Introduction

Electricity and magnetism were initially thought to be separate phenomena. However, 19th-century experiments showed that electric currents produce magnetic fields (Oersted, Ampere). Faraday and Henry then demonstrated the converse: a changing magnetic field induces an electric field (electromagnetic induction).

James Clerk Maxwell unified these phenomena with a set of equations (Maxwell's equations) showing that not only electric currents but also a **time-varying electric field** gives rise to a magnetic field. This was a crucial insight, requiring the concept of **displacement current** to ensure consistency in Ampere's circuital law when applied to situations like a charging capacitor.

From Maxwell's equations emerged the prediction of **electromagnetic waves**, which are coupled, time-varying electric and magnetic fields that propagate through space. The speed of these predicted waves in vacuum was remarkably close to the speed of light, leading to the conclusion that **light is an electromagnetic wave**. This unification of electricity, magnetism, and optics was a major scientific achievement.

Heinrich Hertz experimentally verified the existence of electromagnetic waves in 1885, opening the door to their technological use in communication (radio waves by Marconi and others), which has revolutionised the world.

This chapter discusses the necessity of displacement current, the nature and properties of electromagnetic waves, and the broad range of electromagnetic waves that constitute the electromagnetic spectrum.

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Displacement Current

To understand the concept of displacement current, consider a charging parallel plate capacitor. A conduction current $i(t)$ flows through the wires connected to the capacitor plates. We can apply Ampere's circuital law ($\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$) to find the magnetic field at a point outside the capacitor.

If we choose an Amperian loop around the wire [Fig. 8.1(a)], the conduction current $i(t)$ passes through the area bounded by the loop, and Ampere's law gives $B(2\pi r) = \mu_0 i(t)$, implying a magnetic field exists at point P.

However, if we choose a different surface with the same boundary loop, such as a surface shaped like a pot with its bottom between the capacitor plates [Fig. 8.1(b)] or a tiffin box shape [Fig. 8.1(c)], no conduction current passes through these surfaces (as there is a gap between the capacitor plates). According to the original Ampere's law, the integral $\oint \vec{B} \cdot d\vec{l}$ would be zero in these cases, leading to a contradiction in the value of $\vec{B}$ at point P. This indicated an inconsistency in Ampere's law when applied to time-varying situations.

Maxwell resolved this by proposing the existence of an additional term in Ampere's law: the **displacement current ($i_d$)**. He argued that a changing electric field must also act as a source of magnetic field.

Between the capacitor plates, there is a changing electric field $\vec{E}$ as the capacitor charges. The electric flux $\Phi_E$ through the surface S between the plates [Fig. 8.1(c)] is $\Phi_E = EA$. By Gauss's law, $EA = Q/\epsilon_0$, so $Q = \epsilon_0 E A = \epsilon_0 \Phi_E$.

The conduction current $i = dQ/dt = \frac{d}{dt}(\epsilon_0 \Phi_E) = \epsilon_0 \frac{d\Phi_E}{dt}$. This suggested that $\epsilon_0 \frac{d\Phi_E}{dt}$ acts like a current.

Maxwell defined the **displacement current** $i_d$ as:

$\mathbf{i_d = \epsilon_0 \frac{d\Phi_E}{dt}}$

He proposed that the total current ($i_{total}$) contributing to the magnetic field is the sum of the conduction current ($i_c$) and the displacement current ($i_d$):

$\mathbf{i_{total} = i_c + i_d = i_c + \epsilon_0 \frac{d\Phi_E}{dt}}$

The **Ampere-Maxwell law** (generalised Ampere's circuital law) includes both currents:

$\mathbf{\oint_C \vec{B} \cdot d\vec{l} = \mu_0 (i_c + i_d)_{enc} = \mu_0 \left(i_c + \epsilon_0 \frac{d\Phi_E}{dt}\right)_{enc}}$

where $(i_c + i_d)_{enc}$ is the total current (conduction plus displacement) passing through any surface bounded by the loop C.

Using this generalised law, the contradiction is removed. Outside the capacitor, only conduction current $i_c$ is present, $i_d = 0$. Between the plates, $i_c = 0$, and the changing electric field gives rise to a displacement current $i_d = \epsilon_0 \frac{d\Phi_E}{dt}$. For a charging capacitor, the displacement current between the plates is exactly equal to the conduction current in the wires connected to the plates, $i_d = i_c$. Thus, the total current through any surface bounded by the loop is the same, resolving the inconsistency.

Displacement current has the same physical effect as conduction current in producing a magnetic field. Its existence highlights the symmetry between electric and magnetic phenomena: a changing magnetic field produces an electric field (Faraday's law), and a changing electric field produces a magnetic field (due to displacement current).

Maxwell formulated a set of four equations, known as **Maxwell's equations**, which, along with the Lorentz force formula, describe all classical electromagnetic phenomena. These equations are:

1. $\oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0$ (Gauss's Law for electricity)

2. $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss's Law for magnetism, absence of magnetic monopoles)

3. $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$ (Faraday's Law)

4. $\oint \vec{B} \cdot d\vec{l} = \mu_0 i_{c\_enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Ampere-Maxwell Law)

The most significant consequence of these equations is the prediction of electromagnetic waves.

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Electromagnetic Waves

Maxwell's equations demonstrate that time-dependent electric and magnetic fields can sustain each other and propagate through space as **electromagnetic waves**. This is a consequence of the symmetry between electric and magnetic phenomena (changing $\vec{B}$ creates $\vec{E}$, changing $\vec{E}$ creates $\vec{B}$).

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Sources Of Electromagnetic Waves

Electromagnetic waves are produced by **accelerated charges**. Neither stationary charges (produce only static $\vec{E}$) nor charges in uniform motion (produce static $\vec{B}$ and static $\vec{E}$) radiate electromagnetic waves. An oscillating charge is an example of an accelerated charge; it produces oscillating $\vec{E}$ and $\vec{B}$ fields that propagate as a wave.

The frequency of the electromagnetic wave emitted is equal to the frequency of oscillation of the charge. The energy for the wave comes from the accelerated charge.

Hertz experimentally demonstrated electromagnetic waves in 1887 using low frequencies (radio waves). Higher frequencies (like visible light) are not easily produced by simple circuits. J.C. Bose produced shorter wavelength radio waves. Marconi achieved long-distance communication using radio waves, marking the beginning of wireless communication.

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Nature Of Electromagnetic Waves

From Maxwell's equations, the electric field $\vec{E}$ and magnetic field $\vec{B}$ in an electromagnetic wave are **perpendicular to each other and to the direction of propagation** of the wave. They are coupled time-varying fields.

A simple example is a plane electromagnetic wave propagating in the z-direction. The electric field may be along the x-direction, oscillating as $E_x = E_0 \sin(kz - \omega t)$. The magnetic field is then along the y-direction, oscillating as $B_y = B_0 \sin(kz - \omega t)$. $\vec{E}$ and $\vec{B}$ oscillate in phase with the same frequency and wavelength.

Here, $k$ is the wave number, related to the wavelength $\lambda$ by $k = 2\pi/\lambda$. $\omega$ is the angular frequency. The speed of propagation is $v = \omega/k$. Maxwell's equations show that in vacuum, the speed is $c = 1/\sqrt{\mu_0\epsilon_0}$. This value is equal to the speed of light, confirming that light is an electromagnetic wave.

The speed of electromagnetic waves in vacuum is constant ($c \approx 3 \times 10^8$ m/s) and independent of wavelength. In a material medium with permittivity $\epsilon$ and permeability $\mu$, the speed is $v = 1/\sqrt{\mu\epsilon}$. The refractive index of a medium is related to this speed.

Electromagnetic waves are **transverse waves** because the oscillating fields are perpendicular to the direction of propagation. They exhibit wave properties like diffraction, refraction, and polarisation.

Electromagnetic waves carry **energy** and **momentum**. Energy density in a region with $\vec{E}$ is $\frac{1}{2}\epsilon_0 E^2$, and with $\vec{B}$ is $\frac{1}{2\mu_0} B^2$. In an electromagnetic wave, energy is shared equally between electric and magnetic fields. Because they carry momentum, they exert pressure called **radiation pressure**. The momentum $p$ delivered to a perfectly absorbing surface with total energy transfer $U$ is $p = U/c$.

The ability of electromagnetic waves to carry energy over large distances is fundamentally important (light from the sun, radio signals for communication).

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Electromagnetic Spectrum

The **electromagnetic spectrum** is the classification of electromagnetic waves according to their frequency or wavelength. All electromagnetic waves travel at the speed of light $c$ in vacuum, but they have different wavelengths and frequencies ($\lambda = c/f$). Different regions of the spectrum are given names based on how the waves are produced or detected and their typical interactions with matter. There are no sharp boundaries between these regions.

Regions of the electromagnetic spectrum, from longest wavelength (lowest frequency) to shortest wavelength (highest frequency):

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Radio Waves

Wavelengths > 0.1 m. Frequencies from ~500 kHz to ~1000 MHz (1 GHz). Produced by accelerated charges in wires (electronic circuits, antennas). Used in radio and television broadcasting (AM, FM, TV bands), cellular communication (UHF). Long-distance transmission via reflection from the ionosphere (short waves) or line-of-sight (TV, FM, cellular).

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Microwaves

Wavelengths ~1 mm to 0.1 m. Frequencies ~1 GHz to 300 GHz. Produced by special vacuum tubes (klystrons, magnetrons). Used in radar systems (aircraft navigation, speed guns) due to their short wavelengths allowing good resolution. Also used in microwave ovens (frequency tuned to water molecule resonance for efficient heating).

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Infrared Waves

Wavelengths ~700 nm to 1 mm. Frequencies ~4 $\times 10^{14}$ Hz to $4 \times 10^{11}$ Hz. Produced by hot bodies and molecules (vibrations of atoms/molecules). Often called **heat waves** because they are strongly absorbed by many molecules (especially water, CO$_2$) increasing thermal motion. Used in physical therapy (infrared lamps), thermal imaging (night vision, Earth satellites for monitoring temperature, crops), remote controls (LEDs emitting infrared). Play a role in the greenhouse effect (trapping Earth's re-radiated thermal radiation).

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Visible Rays

Wavelengths ~400 nm (violet) to 700 nm (red). Frequencies ~7 $\times 10^{14}$ Hz to 4 $\times 10^{14}$ Hz. The part of the spectrum detected by the human eye. Produced by electrons in atoms changing energy levels. Provides visual information about the surroundings. Different organisms may be sensitive to different visible ranges or extend into IR/UV.

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Ultraviolet Rays

Wavelengths ~0.6 nm to 400 nm. Frequencies ~5 $\times 10^{16}$ Hz to $7.5 \times 10^{14}$ Hz. Produced by special lamps and hot bodies (like the Sun). Most solar UV is absorbed by the ozone layer in the stratosphere. UV radiation can be harmful; causes tanning (melanin production). Used in LASIK eye surgery (precision beams), water purifiers (killing germs). Ozone layer depletion by CFCs is a major concern.

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X-Rays

Wavelengths ~10$^{-4}$ nm to 10 nm. Frequencies ~3 $\times 10^{18}$ Hz to $3 \times 10^{16}$ Hz. Produced by bombarding metal targets with high-energy electrons (in X-ray tubes) or transitions of inner shell electrons in heavy atoms. Used as diagnostic tools (medical imaging) and in cancer treatment (radiotherapy). Can damage living tissues, requiring careful use and shielding.

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Gamma Rays

Wavelengths < 10$^{-10}$ m to less than 10$^{-14}$ m. Frequencies > 3 $\times 10^{18}$ Hz. Highest frequency/energy part of the spectrum. Produced in nuclear reactions and emitted by radioactive nuclei. Used in medicine (sterilisation, cancer treatment) and industry (gauging, inspection). Highly penetrating and ionising, requiring heavy shielding.

Table 8.1 summarises the production and detection of these different types of electromagnetic waves. Regions overlap.

Type	Wavelength range	Production	Detection
Radio	> 0.1 m	Rapid acceleration and decelerations of electrons in aerials	Receiver’s aerials
Microwave	0.1 m to 1 mm	Klystron valve or magnetron valve	Point contact diodes
Infra-red	1 mm to 700 nm	Vibration of atoms and molecules	Thermopiles, Bolometer, Infrared photographic film
Light	700 nm to 400 nm	Electrons in atoms emit light when they move from one energy level to a lower energy level	The eye, Photocells, Photographic film
Ultraviolet	400 nm to 1 nm	Inner shell electrons in atoms moving from one energy level to a lower level	Photocells, Photographic film
X-rays	1 nm to 10$^{-3}$ nm	X-ray tubes or inner shell electrons	Photographic film, Geiger tubes, Ionisation chamber
Gamma rays	<10$^{-3}$ nm	Radioactive decay of the nucleus	(Same as X-rays)

Summary

This chapter introduces electromagnetic waves, their origin, properties, and the electromagnetic spectrum.

• Maxwell's equations unify electricity and magnetism. They show that a changing electric field induces a magnetic field (displacement current) and a changing magnetic field induces an electric field (Faraday's law).
• **Displacement current** ($i_d = \epsilon_0 d\Phi_E/dt$) is due to a time-varying electric field and acts as a source of magnetic field like conduction current. Ampere's law is generalised to Ampere-Maxwell law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 (i_c + i_d)_{enc}$.
• **Electromagnetic waves** are coupled, time-varying $\vec{E}$ and $\vec{B}$ fields that propagate through space. Produced by accelerated charges (e.g., oscillating charge). Frequency of wave equals frequency of source.
• $\vec{E}$ and $\vec{B}$ are perpendicular to each other and propagation direction ($\vec{E} \times \vec{B}$). They oscillate in phase.
• Speed in vacuum $c = 1/\sqrt{\mu_0\epsilon_0}$. Speed in medium $v = 1/\sqrt{\mu\epsilon}$. $\lambda = c/f$.
• EM waves are transverse, carry energy and momentum (exert radiation pressure).
• **Electromagnetic spectrum:** Classification by frequency/wavelength. Regions (increasing $\lambda$): gamma rays, X-rays, ultraviolet, visible light, infrared, microwaves, radio waves. Different interactions with matter.

Exercises

Questions covering the concepts of displacement current, Maxwell's equations, sources and properties of electromagnetic waves, calculations involving E and B field amplitudes, wavelength, frequency, speed, energy, momentum, and radiation pressure of EM waves. Also includes questions on the electromagnetic spectrum and applications of different types of EM waves.

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ADDITIONAL EXERCISES

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