Reflection and spherical mirrors: the mirror formula

The words you need

Light reflects from a spherical mirror by the two laws of reflection. (1) The angle of reflection equals the angle of incidence. (2) The incident ray, the reflected ray and the normal at the point of incidence lie in one plane. For a spherical mirror, the normal at any point lies along the radius: the line joining that point to the centre of curvature C. The pole P is the centre of the mirror's surface. The principal axis is the line through P and C. Paraxial rays are rays close to the principal axis that make small angles with it. Paraxial rays parallel to the axis meet at the principal focus F (concave mirror), or seem to spread out from it (convex mirror). The distance PF is the focal length f. Simple geometry with paraxial rays gives f = R/2, where R = PC is the radius of curvature.

The New Cartesian sign convention

• Your textbook calls it the Cartesian sign convention. Use it for every mirror, surface and lens in this chapter.
• Measure all distances from the pole of a mirror (or the optical centre of a lens).
• Draw the object on the left, so the incident light travels from left to right.
• Distances measured in the direction of the incident light are positive. Distances measured against it are negative.
• Heights measured upwards from the principal axis are positive. Heights measured downwards are negative.
• So a real object always has u negative. A concave mirror has f and R negative; a convex mirror has f and R positive.
• For a mirror, a real image forms in front of it (v negative). A virtual image forms behind it (v positive).
• Put the signs in only once, when you substitute numbers. Never change the formula itself.

Derivation: the mirror equation and magnification

Take a concave mirror forming a real image A′B′ of an object AB placed beyond C. Use paraxial rays, so the mirror is nearly flat near P.

1. 1
   A ray from the top A of the object, parallel to the axis, strikes the mirror at M and reflects through F. A second ray from A strikes the pole P and reflects at an equal angle.
2. 2
   Triangles A′B′P and ABP are similar (equal angles at P): B′A′/BA = B′P/BP. ...(1)
3. 3
   Triangles A′B′F and MPF are similar, and PM = BA for paraxial rays: B′A′/BA = B′F/FP. ...(2)
4. 4
   From (1) and (2): B′F/FP = B′P/BP. Since B′F = B′P − FP, this gives (B′P − FP)/FP = B′P/BP. ...(3)
5. 5
   Apply the sign convention: B′P = −v, FP = −f and BP = −u.
6. 6
   Then (−v + f)/(−f) = (−v)/(−u), which gives (v − f)/f = v/u.
7. 7
   Divide both sides by v and rearrange: 1/f − 1/v = 1/u.
8. 8★
   So, the mirror equation is 1/v + 1/u = 1/f.
9. 9★
   In (1), put B′A′ = −h′ (the image is below the axis) and BA = h: −h′/h = (−v)/(−u). So, m = h′/h = −v/u.

★ key step

How to draw the ray diagram for this derivation

• Draw a horizontal principal axis. Draw a concave mirror on the right as an arc, with its hollow side facing left towards the object.
• Mark P where the mirror meets the axis. Mark F and C on the axis to the left of P, with PC = 2 × PF.
• Draw the object AB as an upright arrow beyond C, with B on the axis.
• From A, draw a ray parallel to the axis to the mirror at M. Reflect it back through F.
• From A, draw a ray to P. Reflect it below the axis, making the same angle with the axis.
• Where the two reflected rays meet, mark A′. Drop A′B′ to the axis: a real, inverted, smaller image between F and C.
• Draw a dotted line from M down to the axis. Label A, B, A′, B′, M, P, F, C, u, v and f, and put arrows on every ray.

Worked example: concave and convex mirrors

Activity: focal length of a concave mirror and of a convex mirror

Two practical experiments on an optical bench, using the no-parallax method. A needle and an image coincide when they stay together as you move your eye sideways.

1. 1
   Concave mirror (u and v): Fix the mirror at one end of the optical bench. Place the object needle beyond F.
2. 2
   Look along the axis at the inverted image. Move the image needle until its tip touches the image tip with no parallax.
3. 3
   Record u and v (both negative) for five or six object positions. Work out f = uv/(u + v) for each, and take the mean.
4. 4
   Result: the mean f is negative, as for any concave mirror. With signs, a graph of 1/v against 1/u is a straight line that cuts both axes at 1/f.
5. 5
   Convex mirror (using a convex lens): With a convex lens L alone, form a real image I of the object needle O. Mark the position of I.
6. 6
   Place the convex mirror between L and I. Slide it until the image of O, after reflection, coincides with O itself with no parallax.
7. 7
   The rays now strike the mirror normally, so I is its centre of curvature. R = distance from the mirror to I, and f = +R/2.
8. 8
   Result: the focal length of the convex mirror is positive and equal to half the measured R.