Ray Optics: Mirrors, Lenses & Instruments

Reflection at Spherical Mirrors

Light reflects off spherical mirrors (concave and convex) following the law of reflection. The mirror formula relates object distance (u), image distance (v), and focal length (f):

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

where f = R/2 (R = radius of curvature). All distances are measured from the pole using the sign convention (distances along the incident ray are positive).

Magnification: m = -v/u = $\frac{h_{i}}{h_{o}}$, where $h_{i}$ and $h_{o}$ are image and object heights respectively. Negative m indicates an inverted image.

Refraction at Spherical Surfaces

When light passes from medium of refractive index n₁ to n₂ through a curved surface of radius R:

$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$

Thin Lens Formula

For a thin lens with focal length f:

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

Lens Maker's Equation (lens of refractive index n₂ in medium n₁):

$\frac{1}{f} = \left(\frac{n_2}{n_1} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Power of a lens: P = 1/f (in dioptres when f is in metres). For combination: P = P₁ + P₂ (thin lenses in contact).

Magnification: m = v/u (note: no negative sign unlike mirrors).

Prism

Deviation by a prism of angle A and refractive index n:

$\delta = (i_1 + i_2) - A$

At minimum deviation (δ_m), i₁ = i₂ and the ray passes symmetrically:

$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

For thin prism (small A): δ = (n - 1)A.

Dispersion: Angular dispersion = ($n_{V}$ - $n_{R}$)A.
Dispersive power ω = ($n_{V}$ - $n_{R}$)/($n_{Y}$ - 1).

Total Internal Reflection (TIR)

When light travels from denser to rarer medium and angle of incidence exceeds the critical angle θ_c:

$\sin\theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$

Applications: optical fibres, mirage, sparkling of diamond (θ_c ≈ 24.4°).

Optical Instruments

Simple Microscope: Magnifying power M = 1 + D/f (image at near point), M = D/f (image at infinity), where D = 25 cm (least distance of distinct vision).

Compound Microscope: M = (L/$f_{o}$)(1 + D/$f_{e}$) where L is tube length, $f_{o}$ and $f_{e}$ are focal lengths of objective and eyepiece.

Astronomical Telescope (normal adjustment): M = -$\frac{f_{o}}{f_{e}}$, tube length = $f_{o}$ + $f_{e}$.

Resolving Power of telescope = D/1.22λ (D = aperture diameter).

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