Foundations — Self-inductance L, mutual inductance M

This page assumes you know nothing. Before you can read $L=N\Phi /I$ you must know what $N$, $\Phi$, and $I$ each mean, what picture each draws, and why the topic needs it. We build them one at a time, each resting on the previous.

0. Electric current $I$ — the flow we start from

Picture it: imagine water in a pipe. The current $I$ is how many litres pass a cross-line every second — not how much water is in the pipe, but how fast it flows.

Why the topic needs it: current is the cause of everything here. No current → no magnetic field → no flux → no inductance. Later the whole story is about what happens when $I$ changes, so we must name it first.

1. Magnetic field $B$ — what a current creates around itself

Picture it: wrap your right hand around a wire with your thumb pointing along the current — your curled fingers show which way $\vec{B}$ circles the wire. Inside a coil, all those circles add up into one strong field running straight down the coil's axis.

Why the topic needs it: $B$ is the invisible middle-man. Current can only "feel" itself and its neighbour through the field it makes. See Ampère's law for how we compute $B$ from $I$.

2. Area vector $\vec{A}$ — describing the loop the field passes through

Picture it: hold a hula-hoop. The area is the disc inside it; the area-arrow $\vec{A}$ is a skewer poked straight through the middle, at right angles to the hoop.

Why the topic needs it: to ask "how much field passes through the loop" we need to know which way the loop faces. Field along the skewer passes through fully; field sliding across the face passes through not at all.

3. Magnetic flux $\Phi$ — how much field pierces the loop

Picture it: think of $\vec{B}$ as rain falling and the loop as a ring held up to catch it. If the ring faces the rain head-on, it catches the most (full flux). Tilt it and it catches less; hold it edge-on and it catches nothing.

Why $\cos$ and not something else? $\cos \theta$ is exactly the fraction of the arrow $\vec{B}$ that lies along the skewer $\vec{A}$. It answers "how much of the field is genuinely going through, versus sliding across?" Inside a solenoid the field runs straight along the axis and the loop faces it, so $\theta =0$ and we simply get $\Phi =BA$ — which is why the parent note drops the cosine. See Magnetic flux.

Why the topic needs it: flux is the star quantity. Inductance is defined as flux-per-amp, so we cannot even write $L$ without $\Phi$.

4. Turns $N$ and flux linkage $N\Phi$ — counting every loop

Picture it: stack $N$ identical rings on the same skewer. Each ring catches the same flux $\Phi$; the coil as a whole "feels" all of them, so its total linkage is $N$ times bigger.

Why the topic needs it: Faraday's law responds to the total linkage $N\Phi$, not one loop. This is the source of the famous $L\propto N^2$: more turns means more field made (one factor of $N$) and more turns to link it (a second factor of $N$).

5. Rate of change $\frac{dI}{dt}$ — the "how fast is it changing?" tool

Picture it: it is the steepness of the graph of $X$ against time. Flat line → zero. Steep uphill → big positive. Downhill → negative.

Why this tool and not just $I$? Because Faraday's law cares about change, not amount. A giant steady current makes a giant steady flux — but nothing changes, so nothing is induced. Only $dI/dt$ (or $d\Phi /dt$) can make a coil fight back. That is precisely why the working formula is $\epsilon =-LdI/dt$ and not $\epsilon =-LI$.

6. Induced EMF $\epsilon$ and the minus sign — the fight-back voltage

Why the minus sign? It is Lenz's law: the induced EMF always points so as to oppose the change that caused it. If the flux is growing, the EMF tries to shrink it; if shrinking, the EMF tries to prop it up. Nature resists sudden change — this is what makes a coil act "stubborn."

Why the topic needs it: $\epsilon$ is the observable consequence. $L$ and $M$ are just the constants that turn a changing current into this measurable voltage. See Faraday's law of induction.

7. Putting it together — where $L$ and $M$ come from

Now every symbol is earned, the definitions read cleanly:

Differentiate in time (the geometry is fixed, so $L$ and $M$ are constants) and use Faraday: ${\epsilon }_{\text{self}}=-L\frac{dI}{dt},{\epsilon }_2=-M\frac{d{I}_1}{dt}$

This is the whole engine of the parent topic.

The prerequisite map

Read it top-down: current makes field, field plus area makes flux, flux times turns makes linkage, linkage per amp is inductance, and inductance times rate of change (with Lenz's minus sign) is the EMF you actually measure.

A worked micro-example to lock the chain

Equipment checklist

Test yourself — cover the right side and answer aloud before revealing.