In classical electromagnetism, Oersted and Ampere established that an electric current produces a magnetic field, while Faraday showed that a time-varying magnetic field induces an electric field. James Clerk Maxwell proposed the converse: a time-varying electric field must also generate a magnetic field. This symmetry unified electricity, magnetism, and light into a single framework.
Consider a parallel plate capacitor $C$ being charged by a time-dependent current $i(t)$. To find the magnetic field at a point $P$ outside the capacitor, we apply Ampere's circuital law along a circular loop of radius $r$ centered on the wire:
$$\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 i(t)$$This yields a non-zero magnetic field $B$ at point $P$:
$$B(2\pi r) = \mu_0 i(t) \implies B = \frac{\mu_0 i(t)}{2\pi r}$$However, if we choose a pot-shaped surface that does not touch the wire but has its bottom passing through the gap between the capacitor plates, no conduction current passes through this surface. Applying Ampere's law to this surface yields:
$$\oint \mathbf{B}\cdot d\mathbf{l} = 0$$This contradiction (where the magnetic field at $P$ is calculated to be both non-zero and zero depending on the surface chosen) indicates that Ampere's circuital law was incomplete and missing a crucial term.
Historical Chronicle: James Clerk Maxwell (1831–1879)
To resolve the inconsistency, Maxwell postulated the existence of an additional current acting in the space between the capacitor plates, termed the displacement current ($i_d$). This current is produced by the time-varying electric field between the plates.
The total current $i$ is the sum of the conduction current ($i_c$) carried by charge carriers in conductors, and the displacement current ($i_d$) due to changing electric fields:
$$i = i_c + i_d = i_c + \epsilon_0 \frac{d\Phi_E}{dt}$$Substituting this total current into Ampere's law yields the generalized Ampere-Maxwell Law :
$$\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 i_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$$Outside the capacitor plates, $i_d = 0$ and $i_c = i$. Inside the plates, $i_c = 0$ and $i_d = i$. This formulation ensures continuity of current and resolves the charging paradox.
Maxwell's equations represent a complete, unified set of four fundamental equations that mathematically express all classical laws of electromagnetism in vacuum:
| Law / Equation Name | Mathematical Form | Physical Significance |
|---|---|---|
| Gauss's Law for Electricity | $$\oint \mathbf{E}\cdot d\mathbf{A} = \frac{Q}{\epsilon_0}$$ | Relates the net electric flux through a closed surface to the enclosed net charge. |
| Gauss's Law for Magnetism | $$\oint \mathbf{B}\cdot d\mathbf{A} = 0$$ | States that the net magnetic flux through any closed surface is zero; magnetic monopoles do not exist. |
| Faraday's Law of Induction | $$\oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$$ | Shows that a time-varying magnetic field induces a spatially varying, non-conservative electric field. |
| Ampere-Maxwell Law | $$\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 i_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$$ | Shows that magnetic fields are generated by both conduction currents and time-varying electric fields. |
The most profound prediction of Maxwell's equations is the existence of electromagnetic waves (EM waves), which are self-sustaining, coupled time-varying electric and magnetic fields propagating through space.
EM waves are transverse waves: the electric field vector $\mathbf{E}$ and magnetic field vector $\mathbf{B}$ are perpendicular to each other and perpendicular to the direction of wave propagation.
For a plane electromagnetic wave propagating along the $+z$ direction:
$$E_x(z, t) = E_0 \sin(kz - \omega t)$$ $$B_y(z, t) = B_0 \sin(kz - \omega t)$$Where:
Historical Chronicle: Heinrich Rudolf Hertz (1857–1894)
The classification of electromagnetic waves according to their frequency or wavelength is known as the electromagnetic spectrum . There are no sharp boundaries between adjacent regions.
| Type of Wave | Wavelength Range | Production Method | Detection Method | Key Applications |
|---|---|---|---|---|
| Radio Waves | $> 0.1 \text{ m}$ | Accelerated motion of charges in conducting wires / antennas. | Receiver aerials. | Radio and television communication, cellular phones (UHF). |
| Microwaves | $0.1 \text{ m}$ to $1 \text{ mm}$ | Special vacuum tubes (klystrons, magnetrons, Gunn diodes). | Point contact diodes. | Radar systems for aircraft navigation, speed guns, microwave ovens. |
| Infrared (Heat) Waves | $1 \text{ mm}$ to $700 \text{ nm}$ | Vibration of atoms and molecules in hot bodies. | Thermopiles, bolometers, infrared photographic film. | Physical therapy, greenhouse effect, remote switches, night vision. |
| Visible Light | $700 \text{ nm}$ to $400 \text{ nm}$ | Electrons in atoms transitioning between energy levels. | Human eye, photocells, photographic film. | Vision, photography, optical instruments. |
| Ultraviolet (UV) Rays | $400 \text{ nm}$ to $1 \text{ nm}$ | Inner shell atomic transitions, very hot bodies (the Sun, arc lamps). | Photocells, photographic film. | Water purification (killing germs), LASIK eye surgery, forensics. |
| X-Rays | $1 \text{ nm}$ to $10^{-3} \text{ nm}$ | Bombarding a high-melting-point metal target with high-energy electrons. | Photographic film, Geiger tubes, ionization chambers. | Medical diagnostics (fractures), cancer treatment, security scanners. |
| Gamma Rays | $< 10^{-3} \text{ nm}$ | Radioactive decay of atomic nuclei, nuclear reactions. | Geiger tubes, ionization chambers, photographic film. | Radiotherapy (destroying cancer cells), nuclear physics research. |
Mathematical Problem Statement (Example 8.1): A plane electromagnetic wave of frequency 25 MHz travels in free space along the $x$-direction. At a particular point in space and time, the electric field is $\mathbf{E} = 6.3 \mathbf{\hat{j}}$ V/m. What is the magnetic field $\mathbf{B}$ at this point?
Step-by-Step Algorithmic Solution Trace:
- Identify the relationship between the amplitudes of the electric and magnetic fields: $$B = \frac{E}{c}$$
- Substitute the given values ($E = 6.3$ V/m and $c = 3 \times 10^8$ m/s): $$B = \frac{6.3}{3 \times 10^8} = 2.1 \times 10^{-8} \text{ T}$$
- Determine the direction of the magnetic field vector. The wave propagates along the $+x$-direction ($\mathbf{\hat{i}}$), and the electric field is along the $+y$-direction ($\mathbf{\hat{j}}$). Since the direction of propagation is given by the cross product $\mathbf{E} \times \mathbf{B}$: $$\mathbf{\hat{j}} \times \mathbf{\hat{k}} = \mathbf{\hat{i}}$$ Therefore, the magnetic field vector must point along the $+z$-direction ($\mathbf{\hat{k}}$).
Final Evaluation Output: $\mathbf{B} = 2.1 \times 10^{-8} \mathbf{\hat{k}} \text{ T}$
Mathematical Problem Statement (Example 8.2): The magnetic field in a plane electromagnetic wave is given by: $$B_y = (2 \times 10^{-7} \text{ T}) \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t)$$ (a) What is the wavelength and frequency of the wave? (b) Write an expression for the electric field.
Step-by-Step Algorithmic Solution Trace:
- Compare the given equation with the standard wave equation $B_y = B_0 \sin(kx + \omega t)$:
- $B_0 = 2 \times 10^{-7}$ T
- $k = 0.5 \times 10^3 \text{ rad/m}$
- $\omega = 1.5 \times 10^{11} \text{ rad/s}$
- Calculate the wavelength $\lambda$ using $k = \frac{2\pi}{\lambda}$: $$\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.5 \times 10^3} \approx 1.26 \times 10^{-2} \text{ m} = 1.26 \text{ cm}$$
- Calculate the frequency $\nu$ using $\omega = 2\pi\nu$: $$\nu = \frac{\omega}{2\pi} = \frac{1.5 \times 10^{11}}{2\pi} \approx 2.39 \times 10^{10} \text{ Hz} = 23.9 \text{ GHz}$$
- Calculate the electric field amplitude $E_0$ using $E_0 = B_0 c$: $$E_0 = (2 \times 10^{-7} \text{ T}) \times (3 \times 10^8 \text{ m/s}) = 60 \text{ V/m}$$
- Determine the direction of the electric field. The wave propagates along the $-x$-direction (due to the $+$ sign in the phase), and the magnetic field is along the $+y$-direction. Since $\mathbf{E}$ must be perpendicular to both, the electric field oscillates along the $z$-axis: $$E_z = 60 \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \text{ V/m}$$
Final Evaluation Output: (a) $\lambda = 1.26 \text{ cm}$, $\nu = 23.9 \text{ GHz}$ (b) $E_z = 60 \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \text{ V/m}$