Question

Derive the mirror formula.what is the corresponding formula for a thin lens

Answer 1

Answer on Question #72290, Physics / Optics

Question. Derive the mirror formula. What is the corresponding formula for a thin lens.

Solution.

Consider the object $OP$ shown in Figure. The center of curvature of the mirror is labeled $C$ and is a distance $R$ from the vertex of the mirror, as marked in the figure. The object and image distances are labeled $d_0$ and $d_i$ , and the object and image heights are labeled $h_0$ and $h_i$ , respectively. Because the angles $\phi$ and $\phi'$ are alternate interior angles, we know that they have the same magnitude. However, they must differ in sign if we measure angles from the optical axis, so $\phi = -\phi'$ . An analogous scenario holds for the angles $\theta$ and $\theta'$ . The law of reflection tells us that they have the same magnitude, but their signs must differ if we measure angles from the optical axis. Thus, $\theta = -\theta'$ . Taking the tangent of the angles $\theta$ and $\theta'$ , and using the property that $tg(-\theta) = -tg\theta$ , gives us

Similarly, taking the tangent of $\phi$ and $\phi^{\prime}$ gives

Combining these two results gives

$$\frac {d _ {0}}{d _ {i}} = \frac {d _ {0} - R}{R - d _ {i}}$$

or

$$d _ {0} (R - d _ {i}) = d _ {i} (d _ {0} - R) \rightarrow d _ {0} R - d _ {0} d _ {i} = d _ {0} d _ {i} - d _ {i} R \rightarrow 2 d _ {0} d _ {i} = d _ {0} R + d _ {i} R \rightarrow$$ $$\frac {d _ {0} + d _ {i}}{d _ {0} d _ {i}} = \frac {2}{R} \rightarrow \frac {1}{d _ {0}} + \frac {1}{d _ {i}} = \frac {2}{R} \quad (1).$$

No approximation is required for this result, so it is exact. However, in the small-angle approximation, the focal length of a spherical mirror is one-half the radius of curvature of the mirror, or $f = R / 2$ . Inserting this into Eq. 1 gives the mirror equation or formula:

$$\frac {1}{d _ {0}} + \frac {1}{d _ {i}} = \frac {1}{f}.$$

The mirror equation relates the image and object distances to the focal distance and is valid only in the small-angle approximation. Although it was derived for a concave mirror, it also holds for convex mirrors. We can extend the mirror equation to the case of a plane mirror by noting that a plane mirror has an infinite radius of curvature. This means the focal point is at infinity, so the mirror equation simplifies to

$$d _ {0} = - d _ {i}.$$

As done for spherical mirrors, we can use ray tracing and geometry to show that, for a thin lens,

$$\frac {1}{d _ {0}} + \frac {1}{d _ {i}} = \frac {1}{f}.$$

where $f$ is the focal length of the thin lens. This is the thin-lens equation.

(More detail see Max Born & Emil Wolf Principles of Optics)

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