf in seconds. Never look at the sun through a lens, and keep lenses away from anything that can burn. Store them covered, away from windows.
How to check your answer
Sign test. Object inside f gives a negative dᵢ, an upright virtual image bigger than the object. Object outside f gives a positive dᵢ, a real inverted image.
Plug back. Compute
1/dₒ + 1/dᵢand see if it equals1/f.Draw it. The ray diagram must agree with the algebra.
Units. Use the same length unit for all three distances.
Mistakes that cost marks
Forgetting to invert at the end: the equation gives
1/dᵢ, not dᵢ.Ignoring the minus sign on dᵢ or m, which carry real meaning.
Saying a virtual image can be shown on a screen. It cannot.
Mixing up the object distance and the lens thickness.
Where you will use it next
Cameras, glasses and telescopes use the same equation, often with two lenses together. The speed change that drives the bending is explained in how waves carry energy.
Frequently asked questions
Is a magnifying glass the same as a camera lens?
Both use convex lens action, but a camera forms a real image on a sensor, while a magnifier forms a virtual image for your eye.
Can a lens make a bigger image without getting farther away?
Inside f, bringing the object closer to f makes the virtual image larger. A shorter focal length lens also magnifies more.
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