Refraction Total Internal

9.3 Refraction of Light

When a ray of light travels from one transparent medium into another, its speed changes and, unless it strikes the boundary at 90°, its direction also changes. This bending of light on crossing a boundary is called refraction.

The absolute refractive index of a medium is

where \(c = 3\times 10^8\) m/s is the speed of light in vacuum and \(v\) its speed in the medium. Since \(v < c\) always, \(n > 1\). The relative refractive index of medium 2 with respect to medium 1 is

• When light passes from a rarer to a denser medium (air → glass), it bends towards the normal — \(\theta_2<\theta_1\).
• When it goes from denser to rarer (glass → air), it bends away from the normal — \(\theta_2>\theta_1\).

Apparent Depth

An object submerged in water appears to lie closer to the surface than it really is. If the object is at real depth \(h\) in a medium of refractive index \(n_2\) and the observer looks from medium of index \(n_1\),

For a coin at the bottom of a swimming pool 2 m deep (\(n_{\text{water}}=1.33\)), the apparent depth is only \(2/1.33 \approx 1.5\) m. A straw dipped in a glass of water looks bent at the water-air interface for the same reason.

9.4 Total Internal Reflection (TIR)

When light travels from a denser to a rarer medium, the refracted ray bends away from the normal. As the angle of incidence is increased, the refracted angle approaches 90°. At a particular angle of incidence — the critical angle \(\theta_c\) — the refracted ray just grazes the boundary. Beyond that, no refraction is possible: all of the light is reflected back into the denser medium.

Everyday Applications of TIR

1. Mirage on a hot road: a layer of hot (hence less dense, lower-n) air sits over cooler air above. Light from the sky grazing downward into the hotter layer undergoes TIR, so the driver sees an inverted image of the sky that looks like a patch of water.
2. Brilliance of a diamond: diamond has \(n=2.42\), giving a small critical angle \(\theta_c \approx 24.4°\). Rays that enter through the faceted top are trapped by multiple total internal reflections and emerge only through the top facets, producing the "fire" of the stone.
3. Totally reflecting prisms in binoculars, periscopes and SLR cameras: right-angle glass prisms use TIR at 45° (greater than the glass critical angle of about 42°) — far more efficient than a silvered mirror.
4. Optical fibres: a thin glass core of high \(n\) is surrounded by cladding of slightly lower \(n\). Light entering at a small angle bounces by repeated TIR along the length of the fibre with negligible loss — the backbone of modern telecommunications, endoscopy and fibre-optic sensors.

9.5 Refraction at Spherical Surfaces and Lenses

For a single spherical refracting surface of radius \(R\) separating media of refractive indices \(n_1\) (object side) and \(n_2\) (image side), the Cartesian-sign-convention formula is

Lens Maker's Formula

A thin lens is two refracting spherical surfaces back-to-back. Applying the surface formula twice and combining:

where \(n\) is the refractive index of the lens material relative to the surrounding medium, and \(R_1,R_2\) are the radii of the two surfaces (with sign).

Thin-Lens Formula and Magnification

Note the sign pattern: unlike the mirror formula which has "\(+\)" between the two reciprocals, the lens formula has "\(-\)".

Combination of Thin Lenses in Contact

When two thin lenses of focal lengths \(f_1\) and \(f_2\) are placed in contact, the system behaves like a single lens of focal length \(f\) given by

Power of a Lens

The power \(P\) of a lens is the reciprocal of its focal length in metres:

For lenses in contact, powers add: \(P = P_1 + P_2\). A converging lens has \(P > 0\); a diverging lens has \(P < 0\).

Convex vs Concave Lens

Six Cases of Image Formation by a Convex Lens

Worked Examples — Refraction and Lenses