Introduction

This chapter introduces the concept of electromagnetic waves, a groundbreaking prediction by James Clerk Maxwell. Maxwell unified electricity and magnetism by proposing that a changing electric field creates a magnetic field, and vice versa. This led to the understanding that light itself is an electromagnetic wave. The chapter discusses the crucial role of displacement current, the nature of electromagnetic waves, and the electromagnetic spectrum.

Maxwell's equations, which encapsulate the fundamental laws of electromagnetism, predict the existence of self-propagating, time-varying electric and magnetic fields that travel at the speed of light. These waves carry energy and are responsible for phenomena ranging from radio communication to light and X-rays.

Displacement Current

Maxwell identified an inconsistency in Ampere's circuital law when applied to charging capacitors. He proposed the existence of a displacement current ($I_d$), which is proportional to the time rate of change of electric flux ($\Phi_E$):

$I_d = \epsilon_0 \frac{d\Phi_E}{dt}$

This displacement current acts as a source of magnetic field, similar to the conduction current ($I_c$) due to moving charges. The generalized Ampere-Maxwell law includes both:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_c + I_d) = \mu_0 (I_c + \epsilon_0 \frac{d\Phi_E}{dt})$

The existence of displacement current explains how changing electric fields generate magnetic fields, thus bridging the gap in Maxwell's original formulation and predicting the existence of electromagnetic waves.

Figure 8.1 illustrates the problem with Ampere's law for a charging capacitor and the introduction of displacement current. Figure 8.2 shows the electric and magnetic fields between capacitor plates.

Electromagnetic Waves

Sources Of Electromagnetic Waves

Electromagnetic (EM) waves are produced by accelerated charges. A charge oscillating harmonically generates EM waves of the same frequency. A simple oscillating electric dipole is a basic source of EM waves. Importantly, neither stationary charges nor charges moving with constant velocity (steady currents) produce EM waves.

Nature Of Electromagnetic Waves

Electromagnetic waves consist of oscillating electric ($\vec{E}$) and magnetic ($\vec{B}$) fields that are perpendicular to each other and also perpendicular to the direction of wave propagation. They are transverse waves.

The electric and magnetic fields oscillate sinusoidally with space and time, described by:

$E_x = E_0 \sin(kz - \omega t)$

$B_y = B_0 \sin(kz - \omega t)$

where $k$ is the wave number, $\omega$ is the angular frequency, $E_0$ and $B_0$ are the amplitudes, and $c = \omega/k = 1/\sqrt{\mu_0\epsilon_0}$ is the speed of the wave in vacuum. The relationship between the amplitudes is $E_0 = cB_0$. EM waves propagate through vacuum, carrying energy and momentum.

Example 8.1 calculates the magnetic field vector given the electric field vector and propagation direction. Example 8.2 determines wavelength, frequency, and the electric field expression from the magnetic field equation.

Figure 8.3 illustrates a linearly polarized plane EM wave.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all possible frequencies (or wavelengths) of electromagnetic radiation. Different regions of the spectrum are classified based on their production and detection methods, although the boundaries are not always sharp.

Radio Waves

Wavelengths > 0.1 m; produced by accelerated charges in aerials; used in radio, TV, and mobile communication.

Microwaves

0.1 m to 1 mm; produced by klystrons, magnetrons; used in radar, microwave ovens, and communication.

Infrared Waves

1 mm to 700 nm; produced by vibrating atoms and molecules; detected as heat; used in remote controls, thermal imaging, and therapy.

Visible Rays

700 nm to 400 nm; produced by electronic transitions in atoms; detected by the human eye; comprises the colours we see.

Ultraviolet Rays

400 nm to 1 nm; produced by inner-shell electrons in atoms or special lamps; harmful in excess but used in sterilization and some medical treatments.

X-Rays

1 nm to $10^{-3}$ nm; produced by bombarding metal targets with high-energy electrons; used in medical imaging and cancer therapy.

Gamma Rays

< $10^{-3}$ nm; produced by radioactive decay of nuclei; highly energetic and penetrating; used in cancer treatment and medical imaging.

Figure 8.4 shows the electromagnetic spectrum, and Table 8.1 summarizes the types, wavelengths, production, and detection methods of EM waves.

Example 8.1 relates electric field amplitude to magnetic field amplitude for an EM wave. Example 8.2 analyzes parameters of an EM wave from its magnetic field equation.

Example 8.3 classifies various diagrams as representing correct or incorrect magnetic or electrostatic field lines.

Example 8.4 addresses conceptual questions about magnetic field lines and sources.

Example 8.5 calculates properties of an EM wave given its frequency and electric field amplitude.

Example 8.6 calculates wavelength, frequency, amplitudes, and energy densities for an EM wave.

Example 8.7 deals with calculating photon energies for different parts of the EM spectrum.

Example 8.8 calculates various parameters for a series LCR circuit at resonance. Example 8.9 explains the working principle of a metal detector.

Example 8.10 asks about the physical quantity that is the same for X-rays, red light, and radio waves (speed of light in vacuum).

Example 8.11 probes understanding of wave properties and calculations.

Example 8.12 concerns the energy densities of electric and magnetic fields in an EM wave.

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