← CA-5 · CAPACITORS, INDUCTORS & FIRST-ORDER CIRCUITS

Capacitors

YOU WILL BE ABLE TO

Describe what a capacitor is and state q = Cv.
Use the element law i = C·dv/dt in both directions.
Compute stored energy, and combine capacitors in series and parallel.
Explain why capacitor voltage cannot change instantaneously.

Everything so far dissipates energy the instant it arrives. This unit introduces the two elements that STORE it — and with storage comes memory, time, and the transient behaviour that makes timers, filters, and every power supply work. First: the capacitor — two conductive plates separated by an insulator. Push charge onto one plate and equal opposite charge is drawn to the other; the stored charge is proportional to the voltage across the plates.

q = Cv

capacitance, farads

C is the capacitance in farads (F): one coulomb per volt. A full farad is enormous; real parts live in microfarads (µF), nanofarads (nF), and picofarads (pF). Differentiate q = Cv with respect to time and, since i = dq/dt, the capacitor's element law appears — its current depends not on its voltage, but on how fast that voltage CHANGES.

i = C\,\frac{dv}{dt}

capacitor element law

The capacitor under the passive sign convention: i = C dv/dt. Steady voltage means zero current.

Three consequences fall straight out of the element law. First, under DC steady state, dv/dt = 0, so i = 0: a fully settled capacitor is an OPEN CIRCUIT — no current flows through the insulator, ever; charge only marches on and off the plates. Second, capacitor voltage cannot jump: an instantaneous change would demand dv/dt → ∞ and hence infinite current. Voltage is the capacitor's continuous, unbreakable state variable. Third, moving charge against a growing voltage stores energy in the electric field between the plates, recoverable in full:

w = \tfrac{1}{2}Cv^2

stored energy, joules

WORKED EXAMPLE

Current from a voltage ramp

A 10 µF capacitor's voltage ramps as v(t) = 5t volts (t in seconds).
i = C dv/dt = 10 µF × 5 V/s = 50 µA, constant while the ramp lasts.
Steady current in, steadily rising voltage — this linear charging is the heart of every timing circuit.

i = 50 µA

WORKED EXAMPLE

Energy in a photo flash

A camera-flash capacitor: 100 µF charged to 12 V.
w = ½Cv² = ½ × 100×10⁻⁶ × 144 = 7.2 mJ.
Released in about a millisecond, that is roughly 7 W of momentary power from a part the size of a fingertip — capacitors trade energy density for delivery speed, the opposite trade to a battery.

7.2 mJ

PRACTICE

How much charge is stored on a 47 µF capacitor at 5 V?

A 1 µF capacitor sees its voltage rising at 2000 V/s. What current flows?

Two 10 µF capacitors: find the equivalent capacitance in parallel and in series, and contrast with two 10 Ω resistors.

Two short conceptual checks: (a) why can't a capacitor's voltage change instantaneously? (b) after a circuit has been on for a long time, how do you treat every capacitor when computing DC values?