Ray Optics

Reflection: The angle of incidence equals the angle of reflection (measured from the normal). A plane mirror produces a virtual, erect, laterally inverted image of the same size, at the same distance behind the mirror.

Spherical mirrors — concave (converging, f < 0 by Cartesian convention) and convex (diverging, f > 0). Focal length f = R/2, where R is the radius of curvature; [f] = [L], SI unit: metre (m).

The mirror formula: 1/v + 1/u = 1/f, where u = object distance (always negative, object on left), v = image distance, and f = focal length. Magnification m = -v/u = $\frac{h_{i}}{h_{o}}$; m > 0 means erect, m < 0 means inverted; |m| > 1 means magnified, |m| < 1 means diminished. Note: [v], [u], [f] all have [L], SI unit: m.

Cartesian sign convention (CRITICAL for NEET): (i) All distances measured from the pole (mirror) or optical center (lens). (ii) The principal axis is the x-axis; object is placed to the left, incident light travels left to right. (iii) Distances in the direction of incident light (left to right) are POSITIVE. (iv) Distances against the direction of incident light (right to left) are NEGATIVE. (v) Heights above the principal axis are positive; below are negative. For mirrors: u is always negative. Concave mirror: f is negative. Convex mirror: f is positive.

Refraction — Snell's law: $n_{1}$ sin $\theta_{1}$ = $n_{2}$ sin $\theta_{2}$, where n = c/v is the refractive index (dimensionless). Light bends toward the normal when entering a denser medium.

Total internal reflection (TIR): Occurs when light travels from a denser to a rarer medium ($n_{1}$ > $n_{2}$) and the angle of incidence exceeds the critical angle: sin $\theta_{c}$ = $\frac{n_{2}}{n_{1}}$. Both conditions are necessary: (1) denser to rarer medium, (2) angle > $\theta_{c}$.
Applications: optical fibre, mirage, diamond sparkle (n = 2.42, $\theta_{c}$ = 24.4 deg), totally reflecting prisms.

Thin lens formula: 1/v - 1/u = 1/f.
Magnification m = v/u (NO negative sign, unlike mirrors). For lenses: u is always negative (object on left), convex lens f > 0, concave lens f < 0.
The lensmaker's equation: 1/f = (n - 1)(1/$R_{1}$ - 1/$R_{2}$), where $R_{1}$ is the radius of the surface facing the object and $R_{2}$ is the other surface; for a biconvex lens, $R_{1}$ > 0 and $R_{2}$ < 0.

Power of a lens: P = 1/f (f in metres); [P] = [$L^{-1}$], SI unit: dioptre (D). Convex lens: P > 0; concave lens: P < 0.
For thin lenses in contact: P = $P_{1}$ + $P_{2}$, or equivalently 1/f = 1/$f_{1}$ + 1/$f_{2}$.

Prism: Deviation $\delta$ = (i + e) - A, where i = angle of incidence, e = angle of emergence, A = angle of the prism.
At minimum deviation: i = e, r = A/2, and n = sin((A + $\delta_{m}$)/2)/sin(A/2).
For a thin prism (small A): $\delta$ = (n - 1)A.
Dispersion is the splitting of white light; dispersive power $\omega$ = ($n_{v}$ - $n_{r}$)/($n_{y}$ - 1).

Optical instruments: Compound microscope magnifying power M = (v/u)(D/$f_{e}$) (image at D) or M = -(L/$f_{o}$)(D/$f_{e}$) (image at infinity); the objective has SHORT focal length, eyepiece has relatively longer focal length.
Astronomical telescope: M = -$\frac{f_{o}}{f_{e}}$ (normal adjustment), length L = $f_{o}$ + $f_{e}$.

The key testable concept is the mirror and lens formulae with rigorous sign convention application, which accounts for the most common errors and the majority of numerical questions in NEET ray optics.

Solved Numericals

N₁. An object is placed 30 cm in front of a concave mirror of focal length 20 cm. Find the image position, magnification, and nature of image.

Sign convention: Object is on the left. u = -30 cm. Concave mirror: f = -20 cm.

Mirror formula: 1/v + 1/u = 1/f 1/v + 1/(-30) = 1/(-20) 1/v = -$\frac{1}{20}$ + $\frac{1}{30}$ = (-3 + 2)/60 = -$\frac{1}{60}$ v = -60 cm.

The negative value of v means the image is on the same side as the object (in front of the mirror) — hence real.

Magnification: m = -v/u = -(-60)/(-30) = -$\frac{60}{30}$ = -2.

m = -2: negative sign means inverted; |m| = 2 > 1 means magnified (twice the object size).

Nature: Real, inverted, magnified. (Object is between F and 2F; image is beyond 2F.)

N₂. A glass prism of refractive index 1.5 has angle A = 60 deg. Find the minimum deviation.

At minimum deviation: n = sin((A + $\delta_{m}$)/2) / sin(A/2) 1.5 = sin((60 + $\delta_{m}$)/2) / sin(30 deg) 1.5 = sin((60 + $\delta_{m}$)/2) / 0.5 sin((60 + $\delta_{m}$)/2) = 0.75 (60 + $\delta_{m}$)/2 = arcsin(0.75) = 48.59 deg 60 + $\delta_{m}$ = 97.18 deg $\delta_{m}$ = 37.18 deg ~ 37.2 deg.

For thin prism with same n and A = 10 deg: $\delta$ = (n - 1)A = (1.5 - 1) x 10 = 0.5 x 10 = 5 deg.

Note: The thin prism formula is valid only when A is small (typically < 10 deg). For A = 60 deg, the exact formula must be used.

N₃. The refractive index of glass is 1.5. Find the critical angle for the glass-air interface. Will TIR occur at 45 deg? At 40 deg?

Critical angle: sin $\theta_{c}$ = $\frac{n_{2}}{n_{1}}$ = $\frac{n_{air}}{n_{glass}}$ = $\frac{1}{1}$.5 = $\frac{2}{3}$. $\theta_{c}$ = arcsin($\frac{2}{3}$) = 41.81 deg ~ 41.8 deg.

At 45 deg: Since 45 deg > $\theta_{c}$ (41.8 deg), total internal reflection WILL occur (assuming light goes from glass to air). No refracted ray exists.

At 40 deg: Since 40 deg < $\theta_{c}$ (41.8 deg), TIR will NOT occur. The light will be partially reflected and partially refracted into air.

Important: TIR requires light travelling from denser to rarer medium. If light goes from air to glass, TIR can NEVER occur regardless of angle.