Capacitors
A capacitor stores energy in an electric field between two conductors. Its defining behaviour, resisting sudden changes in voltage, makes it essential for filtering, timing, and energy storage.
Structure and capacitance
A capacitor consists of two conducting plates separated by an insulator, the dielectric. Applying a voltage moves charge onto the plates, +q on one and -q on the other. Capacitance is the charge stored per volt: \(C = \frac{q}{v}\), measured in farads (F). For parallel plates, \(C = \frac{\varepsilon A}{d}\): larger plate area A, smaller separation d, and higher-permittivity dielectric all increase capacitance.
Current-voltage relationship
Since \(q = Cv\), the current is \(i = C\frac{dv}{dt}\). A capacitor passes current only while its voltage changes. With constant (DC) voltage, no current flows in steady state, so a capacitor acts as an open circuit to DC. Its voltage cannot change instantaneously, as that would need infinite current, which is why capacitors smooth voltage fluctuations.
Energy storage
A capacitor stores energy \(w = \frac{1}{2}Cv^2\) in its electric field. A 1,000 \(\mu\)F capacitor charged to 50 V stores \(\frac{1}{2} \times 10^{-3} \times 2{,}500 = 1.25\) J. Large capacitors in power supplies and camera flashes can deliver dangerous shocks after power is removed, so they must be discharged safely before handling.
Combining capacitors
Capacitors in parallel add like resistors in series: \(C_{eq} = C_1 + C_2 + \dots\), since their plate areas effectively add. Capacitors in series combine like resistors in parallel: \(\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots\), which reduces capacitance but increases the total voltage rating.
Types of capacitor
Ceramic capacitors are small and cheap, from picofarads to tens of microfarads, used for decoupling and high frequencies. Electrolytic capacitors, aluminium or tantalum, offer large capacitance in small volumes but are polarised and fail, sometimes violently, if connected backwards. Film capacitors are stable and precise. Supercapacitors reach thousands of farads for short-term energy storage.
Practical ratings
Every capacitor has a maximum voltage rating, which should not be exceeded, and a tolerance, often 10% to 20% for electrolytics. Real capacitors also have equivalent series resistance (ESR) and leakage current. Electrolytic capacitors age, particularly at high temperatures, and dried-out capacitors are a common cause of failing power supplies.
Uses of capacitors
Decoupling capacitors placed close to integrated circuits supply brief current surges and absorb noise on power rails. Filter capacitors smooth rectified DC in power supplies. Coupling capacitors pass AC signals between amplifier stages while blocking DC. Capacitors set timing in oscillators, store energy in flashes and defibrillators, and touchscreens sense the capacitance of a finger.
Worked example
A 10 \(\mu\)F capacitor's voltage rises steadily from 0 to 5 V in 2 ms. The rate of change is \(\frac{5}{0.002} = 2{,}500\) V/s, so the current is \(i = C\frac{dv}{dt} = 10^{-5} \times 2{,}500 = 25\) mA. The final stored energy is \(\frac{1}{2} \times 10^{-5} \times 25 = 125\) \(\mu\)J, and the stored charge is \(q = Cv = 50\) \(\mu\)C.
Key takeaways
A capacitor stores charge on plates separated by a dielectric; C = q/v in farads, and C = epsilon A / d for parallel plates. Current is i = C dv/dt, so capacitors block steady DC and resist sudden voltage changes. Stored energy is w = Cv^2 / 2, which can remain after power-off. Parallel capacitances add; series capacitances combine reciprocally. Ceramic, electrolytic (polarised), film, and supercapacitors suit different needs. Respect voltage ratings, tolerance, ESR, and ageing. Capacitors decouple, filter, couple signals, set timing, and store energy.
Flashcard
Two conductors separated by a dielectric
Check your understanding
Why can't the voltage across a capacitor change instantly?