Capacitors, inductors and energy

How a camera flash works

• A phone's camera flash collects energy for a second and releases it in about a millisecond. No arrangement of resistors can do that: resistors only burn energy.
• Two new elements store it. That one change is what makes circuits change with time.

The capacitor: current follows the slope of voltage

• A capacitor 电容器 stores charge on two plates separated by an insulator: $q = Cv$, with capacitance 电容 $C$ in farads. Differentiating gives $i = C\dfrac{dv}{dt}$.
• It carries current only while its voltage changes. At steady DC, $\dfrac{dv}{dt} = 0$, so the capacitor is an open circuit.

The inductor: voltage follows the slope of current

• An inductor 电感器 stores energy in the magnetic field of a coil: $v = L\dfrac{di}{dt}$, with inductance 电感 $L$ in henrys.
• It develops voltage only while its current changes. At steady DC, $\dfrac{di}{dt} = 0$, so the inductor is a short circuit.

Combinations are mirror images

• Parallel capacitors share one voltage, so their charges and capacitances add. Two 47 µF capacitors in series give $\dfrac{47 \times 47}{94} = 23.5\,\mu\text{F}$.
• Inductors do the opposite: in series they add, in parallel they combine by the product-over-sum rule. The storage laws swap the roles of $v$ and $i$, so the resistor rules swap too.

Stored energy, and the rate that matters

• Energy depends only on the present state: $w_{C} = \dfrac{1}{2}Cv^{2}$ and $w_{L} = \dfrac{1}{2}Li^{2}$.
• The amounts are small; the rate makes them dramatic. That is why large capacitors are discharged before anyone touches them.

Stored quantities cannot jump

• Capacitor voltage cannot jump (that would need infinite current); inductor current cannot jump (that would need infinite voltage). This continuity is the key to every switching problem.
• Interrupting an inductor is therefore dangerous: stopping 1.5 A in 1 ms through 0.1 H gives $v = 0.1 \times 1500 = 150\,\text{V}$, the arc across the opening contact.

Three common slips

Recap