CBSE Class 12 Physics Electromagnetic Waves Notes Set 05

1. Displacement Current

Ampere's circuital law states that the line integral of the magnetic field around a closed loop equals μ₀ times the current passing through any surface bounded by that loop:
\( \oint \vec{B} \cdot d\vec{l} = \mu_0 i \)

The difficulty: Consider a capacitor being charged by a current i. Take a small circular loop around the wire. If the surface bounded by the loop is a flat disc cut by the wire, the current through it is i. If instead the surface is a pouch that passes between the capacitor plates, no conduction current crosses it, giving zero. The same loop gives two different answers, so Ampere's law in this form is incomplete when currents change with time.

Maxwell's correction: Between the plates the charge Q on the plates is changing, so the electric field E and the electric flux Φ_E = EA = \( \frac{Q}{\varepsilon_0} \) are changing. Maxwell showed that a changing electric flux acts as a source of magnetic field, just like a current. He called it the displacement current:
\( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \)
Since \( \frac{d\Phi_E}{dt} = \frac{1}{\varepsilon_0}\frac{dQ}{dt} \), we get i_d = \( \frac{dQ}{dt} \) = i. The displacement current between the plates equals the conduction current in the connecting wires, so current is continuous through the whole circuit.

Ampere–Maxwell law:
\( \oint \vec{B} \cdot d\vec{l} = \mu_0 \left( i_c + \varepsilon_0 \frac{d\Phi_E}{dt} \right) = \mu_0 (i_c + i_d) \)

Key points:
1. Displacement current is not a flow of charges; it exists wherever the electric field changes with time, even in vacuum.
2. Its SI unit is the ampere, the same as conduction current.
3. Inside a conductor carrying steady current, i_d = 0; between the plates of a charging capacitor, i_c = 0 and only i_d is present.
4. For a steady (DC) state, the electric field does not change, so i_d = 0 and a fully charged capacitor carries no current.
5. Faraday's law says a changing magnetic field produces an electric field. The displacement current idea says the reverse: a changing electric field produces a magnetic field. This symmetry is what makes electromagnetic waves possible.

2. Maxwell's Equations

Together, the following four equations describe all of classical electricity and magnetism (qualitative understanding is enough at this level):
1. Gauss's law for electricity: \( \oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0} \)
2. Gauss's law for magnetism: \( \oint \vec{B} \cdot d\vec{A} = 0 \) (there are no isolated magnetic poles)
3. Faraday's law: \( \oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \)
4. Ampere–Maxwell law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 i_c + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt} \)
Along with the Lorentz force F = q(E + v × B), these equations predicted that electric and magnetic fields can travel through space as waves with speed \( \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \), which turned out to be the speed of light. Light is therefore an electromagnetic wave.

3. Sources of Electromagnetic Waves

1. A charge at rest produces only an electric field. A charge moving with uniform velocity produces a steady current and a steady magnetic field, but no waves.
2. An accelerated charge produces electromagnetic waves. An oscillating charge, for example, produces an oscillating electric field, which produces an oscillating magnetic field, which in turn produces an oscillating electric field, and so on. The fields regenerate each other and travel outward as a wave.
3. The frequency of the wave equals the frequency of oscillation of the charge.
4. For visible light, the oscillating charges are electrons in atoms; for radio waves, they are electrons surging back and forth in an antenna.
5. Hertz first produced and detected electromagnetic waves in the laboratory (wavelength of a few metres) in 1887. J.C. Bose produced waves of much shorter wavelength (25 mm to 5 mm), and Marconi used such waves to send signals over long distances.

4. Nature of Electromagnetic Waves

For a plane electromagnetic wave travelling along the z-direction:
E_x = E₀ sin(kz − ωt) and B_y = B₀ sin(kz − ωt)
where k = \( \frac{2\pi}{\lambda} \) is the propagation constant (wave number) and ω = 2πν is the angular frequency.

Important features:
1. E and B are perpendicular to each other and to the direction of propagation, so electromagnetic waves are transverse. The direction of travel is along \( \vec{E} \times \vec{B} \).
2. E and B oscillate in phase: both reach their maximum and zero values at the same place and time.
3. Speed in vacuum: c = \( \frac{\omega}{k} \) = νλ = \( \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \) ≈ 3 × 10⁸ m s⁻¹, with μ₀ = 4π × 10⁻⁷ T m A⁻¹ and ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².
4. Ratio of amplitudes: \( \frac{E_0}{B_0} = c \), and at every instant E = cB. Because c is so large, the magnetic field of a light wave is numerically very small compared with its electric field.
5. Electromagnetic waves need no material medium; they travel through vacuum. All of them, from radio waves to gamma rays, travel at the same speed c in vacuum.
6. In a material medium of permittivity ε and permeability μ, the speed is v = \( \frac{1}{\sqrt{\mu \varepsilon}} \), which is less than c. The refractive index is n = \( \frac{c}{v} \) = \( \sqrt{\mu_r \varepsilon_r} \).
7. When a wave passes from one medium into another, its frequency stays the same; its speed and wavelength change.

5. Energy and Momentum of Electromagnetic Waves

1. Like other waves, electromagnetic waves carry energy. In the region of the wave, energy is stored in both the electric and the magnetic field:
energy density u_E = \( \frac{1}{2}\varepsilon_0 E^2 \) and u_B = \( \frac{B^2}{2\mu_0} \). In an electromagnetic wave these two are equal on average, so the energy is shared equally between the electric and magnetic fields.
2. Electromagnetic waves also carry momentum. If a wave delivers energy U to a surface that absorbs it completely, the momentum delivered is
p = \( \frac{U}{c} \)
For a perfectly reflecting surface the momentum delivered is \( \frac{2U}{c} \).
3. Since the wave delivers momentum, it exerts a force and hence a pressure on the surface, called radiation pressure. For sunlight on Earth it is very small (of the order of 10⁻⁶ N m⁻²), which is why we do not feel it.
4. The warmth felt in sunlight is direct evidence that electromagnetic waves transport energy from the Sun to the Earth.

6. Electromagnetic Spectrum

The full range of electromagnetic waves, arranged in order of wavelength or frequency, is called the electromagnetic spectrum. The different regions differ only in wavelength (and frequency); they are produced and detected in different ways, and the boundaries between them are not sharp.

(i) Radio waves: Used in radio and TV communication. AM radio uses 530 kHz to 1710 kHz; short-wave bands go up to about 54 MHz; TV uses about 54 MHz to 890 MHz; FM radio uses 88 MHz to 108 MHz. Cellular phones use radio waves in the ultra high frequency (UHF) band.
(ii) Microwaves: Short-wavelength radio waves used in radar (aircraft navigation, speed guns) and in microwave ovens. In an oven, the microwave frequency (about 2.45 GHz) is chosen to match a resonant frequency of water molecules, so the energy goes into the water in food and heats it.
(iii) Infrared waves: Called heat waves, because water molecules and many other molecules in most materials absorb them readily and warm up. Used in physical therapy, infrared lamps, remote controls of TVs and other devices, and night-vision equipment. Infrared radiation trapped by the atmosphere keeps the Earth warm (greenhouse effect).
(iv) Visible light: The narrow band our eyes can detect, from violet (about 400 nm) to red (about 700 nm). Some animals see beyond this: snakes can detect infrared, and many insects can see into the ultraviolet.
(v) Ultraviolet rays: The Sun is an important source. Most of the Sun's UV is absorbed by the ozone layer in the atmosphere at about 40–50 km altitude. Too much UV is harmful to humans and causes skin damage. Because of their short wavelength, UV rays can be focused into very fine beams for precise uses such as LASIK eye surgery. UV lamps are used to kill germs in water purifiers. Ordinary glass absorbs much of the UV, so welders wear special glass goggles to protect their eyes.
(vi) X-rays: Produced by bombarding a metal target with high-energy electrons. Used as a diagnostic tool in medicine and to treat some forms of cancer. They can damage living tissue, so exposure must be kept as low as possible.
(vii) Gamma rays: The highest frequency, shortest wavelength region. Produced in nuclear reactions and emitted by radioactive nuclei. Used in medicine to destroy cancer cells.

Points to remember:
1. From radio waves to gamma rays, the wavelength decreases and the frequency (and the energy carried per photon) increases.
2. The way a wave interacts with matter depends on its wavelength. Radio and microwaves interact with whole circuits or molecules; infrared with molecular vibrations; visible and UV with outer electrons of atoms; X-rays and gamma rays with inner electrons and nuclei.
3. All regions travel at c in vacuum and all are transverse waves of oscillating E and B fields.

7. Quick Formula Recap