Inductance Formulas

Why This Matters

Inductance is the electromagnetic equivalent of inertia—it's how circuits resist changes in current, and understanding it unlocks everything from transformer design to the transient behavior of RL circuits. You're being tested on your ability to connect geometry, material properties, and time-varying currents to the magnetic energy stored in coils and the voltages they induce. These formulas aren't isolated equations; they're different windows into 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 connects everything. The exam will ask you to compare configurations, predict energy storage, and analyze transient behavior—so understand what each formula tells you about the underlying physics.

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Fundamental Inductance Relationships

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

Self-Inductance Formula

• $L = \frac{N^2 \mu A}{l}$—the master equation showing inductance scales with the square of turns, making coil winding extremely effective
• Permeability $\mu$ determines how easily the core material supports magnetic field lines; iron cores dramatically increase $L$
• Geometry matters: larger cross-sectional area $A$ captures more flux, while longer length $l$ spreads turns apart, reducing inductance

Inductance of a Solenoid

• $L = \frac{\mu N^2 A}{l}$—identical to the general self-inductance formula because solenoids are the idealized case
• Uniform field assumption: this formula assumes a long solenoid where edge effects are negligible and $B$ is constant inside
• Core material choice is critical in applications; air-core solenoids avoid saturation while ferromagnetic cores maximize inductance

Inductance of a Toroid

• $L = \frac{\mu N^2 h}{2\pi} \ln\left(\frac{r_2}{r_1}\right)$—accounts for the curved geometry where field strength varies with radius
• Logarithmic dependence on the ratio $\frac{r_2}{r_1}$ reflects how magnetic field decreases as $\frac{1}{r}$ inside the toroid
• Zero external field: toroids confine all flux internally, making them ideal for sensitive circuits where leakage causes interference

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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

• $\varepsilon = -N\frac{d\Phi}{dt}$—induced EMF equals the negative rate of change of total magnetic flux through the coil
• The negative sign isn't just mathematical; it encodes Lenz's law and ensures energy conservation
• Multiply by $N$ because each turn contributes 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 that opposes any change in current through the inductor
• This is Faraday's law rewritten using $L$ instead of flux; it's equivalent because $\Phi = LI$ for a linear inductor
• Back-EMF is the practical name for this effect—it's why inductors smooth current changes and resist sudden switching

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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, making high-current inductors significant energy reservoirs
• Field energy perspective: this energy is physically stored in the magnetic field itself, with energy density $u = \frac{B^2}{2\mu}$

Mutual Inductance Formula

• $M = k\sqrt{L_1 L_2}$—mutual inductance depends on both self-inductances and how well the coils share magnetic flux
• Coupling coefficient $k$ ranges from 0 (coils completely isolated) to 1 (all flux from one coil passes through the other)
• Transformer principle: power transfer between coils relies on high $k$; tightly wound coils on a shared iron core approach $k \approx 1$

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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 ~63% of its final value (or decay to ~37%)
• Larger $L$ means more "inertia" against current change; larger $R$ dissipates energy faster, speeding response
• Transient analysis: current follows $I(t) = I_0(1 - e^{-t/\tau})$ for growth or $I(t) = I_0 e^{-t/\tau}$ for decay

Inductors in Series

• $L_{\text{total}} = L_1 + L_2 + L_3 + \cdots$—inductances add directly, just like resistors in series
• Same current flows through each inductor, so each contributes its full inductance to opposing current changes
• Assumes no mutual coupling: if coils are magnetically linked, you must add or subtract $2M$ depending on field orientation

Inductors in Parallel

• $\frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \cdots$—reciprocals add, like resistors in parallel
• Total inductance decreases because current can split, reducing the effective opposition to change
• Quick calculation: for two inductors, $L_{\text{total}} = \frac{L_1 L_2}{L_1 + L_2}$—memorize this shortcut for exam speed

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Quick Reference Table

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Self-Check Questions

1. 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?

2. 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?

3. 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?

4. 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?

5. 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 $k = 1$?