Mirror Formula or Mirror Equation:-

1. A ray coming from object hits the concave mirror at a height at point A .

2. After reflection passes through point I which is on the principal axis.

3. Here AC is the normal, and here angle of incidence and angle of reflection

  Are equal and are denoted by θ

4. Three right angle triangles we can observe from the figure.

   They are ΔACP', ΔAIP'.

5. Let the angle at vertices O,C,I are α, β,γ.

6. We know, sum of the interior angles is equal to exterior angles.

      Δ AOC, β = α + θ

              θ= β - α ------ (1)

      Δ ACI,  γ = β + θ

              γ = β + β - α

              2β = α + γ ------(2)

When h becomes very small, P' is taken as P

     P'O = PO, P'C=PC, P'I= PI

In right angle triangle, Tan value is the acute angle which is the ratio

Of opposite side toothed adjacent side.

Tan α = P'A/P'O = h/PO

Tan β = P'A/P'C = h/PC

Ran γ = P'A/P'I= h/PI

When angle θ is very small then Tanθ value become close to θ.

 α = h/PO, β = h/PC, γ = h/ PI

Substitute in equation (2)

  2h/PC = h/PO + h/PI

   2/PC = 1/PO + 1/PI ------(3)

Sign Convention:

1. All the distance should be measured from Pole.

2. The distance is measured in the direction of incident ray is taken as positive, for opposite direction is taken as negative.

3. Height is taken as positive when measured upwards and taken as negative when measured downwards.

Here    PC is radius of curvature, taken as ‘-R’ (opposite to incident ray)

         PO is object distance, taken as ‘-u’

         PI is image distance, taken as ‘-v’

 

              2/R = 1/u + 1/v

Since radius of curvature is twice the focal length(R=2f)

              2/2f = 1/u + 1/v

        

                                  Mirror Equation:  [1/f = 1/u + 1/v]