🎢Principles of Physics II

Inductance Formulas

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Why This Matters

Inductance is the electromagnetic equivalent of inertia: it's how circuits resist changes in current. Understanding it connects everything from transformer design to the transient behavior of RL circuits. The formulas in this guide aren't isolated equations. They're different ways of looking at the same physics of changing magnetic flux.

Don't just memorize $L = \frac{N^2 \mu A}{l}$ and call it a day. Know why more turns means more inductance (each turn adds flux that links through all the others), why the time constant $\tau = \frac{L}{R}$ governs circuit response, and how Faraday's law ties everything together. Expect exam questions that ask you to compare configurations, predict energy storage, and analyze transient behavior.

Fundamental Inductance Relationships

Inductance measures how effectively a current-carrying conductor creates magnetic flux that links back through itself. These formulas define what inductance is and how it depends on geometry and materials.

Self-Inductance Formula

$L = \frac{N^2 \mu A}{l}$ — inductance scales with the square of the number of turns $N$ , making coil winding extremely effective at boosting $L$
Permeability $\mu$ describes how easily the core material supports magnetic field lines. Iron cores dramatically increase $L$ compared to air cores.
Geometry matters: larger cross-sectional area $A$ captures more flux, while a longer length $l$ spreads turns apart and reduces inductance

Inductance of a Solenoid

$L = \frac{\mu N^2 A}{l}$ — this is the same as the general self-inductance formula because the solenoid is the idealized case
This formula assumes a long solenoid where edge effects are negligible and the magnetic field $B$ is uniform inside
Core material choice matters in practice: air-core solenoids avoid magnetic saturation, while ferromagnetic cores maximize inductance

Inductance of a Toroid

$L = \frac{\mu N^2 h}{2\pi} \ln\left(\frac{r_2}{r_1}\right)$ — this accounts for the curved geometry where field strength varies with radial distance
The logarithmic dependence on $\frac{r_2}{r_1}$ comes from the fact that the magnetic field inside a toroid falls off as $\frac{1}{r}$ . You have to integrate over the cross-section rather than just multiply by area.
Zero external field: toroids confine all flux internally, making them ideal for sensitive circuits where field leakage causes interference

Compare: Solenoid vs. Toroid — both depend on $N^2$ and $\mu$ , but solenoids have field leakage at the ends while toroids confine flux completely. If a problem asks about minimizing electromagnetic interference, the toroid is your answer.

Electromagnetic Induction Laws

These formulas describe how changing magnetic conditions create voltage. Faraday's law is the foundation; Lenz's law tells you the direction.

Faraday's Law of Induction

$\varepsilon = -N\frac{d\Phi}{dt}$ — the induced EMF equals the negative rate of change of total magnetic flux through the coil
The negative sign encodes Lenz's law: the induced EMF always acts to oppose the change in flux, which is required by energy conservation
The factor of $N$ appears because each turn experiences the same flux change, so more turns amplify the induced voltage proportionally

Lenz's Law (Inductor Form)

$\varepsilon_{\text{induced}} = -L\frac{dI}{dt}$ — the self-induced EMF opposes any change in current through the inductor
This is Faraday's law rewritten using $L$ instead of flux. The two forms are equivalent because $\Phi_{\text{total}} = LI$ for a linear inductor.
Back-EMF is the practical name for this voltage. It's why inductors smooth out current changes and resist sudden switching.

Compare: $\varepsilon = -N\frac{d\Phi}{dt}$ vs. $\varepsilon = -L\frac{dI}{dt}$ — same physics, different variables. Use the flux form when you know $\frac{d\Phi}{dt}$ directly; use the inductance form when analyzing circuits with known $L$ and changing current.

Energy Storage and Coupling

These formulas address what happens to energy in magnetic systems and how inductors interact with each other.

Energy Stored in an Inductor

$U = \frac{1}{2}LI^2$ — energy stored in the magnetic field, analogous to $\frac{1}{2}CV^2$ for capacitors
Quadratic in current: doubling the current quadruples the stored energy, so high-current inductors become significant energy reservoirs
This energy is physically stored in the magnetic field itself. The energy density at any point in space is $u = \frac{B^2}{2\mu}$ .

Mutual Inductance Formula

$M = k\sqrt{L_1 L_2}$ — mutual inductance depends on both self-inductances and how well the two coils share magnetic flux
The coupling coefficient $k$ ranges from 0 (coils completely isolated, no shared flux) to 1 (all flux from one coil passes through the other)
Transformer principle: efficient power transfer between coils requires high $k$ . Tightly wound coils on a shared iron core approach $k \approx 1$ .

Compare: Self-inductance vs. Mutual inductance — $L$ describes a single coil's interaction with its own field, while $M$ describes the interaction between two coils. Transformers exploit $M$ ; single inductors in filters rely on $L$ .

Circuit Behavior and Combinations

These formulas govern how inductors behave in real circuits: transient response and equivalent inductance calculations.

Time Constant for RL Circuits

$\tau = \frac{L}{R}$ — the characteristic time for current to reach about 63% of its final value (or decay to about 37%)
Larger $L$ means more "inertia" against current change. Larger $R$ dissipates energy faster, which speeds up the response.
Transient current follows $I(t) = I_0(1 - e^{-t/\tau})$ for growth and $I(t) = I_0 e^{-t/\tau}$ for decay. After about $5\tau$ , the transient is essentially complete (current reaches ~99% of its final value).

Inductors in Series

$L_{\text{total}} = L_1 + L_2 + L_3 + \cdots$ — inductances add directly, just like resistors in series
The same current flows through each inductor, so each one contributes its full inductance to opposing current changes
This assumes no mutual coupling. If coils are magnetically linked, you must add or subtract $2M$ depending on whether the fields reinforce or oppose each other.

Inductors in Parallel

$\frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \cdots$ — reciprocals add, just like resistors in parallel
Total inductance decreases because current can split across paths, reducing the effective opposition to change
Two-inductor shortcut: $L_{\text{total}} = \frac{L_1 L_2}{L_1 + L_2}$ . Worth memorizing for exam speed.

Compare: Series vs. Parallel inductors — series increases total inductance (more opposition to current change), parallel decreases it. This mirrors resistor behavior but is opposite to how capacitors combine.

Quick Reference Table

Concept	Key Formulas / Examples
Geometry dependence of $L$	Solenoid formula, Toroid formula, Self-inductance
Electromagnetic induction	Faraday's law, Lenz's law (inductor form)
Energy storage	$U = \frac{1}{2}LI^2$ , energy density $u = \frac{B^2}{2\mu}$
Coil coupling	Mutual inductance $M = k\sqrt{L_1 L_2}$
Transient response	RL time constant $\tau = \frac{L}{R}$
Circuit combinations	Series inductors, Parallel inductors
Material properties	Permeability $\mu$ in all inductance formulas

Self-Check Questions

Both the solenoid and toroid formulas contain $N^2$ and $\mu$ . Why does the toroid formula include a logarithm while the solenoid formula doesn't?
If you double both the number of turns and the length of a solenoid, what happens to its inductance? What if you double turns but halve the length?
Compare the energy stored in an inductor ( $U = \frac{1}{2}LI^2$ ) to energy stored in a capacitor ( $U = \frac{1}{2}CV^2$ ). What plays the role of current in the capacitor case, and why?
An RL circuit has $L = 4 \text{ H}$ and $R = 2 \text{ Ω}$ . How long does it take for the current to reach approximately 95% of its final value after the switch closes?
Two identical inductors can be connected in series or parallel. If each has inductance $L$ , what is the ratio of total inductance in series to total inductance in parallel? How would your answer change if the series inductors had mutual coupling with $k = 1$ ?