The mirror formula
An object stands 30 cm from a concave mirror, f = 10 cm. Where is the image?
- Distances are measured from the pole, P.
- f = −10 cm: measured against the light's travel.
- The formula gives v = −15 cm.
- u = −30 cm: where the object stands.
The image is 15 cm in front of the mirror. In the question, 10 cm is the size of f. Its sign comes from the convention below.
The sign convention
| Quantity | Rule |
|---|---|
| Where from | Every distance is measured from the pole, along the axis. |
| Sign | Positive along the way the light travels, negative against it. |
| Heights | Above the axis, positive; below, negative. |
The light comes from the object towards the mirror. That way is positive. So a point in front of the mirror is negative. A point behind it is positive.
u is the object distance. v is the image distance.
m is the magnification, the image height divided by the object height. A minus m means an inverted image.
A real object is in front, so u is negative. A concave mirror has f negative. A convex mirror has f positive. The focus of a concave mirror is in front of it. The focus of a convex mirror is behind it. A real image in front has v negative.
The rule
1/v + 1/u = 1/f · m = −v/u
This holds for small spherical mirrors, for any position of the object.
Worked example
Take f = −10 cm and u = −30 cm.
Rearrange the formula first: 1/v = 1/f − 1/u.
1/v = 1/f − 1/u = −1/10 + 1/30 = −1/15, so v = −15 cm.
Use 30 as the common bottom number: −1/10 + 1/30 = −3/30 + 1/30 = −2/30 = −1/15.
m = −v/u = −(−15)/(−30) = −0.5
The minus on v means the image is in front. The minus on m means inverted. The 0.5 means half the size.
The image is 15 cm in front of the mirror. It is real, inverted and half the size.
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Watch out!
Writing f = +10 cm for a concave mirror is a mistake. It gives v = +7.5 cm, not −15 cm.