Geometric Optics
Core concepts
Geometric optics uses ray diagrams and the mirror/lens equation $1/f=1/d_o+1/d_i$ with magnification $m=-d_i/d_o=h_i/h_o$. Sign conventions matter: for mirrors, $f>0$ for concave; for lenses, $f>0$ for converging. Real images form where rays actually converge and can be projected; virtual images cannot. Snell's law governs refraction: $n_1\sin\theta_1=n_2\sin\theta_2$, with $n=c/v$. Total internal reflection occurs when light travels from higher to lower index and the angle exceeds the critical angle $\sin\theta_c=n_2/n_1$. Thin-film and dispersion effects follow from index differences.
Traps to watch
- Sign errors in $d_i$: negative $d_i$ means a virtual image on the same side as the object (lens) or behind the mirror.
- A convex mirror and a diverging lens both produce only virtual, upright, reduced images for real objects.
- Total internal reflection requires going from high to low index-never the reverse.
- Magnification sign tells orientation: negative $m$ means inverted.
- Frequency stays constant across a boundary; speed and wavelength change.
Worked example
An object sits $30,\mathrm{cm}$ from a converging lens with $f=10,\mathrm{cm}$. Then $1/d_i=1/10-1/30=2/30$, so $d_i=15,\mathrm{cm}$ (real, opposite side). Magnification $m=-15/30=-0.5$: the image is inverted and half-size.
Rule of thumb: object outside $2f$ gives a real, inverted, reduced image; object inside $f$ gives a virtual, upright, enlarged image.