Spherical mirrors and sign convention
A spherical mirror is part of a hollow sphere. A concave mirror reflects from the inner surface and converges light; a convex mirror reflects from the outer surface and diverges light. Key terms: pole (P), centre of curvature (C), radius of curvature (R), principal axis and principal focus (F). For small apertures, f = R/2.
Cartesian sign convention: distances are measured from the pole; those in the direction of incident light are positive and those against it are negative; heights above the axis are positive. With light coming from the left, the object distance u is negative. The focal length of a concave mirror is negative and that of a convex mirror is positive.
Mirror formula and magnification
1/v + 1/u = 1/f m = h′/h = –v/u A negative m means a real, inverted image; a positive m means a virtual, erect image; |m| > 1 means magnified.
Image formation by a concave mirror
- Object beyond C: real, inverted, diminished image between F and C.
- Object at C: real, inverted, same size, at C.
- Object between C and F: real, inverted, magnified, beyond C.
- Object at F: image at infinity.
- Object between F and P: virtual, erect, magnified, behind the mirror.
A convex mirror always forms a virtual, erect, diminished image between P and F, which is why it is used as a rear-view mirror (wide field of view).
Worked example
A concave mirror has f = –15 cm and an object is at u = –20 cm. Then 1/v = 1/f – 1/u = –1/15 + 1/20 = –1/60, so v = –60 cm. m = –v/u = –3: the image is real, inverted and three times larger.
Key points
- f = R/2.
- 1/v + 1/u = 1/f; m = –v/u.
- Concave f < 0; convex f > 0.
- Convex mirror: always virtual, erect, diminished.
Common mistakes
- Dropping the minus sign on u.
- Using m = v/u (the lens formula) for mirrors.