Geometrical optics: Class 10 Physics

Key points

Geometrical optics treats light as straight lines called rays. It explains how light is reflected and refracted by mirrors, prisms and lenses. Refraction is the bending of light when it passes from one transparent medium to another, because the speed of light changes in the new medium.

Laws of refraction:

• The incident ray, the refracted ray and the normal at the point of incidence lie in the same plane.
• Snell's law: the ratio sin i / sin r is a constant for a given pair of media, where i is the angle of incidence and r is the angle of refraction. This constant is the refractive index of the second medium relative to the first.

The absolute refractive index of a medium is n = c / v, where c is the speed of light in vacuum (3 × 10⁸ m/s) and v is the speed of light in the medium. It has no unit. For water n is about 1.33 and for glass about 1.5. A ray entering a denser medium bends toward the normal. A ray entering a rarer medium bends away from the normal. Refraction explains why a pool looks shallower than it is and why a straw in water looks bent.

Total internal reflection occurs when light goes from a denser to a rarer medium. As the angle of incidence grows, the refracted ray bends further from the normal. At the critical angle C, the refracted ray travels along the surface (angle of refraction 90°). Beyond C all the light is reflected back into the denser medium. The two conditions are: light travels from a denser to a rarer medium, and the angle of incidence is greater than the critical angle. The critical angle is given by sin C = 1/n.

Total internal reflection has important uses. Prisms in binoculars and periscopes turn light through 90° or 180° with almost no loss. An optical fibre is a thin glass strand with a core of high refractive index and a cladding of lower index. Light entering one end keeps reflecting inside and comes out of the other end, even when the fibre is bent. Fibres carry telephone and internet signals and are used by doctors in endoscopes to see inside the body. Diamonds sparkle because their critical angle is small (about 24°).

A prism is a transparent block with two inclined faces. A ray passing through it bends twice, and the emerging ray is deviated toward the base. White light splits into the colours of the spectrum (red, orange, yellow, green, blue, indigo, violet) because each colour has a different refractive index in glass. Violet bends the most and red the least. This splitting is called dispersion. A rainbow is formed by dispersion and internal reflection of sunlight in water drops.

Lenses and their rays:

• A convex (converging) lens is thicker in the middle. It brings parallel rays to a focus, so its focal length is positive.
• A concave (diverging) lens is thinner in the middle. It spreads parallel rays out as if they came from a point, so its focal length is negative.
• A ray through the optical centre goes straight. A ray parallel to the principal axis passes through the focus after refraction. A ray through the focus emerges parallel to the axis.

A convex lens forms a real, inverted image when the object is beyond the focus. The image is smaller when the object is beyond 2f, the same size at 2f and larger between f and 2f. When the object is inside the focal length the image is virtual, upright and magnified, which is how a magnifying glass works. A concave lens always gives a virtual, upright and diminished image.

The lens formula is 1/p + 1/q = 1/f, where p is the object distance, q is the image distance and f is the focal length. We use the real-is-positive sign convention: distances of real objects and real images are positive, and virtual image distances are negative. Magnification m = image height / object height = q / p. The power of a lens is P = 1/f, with f in metres. Its unit is the dioptre (D). A convex lens has positive power and a concave lens negative power.

The human eye forms a real, inverted image on the retina through the cornea and a convex lens. The ciliary muscles change the focal length of the lens so that near and far objects can be focused (accommodation). In short sight (myopia) distant objects are blurred because the image forms in front of the retina. A concave lens corrects it. In long sight (hypermetropia) near objects are blurred because the image forms behind the retina. A convex lens corrects it.

A simple microscope (magnifying glass) is a convex lens of short focal length. Its angular magnification is M = 1 + d/f, when the final image is at the near point d = 25 cm. A compound microscope uses two convex lenses, an objective and an eyepiece, to give greater magnification of tiny objects. An astronomical telescope uses a long focal length objective and a short focal length eyepiece to view distant objects, with magnification M = f₀ / fₑ.

Study tip: Practise ray diagrams for a convex lens for all object positions, and learn the nature of each image in a small table. In numericals write the lens formula, keep distances in the same unit, and check the sign of q to name the image as real or virtual. Draw a neat diagram of an optical fibre and of total internal reflection with labels.

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Common mistakes

• Saying total internal reflection can happen when light goes from a rarer to a denser medium. It cannot.
• Using f in centimetres when calculating power. Convert to metres first.
• Giving a concave lens a positive focal length and power. Both are negative.
• Confusing myopia and hypermetropia and their correcting lenses.
• Drawing refracted rays bending the wrong way at the boundary. Into a denser medium the ray bends toward the normal.

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