Every component so far has been a resistor, which does one thing with energy: converts it to heat immediately. A capacitor and an inductor do something different: they store energy and give it back. That storage takes time to fill and empty, which is how time first enters a circuit. Until now, nothing depended on how long you waited.
The capacitor
A capacitor is two metal plates facing each other with an insulating gap between them: air, plastic, or ceramic. No electrons cross the gap.
Connect a battery. It pushes electrons onto the negative plate. Those electrons pile up and their combined electric field reaches across the gap, pushing electrons off the positive plate. Those ejected electrons flow out through the other terminal, through the circuit, and back to the battery. Current flows in the whole circuit even though nothing crosses the gap: the field couples the two halves.
Both terminals are essential. The second terminal is the escape route for the electrons being pushed off the positive plate. Without it, those electrons have nowhere to go, pile up immediately, and block any further charging; like a clogged pipe, pressure builds but nothing flows.
As the capacitor charges, its own voltage grows and opposes the battery. The voltage left to push current through the resistor shrinks. Less current means slower charging. The fuller the capacitor, the slower it fills. Eventually the capacitor voltage matches the battery exactly: zero voltage across the resistor, zero current, charging stops.
At full charge the capacitor holds its voltage indefinitely but passes no current. It is the opposite of a wire: a wire passes current and drops no voltage; a fully charged capacitor drops the full voltage and passes no current. Put one in series in a DC circuit and it charges up, then the circuit is effectively broken.
For alternating current the picture reverses. The voltage keeps changing direction, so the capacitor never fully charges in one direction before the voltage reverses and starts discharging it. The cycle of charge and discharge repeats continuously. To AC a capacitor looks like it conducts, because current flows in and out of the plates continuously, even though nothing ever crosses the gap.
Capacitance, measured in farads, describes how much charge is needed to reach a given voltage. Large plates close together: strong field coupling, large capacitance. Small plates far apart: weak coupling, small capacitance. Real capacitors use thin dielectric materials between the plates to pack large plate area into a small volume.
In practice. The most common capacitor faults an electrician encounters are motor start and run capacitors. A single-phase motor cannot start on its own: the magnetic field does not rotate. A start capacitor shifts the phase of current in a second winding, creating enough rotation to get the motor spinning, then disconnects. A run capacitor stays in circuit permanently, improving efficiency and torque; when it fails, the motor hums but does not start. Suppression capacitors across contactor and relay coils absorb the voltage spike produced when the coil de-energises, protecting other components. Power factor correction capacitor banks on commercial and industrial sites counteract the lagging current drawn by motors and transformers.
How quickly a capacitor charges: the RC time constant
The current at any moment is set by the voltage still available to drive it:
That current adds charge to the capacitor, raising \(V_{\text{cap}}\), which reduces the current, which slows the charging further. The rate of change of \(V_{\text{cap}}\) is proportional to how much gap remains between \(V_{\text{cap}}\) and \(V_s\). That relationship, rate proportional to remaining gap, always produces an exponential curve.
\(\tau\) (tau) is the time constant. After one \(\tau\) the capacitor reaches 63% of the supply voltage, always, regardless of the values. After \(5\tau\) it is at 99.3%, close enough to call done.
\(R\) and \(C\) both stretch \(\tau\) in the same direction. A larger \(R\) limits current, slowing charging. A larger \(C\) needs more charge to reach the same voltage, also slowing charging. Their product gives the timescale in seconds (ohms × farads = seconds).
On discharge through a resistor, the same curve runs in reverse. The voltage falls exponentially with the same \(\tau\), and the current flows in the opposite direction to charging; the capacitor now acts as the source, driving current from its positive plate, through the external circuit, back to its negative plate.
The inductor
An inductor is a coil of wire. To understand why it behaves the way it does, start with a single electron moving through a straight wire: it creates a tiny magnetic field in rings around itself. Weak and spread out in every direction.
Now bend that wire into a loop. The electron travels around the loop and the rings of magnetic field it produces through the inside of the loop all point the same direction, straight through the centre. On the outside they spread and cancel. The field is concentrated through the middle of the loop.
Stack many loops into a coil. Every electron passing through every turn contributes its field through the centre, all pointing the same way, all adding up. More turns, more current, stronger combined field through the middle. The coil is a magnet whose strength is set by the current flowing through it.
Now switch the current on from zero. The current grows, the field grows with it. But a growing magnetic field induces a voltage in any conductor around it: Faraday's law. The coil itself is that conductor. So the growing field induces a voltage in the very coil that created it, pointing opposite to the battery. Two voltages opposing each other in the same circuit: the battery pushing current up, the coil pushing back.
The result: current cannot jump. It climbs gradually, each step opposed by the field it just produced. As the current approaches its final value, the rate of change slows, the opposing voltage shrinks, and the current settles. Once steady: no more change, no more growing field, no more induced opposition. The coil is just a wire. No charge is consumed; electrons go in one end and come out the other throughout. The energy that went into building the field comes back out as a voltage spike when the current is cut.
Voltage across the inductor equals inductance times rate of change of current. Steady current: zero voltage, invisible. Changing current: opposing voltage proportional to how fast it changes.
This is the mirror of the capacitor:
| Capacitor | Inductor | |
|---|---|---|
| Resists change in | voltage | current |
| Steady DC | blocks (fully charged) | passes (just a wire) |
| Sudden interruption | holds voltage | voltage spike |
Inductance, in henries, measures how strongly the coil opposes change. One henry: a current changing at 1 A/s generates 1 V of opposition. More turns, larger loops, iron core: all increase inductance by strengthening the field per ampere.
In practice. Every motor winding is an inductor; the rotating field that drives the rotor is built from inductors. Transformer windings are inductors coupled through a shared iron core: current changing in one winding creates a changing field that induces voltage in the other. The voltage spike an inductor produces when its current is suddenly interrupted is the arc you see when breaking a circuit under load, and the reason suppression diodes and capacitors are fitted across contactors, relay coils, and motor terminals.
How quickly an inductor's current builds: the RL time constant
The inductor's time constant follows the same logic as the RC circuit, with current in the role voltage played:
At switch-on, the inductor opposes the full supply and current starts at zero. As current builds, the rate of change slows, the opposing voltage drops, and current climbs toward its final value \(V_s/R\) along the same exponential curve. After \(5\tau\), done.
A larger \(L\) resists change more strongly, giving a slower rise. A larger \(R\) means a lower final current and also dissipates energy faster, giving a shorter time constant.
RC and RL transient explorer
Where this comes from
- Capacitors, inductors, and first-order transients: Alexander and Sadiku, Fundamentals of Electric Circuits, ch. 6 to 7.
- Physical intuition: Purcell and Morin, Electricity and Magnetism, ch. 3 and 7.
Check yourself
1. A capacitor is fully charged and the battery is disconnected. Is there voltage across the terminals? Does current flow?
Voltage yes: the capacitor holds its charge and maintains the voltage indefinitely. Current no: with no path to discharge, the charge imbalance stays in place. A fully charged, isolated capacitor is the opposite of a wire: full voltage, zero current.
2. A 470 Ω resistor charges a 100 μF capacitor from a 12 V supply. What is the time constant, and what voltage is across the capacitor after one time constant?
τ = 470 × 100 × 10⁻⁶ = 47 ms. After one τ: V = 12 × 0.632 = 7.58 V.
3. An air conditioning compressor motor starts sluggishly and draws high current but barely turns. The electrician suspects the run capacitor. Why would a failed run capacitor cause this symptom?
The run capacitor shifts the phase of current in the second winding, maintaining a rotating magnetic field. Without it, the rotating field weakens and the motor struggles to develop torque; it draws high current trying to accelerate but cannot. Replacing the run capacitor restores normal starting and running.
4. Why does breaking a circuit under load produce an arc at the contacts, and what does it have to do with inductors?
Any inductive load (motor, relay coil, transformer) stores energy in its magnetic field. When the circuit breaks, the inductor tries to maintain its current by generating a high voltage spike across the opening gap. That voltage ionises the air between the contacts and sustains a brief arc. Suppression components (capacitors, resistors, or diodes across the load) give the inductor's energy somewhere to go other than the contacts.
5. An RL circuit has L = 50 mH and R = 25 Ω, switched onto a 10 V supply. What is the time constant and what is the final current?
τ = L/R = 0.05/25 = 2 ms. Final current I = V/R = 10/25 = 0.4 A, reached in practice after 5τ = 10 ms.