23.9 Inductance

Inductance describes how coils store energy in magnetic fields and resist changes in current. Understanding inductance is essential for analyzing AC circuits, power supplies, and many electromagnetic devices.

Inductance

Inductance of coil configurations

Inductance ($L$) quantifies a coil's ability to store energy in its magnetic field. The unit of measurement is the henry (H). Think of it this way: a coil with higher inductance can store more energy for a given current.

Four main factors determine a coil's inductance:

• Number of turns ($N$): More turns means a stronger magnetic field and higher inductance. Inductance scales with $N^2$, so doubling the turns quadruples the inductance.
• Cross-sectional area ($A$): A larger coil area captures more magnetic flux, increasing inductance.
• Length ($l$): A longer coil spreads the turns out, weakening the field per unit length and reducing inductance.
• Core material permeability ($\mu$): Filling the coil with a material like iron or ferrite concentrates the magnetic field, boosting inductance significantly compared to an air core.

These factors combine into the inductance formula for a solenoid:

$L = \frac{\mu N^2 A}{l}$

Here, $\mu$ is the permeability of the core material, given by $\mu = \mu_0 \mu_r$, where $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \text{ T·m/A}$) and $\mu_r$ is the relative permeability of the material. For vacuum or air, $\mu_r \approx 1$; for iron, $\mu_r$ can be several thousand.

Mutual inductance is a related concept: when the changing magnetic field of one coil induces a voltage in a nearby coil. This is the principle behind transformers.

Energy storage in inductors

An inductor stores energy in its magnetic field whenever current flows through it. The energy stored is:

$U_L = \frac{1}{2} L I^2$

where $U_L$ is energy in joules, $L$ is inductance, and $I$ is the current through the inductor.

Notice the $I^2$ term: the stored energy increases quadratically with current. Doubling the current means four times the stored energy. This is analogous to how a capacitor stores energy as $\frac{1}{2}CV^2$, but here it's current and magnetic fields rather than voltage and electric fields.

Applications that rely on this energy storage include superconducting magnetic energy storage systems and pulse power systems, where large amounts of energy need to be released quickly.

Inductor emf generation

When current through an inductor changes, the magnetic field changes too. By Faraday's law, that changing magnetic field induces an electromotive force (emf) in the inductor itself. The magnitude and direction of this induced emf follow:

$\varepsilon = -L \frac{dI}{dt}$

• $\varepsilon$ is the induced emf
• $L$ is the inductance
• $\frac{dI}{dt}$ is the rate of change of current

The negative sign reflects Lenz's law: the induced emf always opposes the change in current that created it. If current is increasing, the inductor generates an emf that pushes back against the increase. If current is decreasing, the emf acts to sustain it.

This is why inductors resist sudden changes in current. You can't instantly switch the current through an inductor from zero to some value; the back-emf fights the change. This property makes inductors useful for smoothing current fluctuations in circuits, such as in power supply filters and noise reduction.

Electromagnetic induction and related concepts

A few related ideas tie into inductance:

• Electromagnetic induction is the broader process of generating voltage through a changing magnetic field. Inductance is one specific manifestation of this.
• Magnetic flux ($\Phi$) is the total magnetic field passing through a given area. Changes in flux are what drive induced emf.
• Time constant ($\tau$): In an RL circuit (resistor + inductor), $\tau = \frac{L}{R}$. This tells you how quickly current ramps up or decays. After one time constant, current reaches about 63% of its final value.
• Inductive reactance ($X_L$) is the opposition an inductor presents to alternating current. It depends on frequency: $X_L = 2\pi f L$. Higher frequencies face more opposition, which is why inductors are effective at blocking high-frequency signals while passing low-frequency or DC currents.