Sign convention for mirrors and lenses
Measure every distance from the pole or optical centre. Along the incident light is positive, against it negative; up is positive, down negative. So u is always negative for a real object.
The ray diagram
Drag the object or type values. Signs follow the New Cartesian convention: distances are measured from the pole or optical centre, along the incident light is positive, against it negative, and heights above the axis are positive. So u is negative for a real object.
Concave mirror of focal length -10 cm. An object +2 cm tall at u = -15 cm: the image is at v = -30 cm, real, inverted, enlarged.
Show the working
- f = R/2 = -20/2 = -10 cm, u = -15 cm (the object is in front, so u is negative), h = +2 cm.
- 1/v = 1/f - 1/u = 1/(-10) - 1/(-15) = -5/150 = -1/30
- v = -30 cm: negative, so the image is in front of the mirror and real.
- m = -v/u = -(-30)/(-15) = -2: negative, so inverted, and enlarged.
- h′ = m × h = -2 × +2 = -4 cm
Axes on the mirror
New Cartesian sign convention
- The object is always placed to the left of the mirror or lens, so light falls on it from the left.
- All distances parallel to the principal axis are measured from the pole of the mirror or the optical centre of the lens.
- Distances measured to the right of the origin, along the direction of the incident light, are positive (+x).
- Distances measured to the left of the origin, against the incident light, are negative (-x).
- Distances measured upward and perpendicular to the principal axis, such as the height of an erect object or image, are positive (+y).
- Distances measured downward, such as the height of an inverted image, are negative (-y).
Signs of f, v and m
| f | v | m | |
|---|---|---|---|
| Concave mirror | Negative | Negative for a real image, positive for a virtual one | Negative when real, positive when virtual |
| Convex mirror | Positive | Always positive (virtual) | Positive, less than 1 |
| Convex lens | Positive | Positive for a real image, negative for a virtual one | Negative when real, positive when virtual |
| Concave lens | Negative | Always negative (virtual) | Positive, less than 1 |
Quick answers
Why is u always negative?
The object is placed to the left and the light travels to the right. The object distance is measured against the incident light, so it is negative.
Is the focal length of a concave mirror positive or negative?
Negative, because its focus lies in front of the mirror, against the incident light. A convex mirror has a positive focal length.
What sign does the height of an inverted image have?
Negative, because it is measured downward from the principal axis. So the magnification of a real, inverted image is negative.