Understand inductance and the henry: Foundations — Hardware deep dive

The parent note's headline sentence is $ε = - L \frac{d I}{d t}$ . It is packed with symbols — $ε$ , $L$ , $\frac{d I}{d t}$ , and even that lonely minus sign. We are not allowed to use any of them until we have drawn its picture. So we build them one at a time, and only at the very end do we let them stand together in that formula. We are strict: no symbol appears in a formula before it has its own definition above.

1. Current $I$ — flowing charge

Plain words: rate of charge flow.
The picture: arrows of charge marching down a wire — look at the blue arrows in the figure below.
Why the topic needs it: inductance is entirely about what happens when this flow changes, so we must first name the flow itself.

Figure — Understand inductance and the henry

Figure s01 — A wire (grey) carrying charge. The blue dots are charge marching rightward; the blue arrows show the flow direction. The red dashed marker counts how much charge slips past each second — that count is the current $I$ .

2. Time $t$ and the rate of change $\frac{d I}{d t}$

Before we can talk about how fast current changes, we need the thing it changes against: time.

The parent topic's key formula is not about current $I$ but about how quickly $I$ changes as this clock ticks. We need a symbol for "speed of change."

The picture: the steepness of the current-versus-time graph — a steep hill means a large $\frac{d I}{d t}$ , a flat road means zero. See the figure.
Why this tool and not another? We want a number that answers "how fast is current changing right now?" A plain subtraction $(I_{2} - I_{1})$ only gives the average over a chunk of time; the derivative gives the instantaneous slope, which is exactly what Faraday's law demands.

Figure s02 — Current $I$ (blue) plotted against time $t$ . The orange line is the slope while current rises ( $\frac{d I}{d t} > 0$ ); the green line is the flat stretch ( $\frac{d I}{d t} = 0$ ); the red line is the falling stretch ( $\frac{d I}{d t} < 0$ ). The derivative is literally the steepness of the blue curve at each instant.

Recall What does

$\frac{d I}{d t} = 0$ mean physically? Steady, unchanging current — the current graph is a flat horizontal line, so an ideal inductor produces zero voltage and behaves like a plain wire.

3. Magnetic field $B$ — the invisible arrows around a current

The picture: straight orange arrows threading down the middle of the coil (see s03).
Why the topic needs it: the current's whole "inertia" is stored in this field; if there were no field there would be no fight-back.

We will only need the formula for $B$ inside a coil later, once we have named the coil's turns, length, area, and core material. So we postpone $B = μ \frac{N}{ℓ} I$ to section 8 — every symbol in it earns its place first.

4. Magnetic flux $Φ$ — field passing through a loop

First we need a name for "how big the loop is":

The "faced square-on" part also matters, so flux is more than just $B$ times $A$ :

Recall Advanced form (for when the field is uneven)

If $B$ varies across the loop you add up all the tiny patches: $Φ = \int B \cdot d A$ . The stretched-S symbol $\int$ means "add up many tiny pieces" — we meet it properly in section 9. For our uniform-field coil the simple $Φ = B A cos θ$ is all we need.

The picture: field lines (orange) piercing a circular loop — the more that pass through (not past) it, the more flux. See the figure.
Why the topic needs it: flux is the bridge between "current" (which makes $B$ ) and "voltage" (which comes from changing $Φ$ ).

Figure s03 — A wire loop of area $A$ (grey ellipse) held in a magnetic field $B$ (orange arrows). The blue arrow is the loop's normal (perpendicular) direction; the angle $θ$ sits between it and the field. Only the field passing straight through the mouth counts, giving $Φ = B A cos θ$ .

5. Turns $N$ and flux linkage $N Φ$ — counting the loops

A coil is the same wire wound round many times. Each loop feels the same flux $Φ$ , so the total effect adds up.

The picture: three stacked loops, each pierced by the same orange field lines; the linkage is "3 loops' worth."
Why the topic needs it: doubling the loops doubles the caught flux, so $N$ is a knob that controls inductance. (And because more turns also make a stronger field, the effect ends up as $N^{2}$ — that's the parent's "double turns → 4× inductance.")

6. Voltage $V$ , EMF $ε$ , and Faraday + Lenz (the minus sign)

Now we can finally justify the parent's headline formula — no more asserting.

The picture: current graph sloping up (s02) → induced EMF arrow pointing against the current.
Why the topic needs it: this is the entire behaviour of an inductor in one line.

7. Inductance $L$ and the henry — the stubbornness number

Plain words: one number measuring how strongly the coil resists current change.
Why the topic needs it: so we never have to recompute the field from scratch — one measured $L$ turns Faraday's law into the tidy $V = L \frac{d I}{d t}$ .

The dual of this idea — a component that resists voltage change instead — is capacitance.

8. Permeability $μ$ , cross-section $A$ , length $ℓ$ — and why $B = μ \frac{N}{ℓ} I$ then $L = μ \frac{N ^{2} A}{ℓ}$

Now that $B$ , $N$ , and $A$ are all defined, we can name the last two geometry symbols and finally write the coil's field.

Intuition Why

$B = μ \frac{N}{ℓ} I$ ? (Ampère's law, step by step) Ampère's law is a counting rule: walk once around any closed loop, adding up the magnetic field pointing along your path; the total equals $μ$ times the current that pierces the loop. Now choose a clever rectangular walking-path — one long side runs straight down the inside of the coil (where the field is strong and points along our path), and the other long side runs outside (where the field is ~zero). The short ends run across the field, contributing nothing.

What we did: summed the field around the rectangle. Only the inside long side (length $ℓ$ ) contributes, giving $B \times ℓ$ .
Why: outside field is negligible, and the ends are perpendicular so they add nothing — this isolates the inside field.
The current pierced: the path encircles all $N$ turns, each carrying current $I$ , so the threaded current is $N I$ .
Ampère's law then says $B ℓ = μ N I$ , and dividing by $ℓ$ : $B = μ \frac{N}{ℓ} I$ Here $\frac{N}{ℓ}$ is turns per metre: pack more turns into the same length and each amp makes a stronger field. This is the solenoid field result.

Assumptions: (1) the coil is long and thin so the field is uniform inside and negligible outside; (2) $μ$ is a fixed constant — true for air, only approximately true for iron before it saturates; (3) all loops face-on so $cos θ = 1$ .

The picture: a fat, short, iron-filled tube makes a strong field; a thin, long, air tube makes a weak one.
Why the $N^{2}$ ? More turns raise both the field ( $B \propto N$ ) and the number of loops catching it ( $\times N$ ), so the effect squares.

9. Energy $E$ and the $\int$ sign — adding up the build-up

The picture: slicing the area under a graph into thin strips and summing them.
Why this tool and not multiplication? Because the power isn't constant while current builds — it grows as the current grows. Plain "power × time" only works for constant power; the integral handles the smoothly-changing case. See Energy Storage in Fields.

Prerequisite map

Equipment checklist

Current

I

— what is it in one line?

The rate of charge flow past a point, measured in amperes.

Time

t

— what role does it play?

The clock (seconds) that everything is measured "per" — the horizontal axis of the change graphs.

\frac{d I}{d t}

— what does it measure, and can it be negative?

The instantaneous slope of the current-vs-time graph; positive when current rises, negative when it falls.

Magnetic field

B

— scalar or vector, and its unit?

A vector (strength + direction) drawn as arrows, measured in tesla.

Loop area

A

— what is it?

The area of the loop opening (m

^{2}

); the same

A

serves as the coil's cross-section.

Magnetic flux

Φ

— full formula with tilt?

Φ = B A cos θ

; only the field going straight through the loop counts.

Flux linkage

N Φ

— why the

N

Each of the

N

turns catches the flux

Φ

, so the total caught is

N

times as much.

Voltage

V

vs EMF

ε

— how do they relate?

Same unit (volts);

ε

is the push made by changing flux,

V

is the terminal push you measure — equal in size for an ideal inductor.

Faraday's law — state it.

ε = - \frac{d ( N Φ )}{d t}

: induced EMF equals minus the rate of change of flux linkage.

Why the minus sign?

Lenz's law — the induced EMF opposes the change that produced it.

How does Faraday's law become

V = L \frac{d I}{d t}

Sub

N Φ = L I

with

L

constant; pull

L

out of the derivative.

Inductance

L

— one-line definition?

Flux linkage per amp,

L = N Φ/ I

; measured in henries.

Resistance

R

— what is it and where does it appear here?

How strongly a component slows current (ohms); it pairs with

L

τ = L / R

Where does

B = μ \frac{N}{ℓ} I

come from?

Ampère's law on a rectangular path:

B ℓ = μ N I

, so

B = μ \frac{N}{ℓ} I

What does the integral

\int

do here?

Adds the tiny energy bits while current rises, giving

\frac{1}{2} L I^{2}