Mirror formula and magnification

To calculate with mirrors, we need a fixed rule for signs: the New Cartesian Sign Convention. Treat the pole as zero on a number line:

• the object is placed on the left;
• distances measured to the right of P are positive, to the left negative;
• heights above the principal axis are positive, below are negative.

So the object distance u is always negative. A concave mirror's focus is in front of it (on the left), so f is negative. A convex mirror's focus is behind it, so f is positive.

The mirror formula links the three distances:

1/v + 1/u = 1/f

where u is the object distance, v the image distance and f the focal length, all with their signs.

Magnification tells you how big the image is compared with the object:

m = h′/h = –v/u

Read it in two parts:

So m = –1 means real, inverted, same size (object at C). m = +1 means a plane mirror.

Predict before you calculate. If the object is between P and F of a concave mirror, you already know the image is behind the mirror, so v must come out positive. A real, inverted image is always in front of the mirror, so its v is negative. If your answer's sign disagrees with your picture, check your signs.

The book's example: a convex rear-view mirror with R = 3.00 m and a bus 5.00 m away. Then f = +1.50 m and u = –5.00 m, giving v = +1.15 m and m = +0.23. The image is virtual, erect, and about a quarter of the size.