Capacitors and inductors

Capacitors

A capacitor is a device that stores energy in an electric field which is generated by equal and opposite electric charge on opposing electrodes or plates within the capacitor. In turn, the charge Q is proportional to the voltage across the capacitor,

$ Q = CV\frac{}{} $

where C is the capacitance (units of farads (F) ≡ coulombs/volt). The current through the capacitor is the time rate of change of the charge and so the terminal relation of voltage and current for a capacitor is

$ i = \frac{dQ}{dt} = C \frac{dV}{dt} $

In an R-C circuit where a capacitor is charged by through current-limiting resistor from a step voltage source V₀ (the voltage is zero for t<0 and V₀ for t>0),

$ i = \frac{V_0-V_C}{R} = C \frac{dV}{dt} $

for t>0, or

$ V_0-V_C = RC \frac{dV}{dt} $

This first-order differential equation has the well-known solution

$ V_C = A e^{-t/\tau} + B\frac{}{} $

where τ=RC is the "RC" time constant. At time t=0 there is no charge or voltage on the capacitor so V_C(0)=0 and A=−B. Also as t→∞ then V_C→B a constant and so i→0 and the voltage across the resistor also goes to zero. We conclude that B=V₀ and the therefore,

$ V_C = V_0 (1-e^{-t/\tau})\frac{}{} $

Inductors

In contrast to a capacitor, an inductor is a device that generates and stores energy in a magnetic field internally. In turn, the time rate of change of this magnetic field generates a voltage (Faraday's law of induction),

$ v = L \frac{di}{dt} $

where L is the inductance (units of henrys (H) ≡ volt⋅second/ampere). For a constant current i, the voltage across the inductor is identically zero.