Electromagnetic Induction and Maxwell's Equations

Electromagnetic induction is the phenomenon whereby a changing magnetic field produces an electric field, generating an electromotive force (EMF) in a circuit. Discovered by Michael Faraday in 1831, it is the principle behind generators, transformers, induction cooktops, and magnetic brakes.

Magnetic flux

Magnetic flux through a surface counts how many field lines cross it: $$\Phi_B = \int_S\vec{B}\cdot d\vec{A} = BA\cos\theta$$ where $\theta$ is the angle between $\vec{B}$ and the surface normal.

Magnetic flux can change for three reasons:

1. Variation of $\vec{B}$ intensity over time (e.g., moving a magnet closer)
2. Variation of the circuit area $A$ (e.g., deforming a loop)
3. Variation of orientation (e.g., rotating a coil in a field)

Faraday's law

The law quantifies the induced EMF: $$\mathcal{E} = -\frac{d\Phi_B}{dt}$$ For a coil with $N$ turns: $\mathcal{E} = -N\,d\Phi_B/dt$.

Intuition: Nature "resists" changes in magnetic flux, just as a massive body resists changes in velocity (mechanical inertia). The faster the change, the larger the induced EMF. If the flux is constant (magnet at rest), no induction occurs.

Lenz's law

The negative sign encodes Lenz's law: the induced current flows so as to create a magnetic field that opposes the flux change that produced it.

Example: When bringing a magnet near a coil, the flux increases; the induced current generates a field that repels the magnet (opposing the approach). When pulling it away, the induced current generates a field that attracts the magnet. This principle underlies magnetic brakes in trains and roller coasters: the induced magnetic field slows the vehicle without physical contact.

Differential form

Locally, Faraday's law reads: $$\nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}$$ A time-varying magnetic field generates a rotational electric field (closed field lines), even in the absence of a physical circuit.

Real-world applications

• Electric generators: A turbine (steam, water, wind) rotates a coil in a magnetic field, converting mechanical into electrical energy. In hydroelectric plants, falling water spins turbine blades connected to a rotor.
• Transformers: Two magnetically coupled coils transfer energy between circuits at different voltages (the principle behind power distribution lines).
• Induction cooktops: A coil under the ceramic plate generates a high-frequency varying field, inducing eddy (Foucault) currents in the ferromagnetic pan, heating it by the Joule effect.
• Magnetic card readers: The magnetic stripe, moving past the read head, induces an electrical signal interpreted as data.

Self-inductance is the phenomenon whereby a circuit carrying a time-varying current induces an EMF in itself. Every current-carrying circuit generates a magnetic field; if the current changes, the flux through the circuit itself changes, inducing an EMF that opposes the change.

Inductance

Inductance $L$ measures a circuit's ability to oppose changes in current: $$\mathcal{E}_L = -L\frac{dI}{dt}$$ The unit is the henry ($1\,H = 1\,V\cdot s/A$).

Intuition: An inductor behaves like an electrical flywheel. In rotational motion, a flywheel opposes changes in angular velocity; in an inductor, current is the "velocity" and inductance is the "moment of inertia." Current cannot change instantaneously: when a switch is opened, the inductor generates a high voltage to maintain current flow (the spark phenomenon in switches).

Inductance of a solenoid

For an ideal solenoid of length $l$, cross-section $A$, turn density $n = N/l$ and volume $V = Al$: $$L = \mu_0 n^2 V = \mu_0 \frac{N^2}{l} A$$ Inductance grows with the square of the number of turns: doubling the turns quadruples the inductance.

Stored magnetic energy

An inductor carrying current stores energy in its magnetic field: $$U_L = \frac12 L I^2$$ The energy is not dissipated but returned to the circuit when the current decreases. This is analogous to the kinetic energy $\frac12 mv^2$ of a moving mass.

The magnetic energy density at any point in space is: $$u_B = \frac{B^2}{2\mu_0}$$ In vacuum, $u_B$ depends only on the magnitude of the magnetic field, not its direction.

Mutual inductance

Two nearby circuits interact: the current in one induces an EMF in the other: $$\mathcal{E}_2 = -M\frac{dI_1}{dt}$$ where $M$ is the mutual inductance coefficient, which depends on the geometry and distance between the circuits. Mutual inductance is the operating principle of transformers.

Applications

• Switching power supplies (buck/boost converters): Use inductors to convert DC voltages with high efficiency (typical of chargers and modern power supplies).
• Transformers: Mutual inductance between primary and secondary allows stepping voltage up or down in AC circuits.
• Automotive ignition systems: The ignition coil generates a high voltage (thousands of volts) from a 12V battery by suddenly interrupting current in the primary, creating a spark at the spark plug.
• Filters and resonance: Inductors and capacitors together form resonant circuits used in radio, TV, and telecommunications.

James Clerk Maxwell (1864) unified electricity and magnetism into a single theory by introducing a missing term in Ampère's law: the displacement current.

The problem with Ampère's law

Ampère's law $\nabla\times\vec{B} = \mu_0\vec{J}$ (or $\oint\vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$) works for closed circuits but fails when considering a charging capacitor. Between the plates there is no conduction current ($\vec{J}=0$), but the electric field changes over time. Maxwell realized that a surface passing between the plates gave zero result, while one intersecting the wire gave $\mu_0 I$ — a contradiction.

The solution: displacement current

Maxwell proposed that a time-varying electric field generates a magnetic field just like a current does: $$\vec{J}_d = \varepsilon_0 \frac{\partial\vec{E}}{\partial t}$$ The complete Ampère-Maxwell law becomes: $$\nabla\times\vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$$

Intuition: The time variation of the electric field between a capacitor's plates is magnetically equivalent to an electric current. Magnetic fields can be generated either by moving charges (conduction current) or by changing electric fields (displacement current).

The four Maxwell equations in vacuum

The complete differential-form equations are:

1. Gauss's law for electricity: $\nabla\cdot\vec{E} = \rho/\varepsilon_0$ — electric charges are sources of the electric field.
2. Gauss's law for magnetism: $\nabla\cdot\vec{B} = 0$ — no magnetic monopoles exist; $\vec{B}$ field lines are always closed.
3. Faraday's law: $\nabla\times\vec{E} = -\partial\vec{B}/\partial t$ — a changing magnetic field produces an electric field.
4. Ampère-Maxwell law: $\nabla\times\vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\partial\vec{E}/\partial t$ — a changing electric field produces a magnetic field.

Physical significance

Maxwell's equations reveal a deep symmetry: a time-varying electric field generates a magnetic field (eq. 4), and a time-varying magnetic field generates an electric field (eq. 3). This symmetry implies the existence of electromagnetic waves propagating at the speed of light: $$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 2.998 \times 10^8\,\mathrm{m/s}$$ The fact that $c$ matched the measured speed of light led Maxwell to conclude that light is an electromagnetic wave.

Impact on physics

• Unification: Electricity, magnetism, and optics become different aspects of the same phenomenon: the electromagnetic field.
• EM waves: The theoretical prediction led Hertz to demonstrate them experimentally (1887), and Marconi to develop radio (1895).
• Relativity: Maxwell's equations are only invariant under the Lorentz transformations of Einstein's special relativity (1905), which was born precisely from the need to reconcile electromagnetism with classical mechanics.

RLC circuits combine resistance ($R$), inductance ($L$), and capacitance ($C$). Their behavior under alternating current (AC) features rich phenomena such as resonance and phase shifts, essential for radio, filters, and power systems.

Ideal LC circuit (no resistance)

In an LC circuit, energy oscillates between the capacitor (electrostatic energy $Q^2/(2C)$) and the inductor (magnetic energy $LI^2/2$), like a pendulum exchanging kinetic and potential energy. The natural oscillation frequency is: $$\omega_0 = \frac{1}{\sqrt{LC}}$$ The charge on the capacitor and the current through the inductor oscillate sinusoidally at this angular frequency.

Impedances in AC

In alternating current ($V(t) = V_0\cos(\omega t)$), each component opposes current flow differently:

• Resistor: $Z_R = R$, current and voltage in phase ($\phi = 0$)
• Capacitor: $Z_C = 1/(j\omega C)$, current leads voltage by $90°$ ($\phi = -\pi/2$)
• Inductor: $Z_L = j\omega L$, current lags voltage by $90°$ ($\phi = +\pi/2$)

Impedance generalizes resistance to AC circuits: it is a complex number whose real part is resistance and whose imaginary part is reactance.

Total impedance and resonance

For a series RLC circuit: $$Z = R + j\left(\omega L - \frac{1}{\omega C}\right) = \sqrt{R^2 + (X_L - X_C)^2}\cdot e^{j\phi}$$ where $X_L = \omega L$ and $X_C = 1/(\omega C)$ are the reactances.

At resonance ($\omega_r = 1/\sqrt{LC}$), inductive and capacitive reactances cancel ($X_L = X_C$), the impedance is purely real ($Z = R$), and the current is maximum. It is like pushing a swing at its natural rhythm: the amplitude grows enormously even with small forces.

Power in AC circuits

The average power dissipated by an AC load is: $$P = V_{rms} I_{rms} \cos\phi$$ where $\cos\phi$ is the power factor, measuring how close the real power is to the apparent power ($V_{rms}I_{rms}$). A low power factor (large $\phi$) means high current for the same useful power, causing extra losses in transmission lines. Power companies typically require $\cos\phi > 0.9$.

RMS (root mean square) values are defined as: $V_{rms} = V_0/\sqrt{2}$, $I_{rms} = I_0/\sqrt{2}$ for sinusoidal waveforms.

Applications

• Radio tuners: By varying $C$ (variable capacitor), the RLC circuit's resonant frequency changes, selecting a specific radio station (at the broadcaster's $f_r$).
• Band-pass filters: RLC circuits allow only a band of frequencies to pass, used in loudspeakers (crossovers) and communication systems.
• Power supplies: Rectifiers and LC filters convert AC to DC with reduced ripple.
• Ignition systems: The oscillatory RLC discharge is exploited in capacitive discharge ignition circuits.