RL circuit — growth and decay of current: Foundations

Before you can read the parent note RL circuit — growth and decay of current, every letter in that story must mean something to you. This page builds each one from nothing, in the order they depend on each other — no symbol is used before it is defined.

1. Current $I$ — how fast charge flows

Picture a pipe with water flowing. Current is not how much water sits in the pipe — it is how many litres pass a marked line every second. In a wire the "water" is electrons.

Figure 1 below draws exactly this: violet charges drift along a wire, and the magenta "counting line" is where we tally how many pass each second. Follow the orange flow arrows — the number crossing that line per second is $I$ .

Figure — RL circuit — growth and decay of current

Fig 1 — Current as charge crossing a counting line each second: the magenta line is the tally point, the orange arrows show the charges' flow direction.

2. The rate of change $\frac{d I}{d t}$ — how fast the flow itself is changing

Now the subtle jump. We do not just care about the current — we care about how quickly the current is changing.

Think of a car's speedometer. Position is where you are; speed is how fast position changes. Here:

$I$ = the current (the "position").
$\frac{d I}{d t}$ = how fast that current is climbing or falling (the "speed").

Figure 2 shows the current curve in magenta and, at one instant, the violet dashed tangent line whose steepness is $d I / d t$ . Notice the annotations: where the curve is steep the current changes fast; where it flattens, $d I / d t$ shrinks toward zero.

Fig 2 — The slope of the current curve is $d I / d t$ : steep curve = current changing fast (orange note), flat curve = current barely changing (navy note).

3. Voltage $V$ — the electrical push

Before any law that uses $V$ , we must say what $V$ is.

In the pipe picture, $V$ is the height difference that makes water flow downhill: the bigger the drop, the harder the water is pushed. No height difference → no flow.

4. Resistance $R$ and Ohm's law — the friction

Now that both $V$ (Section 3) and $I$ (Section 1) are defined, Ohm's law ties them to $R$ :

$V = I R$

5. EMF $ε$ — the battery's push

It is just a special voltage: the source voltage, the thing driving the whole circuit. In the pipe picture $ε$ is the pump that lifts the water back up to the top.

6. Inductance $L$ — the "electrical inertia"

This is the star of the topic. See Inductance and self-induction for the full story; here is the working picture.

When current through a coil changes, the coil creates a voltage that fights the change. The size of that fight-back voltage is exactly

$V_{L} = L \frac{d I}{d t} .$

Figure 3 shows the coil (violet) with an orange arrow marking the rising current entering it, and a magenta arrow marking the coil's own fight-back voltage $V_{L}$ pointing the opposite way — the push it creates to oppose that rise.

Fig 3 — The coil answers a rising current (orange arrow) with an opposing voltage $V_{L} = L d I / d t$ (magenta arrow): faster change means a bigger fight-back.

7. Faraday & Lenz — where the fight-back comes from

Two prerequisite laws explain why the coil produces $V_{L}$ and, crucially, which direction it pushes.

Faraday's law of electromagnetic induction — a changing magnetic flux through the coil creates a voltage. Changing current makes changing flux, so changing current makes a voltage. That is the origin of $L d I / d t$ .
Lenz's law — the created voltage always opposes the change that caused it. So while current grows, the coil's voltage points against the battery's push.

8. Kirchhoff's voltage law — the bookkeeping rule

Using the signs we just settled in Section 7:

battery raises potential by $ε$ (plus),
resistor drops it by $I R$ (minus),
inductor opposes with $L d I / d t$ (minus).

Add them to zero:

$ε - I R - L \frac{d I}{d t} = 0.$

This is the single equation the entire parent note solves. Every symbol in it you now own.

9. The exponential $e^{- t / τ}$ — the shape of "easing in"

The solution to that equation contains $e^{- t / τ}$ , where $τ$ is a timing number defined just below. You must recognise this shape.

Figure 4 plots both shapes against time measured in units of $τ$ : the violet decay curve $e^{- t / τ}$ dropping from 1, and the magenta growth curve $1 - e^{- t / τ}$ rising toward 1. The orange dashed line marks $t = τ$ , where the dotted guides read off $63.2%$ and $36.8%$ .

Fig 4 — Growth (magenta) and decay (violet) are mirror curves that cross the $63.2%$ / $36.8%$ marks exactly at $t = τ$ (orange line).

At $t = τ$ : $e^{- 1} \approx 0.368$ . So decay has fallen to $36.8%$ ; growth has risen to $1 - 0.368 = 63.2%$ .

10. The time constant $τ = L / R$ — one number rules all

11. Energy in the coil — where the story ends

The $\frac{1}{2}$ and the square appear because energy accumulates as current builds — a picture you'll unpack in that linked note.

Prerequisite map

Equipment checklist

Recall Am I ready? (hide answers)

What does $I$ measure, and in what unit? ::: Charge flowing per second; amperes (A). What does $\frac{d I}{d t}$ mean in plain words? ::: The slope of the current-vs-time graph — how fast the current is changing right now. What is $V$ , and its unit? ::: Voltage — the potential difference or electrical push between two points; volts (V). State Ohm's law and name each symbol. ::: $V = I R$ ; $V$ voltage (volts), $I$ current (A), $R$ resistance (ohms). What is $ε$ and why is $ε / R$ special? ::: The battery's EMF (push, in volts); $ε / R$ is the final steady current when nothing else fights back. What quantity does an inductor react to, and what voltage does it make? ::: It reacts to the change in current $d I / d t$ ; it makes $V_{L} = L d I / d t$ opposing that change. Where do Faraday and Lenz enter? ::: Faraday: changing current makes a voltage. Lenz: that voltage opposes the change (fixes the sign). Why does the inductor term enter KVL with a minus sign? ::: While current grows, Lenz's law makes the coil push against us, so the potential drops by $L d I / d t$ — a minus. Write the KVL loop equation for the RL circuit. ::: $ε - I R - L \frac{d I}{d t} = 0$ . What shape is $e^{- t / τ}$ and why does RL current follow it? ::: A smooth decay from 1 toward 0; because the rate of change is proportional to the remaining gap. What is $τ$ , its formula, and its unit? ::: The time constant $τ = L / R$ , in seconds. Value of $e^{- 1}$ and the two percentages it gives? ::: $\approx 0.368$ ; decay falls to $36.8%$ , growth rises to $63.2%$ at one $τ$ .

Connections

Inductance and self-induction — defines $L$ and the induced voltage $L d I / d t$ .
Faraday's law of electromagnetic induction — why changing current makes a voltage.
Lenz's law — why that voltage opposes the change.
Kirchhoff's voltage law — the loop-sum rule giving our equation.
RC circuit — charging and discharging — the exponential dual of this topic.
Energy stored in a magnetic field — the $\frac{1}{2} L I^{2}$ endpoint.
Parent topic — where these foundations are used.