Sign convention, mirror formula and magnification
Measure every distance from the pole using the New Cartesian sign convention. Then apply the mirror formula 1/v + 1/u = 1/f and m = −v/u.
New Cartesian sign convention
- Place the mirror so that the pole P is the origin (the zero point) and the principal axis is the x-axis.
- Always put the object on the left of the mirror, so light travels from left to right.
- Measure all distances from the pole.
- Distances to the right of P (in the direction of the incident light) are positive (+). Distances to the left are negative (−).
- Heights above the principal axis are positive. Heights below it are negative.
- So the object distance u is always negative. The focal length of a concave mirror is negative and of a convex mirror is positive.
Symbols and what their signs tell you
Concave mirror: a real image
An object 3 cm tall stands 30 cm in front of a concave mirror of focal length 20 cm. Find the position, size and nature of the image.
- Given (with signs): h = +3 cm, u = −30 cm, f = −20 cm (concave).
- Mirror formula: 1/v + 1/u = 1/f, so 1/v = 1/f − 1/u.
- Substitute: 1/v = 1/(−20) − 1/(−30) = −1/20 + 1/30.
- LCM is 60: 1/v = (−3 + 2)/60 = −1/60, so v = −60 cm.
- Magnification: m = −v/u = −(−60)/(−30) = −2.
- Image height: h′ = m × h = −2 × 3 cm = −6 cm.
The image is 60 cm in front of the mirror (v is negative). It is real, inverted (m is negative) and enlarged: 6 cm tall, twice the size of the object.
Convex mirror: a scooter's rear-view mirror
The rear-view mirror of a scooter is convex, with a radius of curvature of 2.0 m. A car is 4.0 m behind the mirror. Find the position and nature of its image, and the magnification.
- Given: R = +2.0 m, so f = R/2 = +1.0 m (convex). u = −4.0 m.
- 1/v = 1/f − 1/u = 1/1.0 − 1/(−4.0) = 1 + 1/4 = 5/4.
- So v = 4/5 m = +0.8 m.
- m = −v/u = −(0.8)/(−4.0) = +0.2.
The image is 0.8 m behind the mirror. It is virtual and erect (m is positive) and one-fifth the size of the car.
Concave mirror: a make-up mirror
A concave make-up mirror has a focal length of 20 cm. A face is 10 cm from it. Where is the image, and how large is it compared with the face?
- Given: u = −10 cm, f = −20 cm.
- 1/v = 1/f − 1/u = −1/20 + 1/10 = (−1 + 2)/20 = 1/20.
- So v = +20 cm.
- m = −v/u = −(20)/(−10) = +2.
The image is 20 cm behind the mirror. It is virtual, erect and twice the size of the face. This is why make-up and shaving mirrors are concave.
A make-up or shaving mirror is concave, so a face held inside F looks two or three times bigger. The round security mirror in a shop corner is convex: its image is always small and upright, just as a positive m predicts.
1 marks: “What is the sign of the focal length of a convex mirror in the New Cartesian sign convention?” Positive, because its focus lies behind the mirror, to the right of the pole.
2 marks: “The magnification produced by a mirror is −1/3. What does this tell you about the image and the type of mirror?” Negative m: real and inverted. One-third size: diminished. Only a concave mirror forms a real image, so the mirror is concave.
3 marks: “An object is placed 12 cm in front of a concave mirror of focal length 18 cm. Find the position and nature of the image.” u = −12 cm, f = −18 cm; 1/v = −1/18 + 1/12 = (−2 + 3)/36 = 1/36; v = +36 cm (behind the mirror); m = −v/u = +3: virtual, erect, three times enlarged.
Left is less: anything to the left of the pole is negative. The object is always on the left, so u is always minus.