Capacitors and Inductors — V-I Relations and Energy Stored

By the end you should be able to: State the voltage-current relations for capacitors and inductors, calculate the energy stored in each, and combine them in series and parallel.

A resistor dissipates energy. These two store it — one in an electric field, one in a magnetic field — and give it back. That difference makes their behaviour depend on rates of change rather than on instantaneous values.

The capacitor

Charge stored is proportional to voltage:

Q = CV

Differentiating, since $i = dQ/dt$ :

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

Read what this says. Current flows only while the voltage is changing. At steady DC, $dv/dt = 0$ , so $i = 0$ : a capacitor is an open circuit to DC.

It also says voltage cannot change instantaneously. A step change would need infinite $dv/dt$ and therefore infinite current. The voltage across a capacitor is continuous — a fact used constantly in transient analysis.

Energy stored:

W = \frac{1}{2}CV^{2} = \frac{1}{2}\frac{Q^{2}}{C} = \frac{1}{2}QV

Stored in the electric field between the plates.

Combinations. Note these are the opposite way round to resistors:

\text{Series: } \frac{1}{C_{eq}} = \sum\frac{1}{C_i} \qquad \text{Parallel: } C_{eq} = \sum C_i

Parallel adds because it is equivalent to increasing the plate area, and $C = \varepsilon A/d$ .

The inductor

Flux linkage is proportional to current, $\lambda = LI$ , and Faraday's law gives $v = d\lambda/dt$ :

\boxed{\,v = L\,\frac{di}{dt}\,}

The exact dual. Voltage appears only while the current is changing. At steady DC, $di/dt = 0$ , so $v = 0$ : an inductor is a short circuit to DC (ignoring winding resistance).

And the current through an inductor is continuous — it cannot change instantaneously, since that would demand infinite voltage.

This is not a theoretical nicety. Interrupting an inductive current — switching off a motor or a relay coil — forces $di/dt$ to be very large, and the inductor generates a large voltage spike in response. That spike is what arcs across switch contacts and destroys transistors, and it is why a freewheeling diode is fitted across every relay coil to give the current a path as it decays.

Energy stored:

W = \frac{1}{2}LI^{2}

Stored in the magnetic field.

Combinations follow resistors, not capacitors:

\text{Series: } L_{eq} = \sum L_i \qquad \text{Parallel: } \frac{1}{L_{eq}} = \sum\frac{1}{L_i}

The duality worth memorising

	Capacitor	Inductor
Relation	$i = C\,dv/dt$	$v = L\,di/dt$
Stores energy in	electric field	magnetic field
Energy	$\tfrac12 CV^2$	$\tfrac12 LI^2$
At DC	open circuit	short circuit
Cannot change instantly	voltage	current
Series	reciprocals add	values add

Learn one column and swap $v \leftrightarrow i$ , $C \leftrightarrow L$ to get the other. Almost every result about one has a mirror image in the other.