Visual walkthrough — Self-inductance L, mutual inductance M

This deep-dive expands the parent topic. The tools we will borrow — one per step — are Ampère's law, Magnetic flux, and Faraday's law of induction with Lenz's law for the sign.

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Step 0 — Meet the object and every symbol on it

WHAT. A solenoid is just a wire wound into a tight helix — a stack of circular loops. We label four things and nothing else:

• $\ell$ — the length of the coil, in metres (how far the stack reaches).
• $N$ — the number of turns (how many loops we wound), a pure count.
• $A$ — the cross-sectional area of one loop, in ${\text{m}}^2$ (the flat disc the field pokes through).
• $I$ — the current flowing through the wire, in amperes.

WHY. These four are the only geometric facts we control. Everything else — the field, the flux, the inductance — will be computed from them. From these we also build one convenience symbol:

PICTURE. The green loops are the turns; the blue axis is the length $\ell$; the yellow disc is the area $A$.

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Step 1 — Current makes a magnetic field ($B$)

WHAT. Push current $I$ through the wire. Each loop becomes a tiny magnet. Stacked tightly, their fields add up inside and cancel outside, leaving a uniform field pointing straight along the axis. Call its strength $B$ (unit: tesla).

WHY this tool — Ampère's law? We need to turn "current" into "field", and Ampère's law is the exact machine that does it: it says the field circulating around a chosen loop equals ${\mu }_0$ times the current threading that loop. We pick a rectangular path (an "Amperian loop") with one long side inside the solenoid and one far outside where $B=0$. That choice makes the algebra collapse to a single term.

$\underset{\text{field inside}}{\underbrace{B}}\underset{\text{inside side length}}{\underbrace{L_{\text{path}}}}={\mu }_0\underset{\text{turns the path encloses}}{\underbrace{(n{L}_{\text{path}})}}\underset{\text{current in each turn}}{\underbrace{I}}$

The path length $L_{\text{path}}$ appears on both sides and cancels — that is why the rectangle trick works:

PICTURE. The dashed red rectangle is the Amperian loop; the blue arrows are the uniform axial field inside; outside, the field is zero.

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Step 2 — Field through one loop becomes flux ($\Phi$)

WHAT. "Flux" $\Phi$ counts how much field pokes through one loop's disc. Think of the field lines as rain and the loop as a hoop: flux is how many raindrops pass through the hoop.

WHY this tool — Magnetic flux? Faraday's law (coming in Step 4) cares about flux, not field. So we must convert $B$ (field per area) into $\Phi$ (total field through the whole disc) before Faraday can act. The general rule is $\Phi =\int \vec{B}\cdot d\vec{A}$, but here $\vec{B}$ is uniform and points straight through the disc, so the integral becomes a plain multiplication:

$\Phi =\underset{\text{field strength}}{\underbrace{B}}\times \underset{\text{disc area}}{\underbrace{A}}={\mu }_0\frac{N}{\ell }IA$

PICTURE. The yellow disc is the loop; blue arrows pass straight through it; the shaded yellow region is the flux $\Phi =BA$.

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Step 3 — All $N$ loops share the flux: linkage $N\Phi$

WHAT. The same field line that pierces the first loop pierces every loop, because the loops are stacked on the same axis. So the field is "linked" $N$ times over. The flux linkage is the total, $N\Phi$.

WHY. Faraday's law counts flux through the whole circuit, and the circuit here is one wire that loops $N$ times. Each loop is like a turnstile the same field line must pass through; with $N$ turnstiles, the field line is counted $N$ times.

$N\Phi =\underset{\text{loops linked}}{\underbrace{N}}\times \underset{\Phi \text{ from Step 2}}{\underbrace{{\mu }_0\frac{N}{\ell }AI}}={\mu }_0\frac{N^2}{\ell }AI$

Notice the first $N$ from counting the loops meets the second $N$ hiding inside $\Phi$ (because more loops made a stronger field back in Step 1). Two $N$'s multiply → $N^2$. This is the birthplace of the famous square.

PICTURE. One red field line snakes through all the green loops; a counter ticks $1,2,\dots ,N$ as it pierces each one.

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Step 4 — Name the flux-per-amp: that's $L$

WHAT. Look at the last line: $N\Phi$ is just a constant times $I$. Everything except $I$ is fixed geometry. We give that constant a name, $L$.

WHY name it? So we never re-run Steps 1–3 again. Once we know $L$, the flux linkage of any current is instantly $LI$. This is pure bookkeeping — a "flux per amp" price tag stamped on the coil.

PICTURE. The straight-line graph $N\Phi$ versus $I$: its slope is $L$.

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Step 5 — Why we care: $L$ becomes the back-EMF

WHAT. Now let $I$ change in time. Because $N\Phi =LI$ and $L$ is fixed, a changing current means a changing flux linkage — and Faraday's law of induction turns that into an induced voltage (EMF), $\epsilon$.

WHY Faraday's law of induction here? It is the only law that says "changing flux → voltage." We differentiate $N\Phi =LI$ with respect to time. Since $L$ is a constant (geometry doesn't move), it slides out of the derivative:

$\epsilon =-\frac{d(N\Phi )}{dt}=-\frac{d(LI)}{dt}=-L\frac{dI}{dt}$

• $\frac{dI}{dt}$ — the rate of change of current (amps per second), not the current itself.
• The minus sign is Lenz's law: the induced EMF fights the change. Current rising? EMF pushes back to slow it. Current falling? EMF pushes forward to sustain it.

PICTURE. Top: a current ramp $I(t)$. Bottom: the resulting EMF — a flat step during the ramp, zero when the current is level, sign flipping when the current turns around.

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Step 6 — Edge & degenerate cases (never leave the reader stranded)

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The one-picture summary

Everything on one canvas: current makes field (Ampère), field makes flux (integrate), flux is linked $N$ times, the slope is $L$, and changing it makes back-EMF (Faraday–Lenz).

Recall Feynman retelling of the whole walkthrough

Wind a wire into a tube of loops. Send current in — Ampère says a straight field grows inside, strong if the loops are packed tight. That field pokes through each loop's opening; counting how much goes through one loop is the flux. But the same field pokes through all the loops, so we count it $N$ times — the linkage. Because packing more loops both strengthens the field and links it more, we get two factors of $N$, hence $N^2$. Now, the linkage is always the same constant times the current, and we name that constant $L$ so we never do the counting again. Finally, if you try to change the current, the flux changes, and Faraday hands you a voltage that (Lenz insists) fights the change — a heavy current-flywheel that hates being sped up or slowed down. That's self-inductance, drawn from nothing.

Recall Quick self-test

Where do the two factors of $N$ come from? ::: One from counting the $N$ turns being linked (Step 3), one from the field being $N$ times stronger because there are $N$ turns making it (Step 1). Why does $I$ vanish from the final $L$? ::: $L=N\Phi /I$ and $N\Phi$ is already proportional to $I$, so $I$ cancels — $L$ is pure geometry. Why is there a minus sign in $\epsilon =-LdI/dt$? ::: Lenz's law — the induced EMF opposes the change in current that caused it.

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Connections

• Ampère's law — Step 1, turns current into field.
• Magnetic flux — Steps 2–3, the linked quantity.
• Faraday's law of induction — Step 5, changing flux → EMF.
• Lenz's law — the minus sign.
• Energy stored in magnetic field — where $\frac{1}{2}L{I}^2$ comes from next.
• Transformers and RL circuits — where $L$ and $M$ are used.