Worked exercise — Concave mirror

Problem

A concave mirror has radius of curvature $R = 8\;\text{m}$, so $f = R/2 = 4\;\text{m}$. An object is placed at $p = 5\;\text{m}$ from the vertex. Find the position $q$ of the image and the magnification $I$.

Solution

Setup. Focus of the spherical mirror: $f = R/2 = 8/2 = 4\;\text{m}$. We apply the conjugate points equation $\tfrac{1}{p}+\tfrac{1}{q}=\tfrac{1}{f}$.

Position of the image.

$\frac{1}{q} = \frac{1}{f} - \frac{1}{p} = \frac{1}{4} - \frac{1}{5} = \frac{5-4}{20} = \frac{1}{20}$

$\ev{q = 20\;\mathrm{m}}$

Magnification.

$I = -\frac{q}{p} = -\frac{20}{5} = -4 \quad\Rightarrow\quad \ev{I = -4}$

Since $q>0$ the image is real (it can be collected on a screen); the negative sign of $I$ indicates that it is inverted, and $|I|=4$ that it is magnified $4$ times.

Linked atoms

Argomenti: Optics Concetti: Lenses and mirrors