Optics — Applied Ray Optics (Medium)

Quick Recap — Ray Optics in Action

Work with a strict sign convention (distances measured from the pole/optical centre; real-is-positive for mirrors on the object side).

• Mirror: $\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}$, $m=-\dfrac{v}{u}$. Lens: $\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$, $m=\dfrac{v}{u}$.
• Lens-maker: $\dfrac{1}{f}=(n-1)\left(\dfrac{1}{R_1}-\dfrac{1}{R_2}\right)$; in a medium replace $(n-1)$ by $\left(\dfrac{n_{lens}}{n_{med}}-1\right)$ — the sign can flip.
• Two lenses a distance $d$ apart: $P=P_1+P_2-d\,P_1P_2$. Displacement method: $f=\dfrac{D^2-x^2}{4D}$.
• Refraction at a spherical surface: $\dfrac{n_2}{v}-\dfrac{n_1}{u}=\dfrac{n_2-n_1}{R}$. Apparent shift through a slab $=t\left(1-\dfrac{1}{n}\right)$.

Worked mini-example. Object 20 cm before a concave mirror, $f=15$ cm: $\dfrac{1}{v}=\dfrac{1}{f}-\dfrac{1}{u}=-\dfrac{1}{15}+\dfrac{1}{20}=-\dfrac{1}{60}$, so $v=-60$ cm (real) and $m=-\dfrac{v}{u}=-3$ — a real, inverted image magnified three times.