Electromagnetic Waves & Ray Optics

Light as an EM wave · the spectrum · mirrors · refraction · total internal reflection · lenses · Class 12 Physics

1The big idea: light is a wave of fields

This chapter joins two halves of the same story. In the first half, Maxwell's equations tell us that a changing electric field makes a magnetic field and a changing magnetic field makes an electric field — so the two can keep regenerating each other and sail off through empty space as an electromagnetic (EM) wave. When Maxwell worked out how fast that wave travels, the number came out equal to the measured speed of light. The conclusion was irresistible: light is an electromagnetic wave.

The second half — ray optics — then does something that looks like a step backwards but is enormously useful. As long as the objects light meets (mirrors, lenses, water) are far larger than its wavelength, we can forget the wave entirely and pretend light travels in straight rays. Everything about mirrors and lenses follows from just two rules for those rays: they reflect and they refract. That is why one chapter carries both ideas — the wave explains what light is; the ray explains what it does when it hits glass.

2Electromagnetic waves & the speed of light

An electromagnetic wave is a self-sustaining ripple of electric and magnetic fields. Three facts define it, and the exam returns to them again and again.

It is transverse. The electric field E and the magnetic field B both oscillate at right angles to the direction the wave travels — and at right angles to each other. So E, B and the direction of travel form a mutually perpendicular set, in that cyclic order. The wave carries no oscillation along its own path; nothing moves back and forth in the travel direction the way it does in sound.

It needs no medium, and its speed is fixed by two constants. Unlike sound, an EM wave travels through a perfect vacuum. Maxwell showed its speed depends only on the electric constant ε₀ (permittivity of free space) and the magnetic constant μ₀ (permeability of free space):

Put in the numbers — μ₀ = 4π × 10⁻⁷ T m A⁻¹ and ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻² — and out drops c ≈ 3 × 10⁸ m s⁻¹. The fact that a purely electromagnetic calculation reproduces the speed of light is the whole reason we say light is electromagnetic.

Like every wave, it obeys c = f λ. Frequency f times wavelength λ equals the wave speed. In vacuum that speed is always c, so a high frequency must mean a short wavelength. That single relation organises the entire spectrum below.

3The electromagnetic spectrum

Radio waves and gamma rays are the same kind of wave — they differ only in frequency (and therefore wavelength). Laid out from low frequency to high, they form one continuous ribbon. You should be able to name them in order and give one use of each.

• Radio waves (longest λ, up to metres and beyond) — radio and television broadcasting, and mobile-phone signalling.
• Microwaves (cm scale) — radar, satellite links, and microwave ovens (they resonate with water molecules and heat food).
• Infrared (just longer than red light) — TV remote controls, thermal (night-vision) imaging, and physiotherapy heat lamps. Felt as radiant heat.
• Visible light (about 400–700 nm) — the one octave our eyes detect, violet through red; it lets us see.
• Ultraviolet (UV) — sterilising equipment and water, and causing sun-tan; the Sun is a strong source and the ozone layer shields us from most of it.
• X-rays (very short λ) — medical imaging of bones and teeth, and inspecting welds or luggage.
• Gamma rays (shortest λ, highest frequency and energy) — emitted by radioactive nuclei; used to kill cancer cells (radiotherapy) and to sterilise medical supplies.

Notice the pattern: as you move right, λ shrinks, f rises, and — because a photon's energy is proportional to frequency — the radiation gets more penetrating and more dangerous. Radio waves pass through you harmlessly; gamma rays can smash molecules apart.

4The Cartesian sign convention

Before any mirror or lens formula makes sense, you must agree on which distances are positive. Ray optics uses one bookkeeping rule and applies it everywhere:

• Measure all distances from the pole (of a mirror) or the optical centre (of a lens).
• Take the incident light as travelling left to right. Distances measured in that direction are positive; distances measured against it (to the left) are negative.
• Heights measured upward from the axis are positive; downward, negative.

Two consequences you will lean on constantly: a real object sitting to the left has negative object distance u. A converging (convex) lens has positive focal length f; a concave mirror has negative f because its focus lies to the left of the pole. If you plug numbers in with their signs, the formulas below tell you not only how big the image is but which side it is on and whether it is upright — no separate rules to memorise.

5Reflection & spherical mirrors

The law of reflection is the simplest rule in optics: the angle of incidence equals the angle of reflection, both measured from the normal, and the incident ray, reflected ray and normal lie in one plane. Applying that law to a curved (spherical) mirror gives the mirror formula, which links the object distance u, the image distance v and the focal length f (where f = R/2, half the radius of curvature):

The magnification m is the ratio of image height h′ to object height h. Its sign carries meaning: a negative m means an inverted image (typical of a real image), a positive m means an erect one. A concave mirror can magnify or shrink depending on where the object sits; a convex mirror always gives a small, erect, virtual image — which is exactly why it is used as a vehicle's wide-view rear mirror.

6Refraction & the refractive index

When light passes from one medium into another it changes speed, and at the boundary it bends. Slowing down (entering a denser medium like glass) bends the ray toward the normal; speeding up bends it away. How much a medium slows light is captured by its refractive index:

The first form says n is how many times slower light travels in the medium than in vacuum (so n is always ≥ 1; water is 1.33, ordinary glass 1.5, diamond 2.42). The second form — Snell's law for light entering from air — relates the angle of incidence i to the angle of refraction r. The general statement is n₁ sin i = n₂ sin r: the product of index and the sine of the angle is conserved across the boundary. The frequency of the light never changes on refraction — only its speed and wavelength do — which is why the colour you see is unaltered.

7Total internal reflection

Now send light the other way — from a dense medium (glass, water) out toward a rarer one (air). It bends away from the normal, so the refracted ray is more slanted than the incident one. Increase the angle of incidence and the refracted ray tips further and further toward the surface, until at one special angle it grazes along the boundary at r = 90°. That incidence angle is the critical angle θ_c. Push past it and refraction becomes impossible — all the light bounces back inside. This is total internal reflection (TIR).

Find θ_c straight from Snell's law by setting r = 90° (going from medium of index n into air):

A larger index means a smaller critical angle — diamond's n = 2.42 gives θ_c ≈ 24.4°, so light striking almost any inner face bounces around inside, and that trapped, re-emerging light is the diamond's sparkle. TIR also carries every internet signal: an optical fibre is a glass thread so light, entering nearly along the axis, hits the wall beyond the critical angle at every bounce and is guided kilometres with almost no loss. The shimmering "water" on a hot road — a mirage — is the same effect in warm, low-density air near the ground.

8Thin lenses, the lensmaker & power

A lens is two refracting surfaces working together. Trace a ray through both using Snell's law and — for a thin lens with object and image in air — everything collapses into one tidy relation, the thin-lens formula, together with its magnification:

Watch the signs against the mirror formula: a mirror folds light back so its formula has a plus, a lens transmits light forward so its formula has a minus, and the lens magnification is +v/u (no leading minus). A convex lens has f > 0 and converges light; a concave lens has f < 0 and diverges it, always giving a small, erect, virtual image.

The lensmaker's formula

Where does f come from? It is set by the glass and the two surface curvatures — that is what the lensmaker's formula tells you:

Here n is the refractive index of the lens material and R₁, R₂ are the radii of curvature of the two faces (with their own signs). Two lessons fall out immediately. First, a stronger glass (bigger n) or more sharply curved faces (smaller R) give a shorter f — a "fatter" lens. Second, since n depends slightly on colour, f does too: that is the origin of chromatic aberration, the coloured fringing round a cheap lens.

Power of a lens

Opticians do not quote focal length; they quote power, the reciprocal of the focal length in metres:

A short-focus, strongly converging lens has a large positive power; a diverging lens has negative power. The neat payoff: powers of thin lenses in contact simply add, P = P₁ + P₂. A +2 D reading lens has f = 0.5 m; a "−1.5" spectacle prescription is a diverging lens of half-metre-ish focal length for short sight.

9Microscope & telescope, in brief

Both instruments are two convex lenses in a tube — an objective facing the thing you look at and an eyepiece at your eye — but they are tuned for opposite jobs.

Compound microscope. To magnify something tiny and close, both lenses are short-focus. The objective forms a real, enlarged image inside the tube; the eyepiece then magnifies that like a simple magnifier. The magnifications multiply, so the total is roughly

where L is the tube length and D ≈ 25 cm is the near point of the eye. Short f_o and short f_e both help.

Astronomical telescope. To magnify something huge but far away, the objective is instead a long-focus lens that gathers parallel rays from the star into a small real image; the eyepiece magnifies it. In normal (relaxed-eye) adjustment the angular magnification is simply the ratio of focal lengths:

So a telescope wants a long objective and a short eyepiece — the reverse of a microscope. Remember it by the job: a microscope brings the small-and-near up close, a telescope brings the large-and-far in near.

10Common confusions to clear up

• Refractive index is never less than 1. Light in a medium travels slower than in vacuum, so n = c/v ≥ 1. If you get n < 1 you have divided the wrong way round.
• Total internal reflection only happens going dense → rare. Light must be trying to leave the denser medium; there is no critical angle when light goes from air into glass.
• Frequency does not change on refraction — wavelength and speed do. Colour is set by frequency, so a red ray stays red inside glass even though it slows and its wavelength shortens.
• Mirror and lens formulas differ by a sign. Mirror: 1/v + 1/u = 1/f with m = −v/u. Lens: 1/v − 1/u = 1/f with m = +v/u. Mixing them is the single most common slip.
• A positive image distance means a real image (for a lens). Under this convention a real image lands on the far side (v > 0); a virtual image sits on the object's side (v < 0) and is erect.
• In E₀ = c B₀, B is not "weaker". The magnetic field only looks small numerically because c is large; the wave's energy is split equally between the electric and magnetic parts.

11Check yourself

Class 12 Physics · Electromagnetic Waves & Ray Optics · aligned to NCERT Chapters 8–9 · SmartStudy.School