The mirror formula
One equation links u, v and f for every spherical mirror.
Find the image without drawing
Drawing ray diagrams is slow. There is a short rule that links the object distance u, the image distance v and the focal length f of a mirror.
Which quantities does the rule need to find where the image forms?
What this lesson covers
The idea
The mirror formula 1/v + 1/u = 1/f relates object distance u, image distance v (both from the pole) and focal length f; it holds for all spherical mirrors and object positions, using the sign convention.
Find the image without drawing
Drawing ray diagrams is slow. There is a short rule that links the object distance u, the image distance v and the focal length f of a mirror.
Which quantities does the rule need to find where the image forms?
- The object distance and the focal length
- The height of the object and the colour of light
- Only the radius of the mirror
Check the formula
Move the object. Compare 1/v + 1/u with 1/f. Do this for the concave mirror and then for the convex one.
They always match
In both mirrors, at every distance, 1/v + 1/u gave the same number as 1/f. Use the signs: u is negative for an object on the left, and f is negative for a concave mirror.
The mirror formula is 1/v + 1/u = 1/f. It connects the object distance u, the image distance v (both measured from the pole) and the focal length f. It is valid for all spherical mirrors and for all positions of the object, if the sign convention is used.
Notes
1/v + 1/u = 1/f: valid for all spherical mirrors and all object positions, with the sign convention.
Check yourself
Which three quantities does the mirror formula connect?
A concave mirror has f = −10 cm. An object is at u = −30 cm. Use 1/v + 1/u = 1/f to find v, in cm.
Answer: -15 cm
1/v = 1/f − 1/u = −1/10 + 1/30 = −2/30, so v = −15 cm.
A convex mirror has f = +20 cm. An object is at u = −60 cm. Find v, in cm.
Answer: 15 cm
1/v = 1/f − 1/u = 1/20 + 1/60 = 4/60, so v = +15 cm, behind the mirror.
For which mirrors and object positions is the mirror formula valid?
- Object height, image height and focal length. Heights come into the magnification formula. The mirror formula uses distances: u, v and f.
- Radius of curvature, aperture and pole distance. The mirror formula links u, v and f. The radius R comes in only through f = R/2.
- Object distance, image distance and focal length — correct. Yes!
- For all spherical mirrors and all positions of the object, using the sign convention — correct. Yes!
- Only for concave mirrors. The formula is valid for all spherical mirrors, concave and convex.
- Only when the image is real. The formula holds for all positions of the object, whether the image is real or virtual.