The lens formula
A magnifying glass makes small print big. A projector puts a small slide on a big screen.
- An object stands 15 cm from a convex lens, f = 10 cm.
- One ray leaves parallel to the axis. It bends through F.
- A second ray goes straight through the centre.
- Where they cross is the image: v = +30 cm, m = −2.
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The rule
A convex lens is thicker in the middle. It brings parallel rays together. A concave lens is thinner in the middle. It spreads them.
The signs work as for mirrors. Distances are measured from the optical centre. It is the middle point of the lens. The formula is different.
Here u, v, f and m mean what they did for mirrors. Light travels from the object through the lens. That way is positive. A real object is in front, so u is negative. A real image behind the lens has v positive.
1/v − 1/u = 1/f · m = v/u
f is positive for a convex lens. It is negative for a concave lens. The focus of a convex lens is behind it. The focus of a concave lens is in front of it.
A concave lens makes a virtual, upright, smaller image. This is true for a real object, wherever it stands.
Worked example
Take f = +10 cm and u = −15 cm.
Rearrange the formula first: 1/v = 1/f + 1/u.
1/v = 1/f + 1/u = 1/10 − 1/15 = 1/30, so v = +30 cm.
Use 30 as the common bottom number: 1/10 − 1/15 = 3/30 − 2/30 = 1/30.
m = v/u = 30 / (−15) = −2
The minus on m means the image is inverted.
The image is 30 cm behind the lens. It is real, inverted and twice as tall.
Now take u = −5 cm, inside F. Then 1/v = 1/10 − 1/5 = −1/10, so v = −10 cm. And m = −10 / −5 = +2. The image is virtual, upright and twice as tall. That is a magnifying glass.