In spherical mirrors, three primary distances are considered:
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- Introduction
- Stepwise Derivation of the Mirror Formula
Introduction:
- Object Distance (u): The distance of the object from the pole of the mirror.
- Image Distance (v): The distance of the image from the pole of the mirror.
- Focal Length (f): The distance of the principal focus from the pole of the mirror.
These quantities are related by a mathematical expression known as the mirror formula:
Stepwise Derivation of the Mirror Formula:
The derivation of the mirror equation establishes the relationship between the object distance (u), image distance (), and focal length (). In the given setup, the two right-angled triangles A′B′F and MPF are identified as similar triangles.
The triangles B′A′F and PMF are similar because they share common angles. Using the property of similar triangles:
Substituting PM=AB into the equation:
∠APB = ∠A'PB
△A'B'P' ∼ △AB'P
By comparing equation (i) and (ii), we get
B'P = -ν (image distance),
FP = -f (focal length),
BP = -u (object distance).
Substituting these values:
On further solving, we get
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