Understand inductance and the henry

WHAT is inductance?

WHY does a coil oppose current change? (Derive from first principles)

Step 1 — Current makes a magnetic field. Any current $I$ produces magnetic flux $\Phi \propto I$. Wind the wire into $N$ turns and the flux links all $N$ turns, so the total flux linkage is $\lambda = N\Phi$.

Why this step? More turns and more current both mean more magnetic field, so flux linkage scales with both.

Step 2 — Define $L$ as the proportionality constant. Because flux linkage is proportional to current for a fixed geometry: $\lambda = N\Phi = L\,I \quad\Rightarrow\quad L = \frac{N\Phi}{I}$

Why this step? $L$ bundles all the geometry (turns, area, core material) into one number so we don't recompute the field every time.

Step 3 — Faraday's law: changing flux makes voltage. Faraday says an induced EMF equals the rate of change of flux linkage: $\varepsilon = -\frac{d\lambda}{dt}$

Substitute $\lambda = LI$ (with $L$ constant): $\boxed{\;\varepsilon = -L\,\frac{dI}{dt}\;}$

Why the minus sign? Lenz's law — the induced voltage opposes the change that made it. That's the "electrical inertia."

WHY is energy $\tfrac{1}{2}LI^2$? (Derive it)

Power delivered to the inductor: $P = VI = \left(L\dfrac{dI}{dt}\right)I$.

Energy is power integrated over time as current rises from $0$ to $I$: $E=\int P\,dt = \int_0^I L\,I\,dI = \tfrac{1}{2}LI^2$

Why this step? We swapped $dt$ for $dI$ using $\frac{dI}{dt}\,dt = dI$, turning a time integral into a clean current integral. The energy lives in the magnetic field, not in heat — it comes back when current falls.

HOW does geometry set $L$? (Solenoid derivation)

For a long solenoid of $N$ turns, length $\ell$, cross-section area $A$, core permeability $\mu$:

• Field inside: $B = \mu \frac{N}{\ell} I$
• Flux per turn: $\Phi = BA = \mu \frac{N A}{\ell} I$
• Flux linkage: $N\Phi = \mu \frac{N^2 A}{\ell} I$

Divide by $I$: $L = \mu\,\frac{N^2 A}{\ell}$

Why it matters: $L \propto N^2$ — doubling turns quadruples inductance. Iron cores raise $\mu$ hugely, so a small iron-core coil beats a big air coil.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Faraday's Law of Induction — the parent principle giving $\varepsilon=-d\lambda/dt$.
• Lenz's Law — source of the opposing minus sign.
• Capacitance and the Farad — the electrical dual of inductance.
• Magnetic Field of a Solenoid — where $B=\mu NI/\ell$ comes from.
• RL Circuits and Time Constant — where $\tau = L/R$ governs current rise.
• Energy Storage in Fields — $\tfrac12LI^2$ vs $\tfrac12CV^2$.

Concept Map

Hinglish (regional understanding)