Capacitance, inductance and energy storage

Separate charge between conductors

A capacitor separates equal and opposite charges on conductors, storing energy in an electric field. For a constant capacitance C, the signed charge q on the terminal whose voltage is v relative to the other terminal is q=Cv. The magnitude of the separated charge is |q|=C|v|. Capacitance has units of farads, F, or coulombs per volt. A capacitor described as holding charge does not acquire a continually growing net charge as a whole in this two-terminal model.

With current referenced into the positive-voltage terminal, i=C dv/dt. The notation dv/dt means the rate at which voltage changes. A finite current can change an ideal capacitor’s voltage over time; an instantaneous finite voltage jump would require an impulsive current outside the ordinary finite-current assumption. At constant voltage the ideal element draws zero current, although real capacitors can leak.

Relate magnetic storage to changing current

An inductor links current with magnetic field and flux. For a constant inductance L and passive terminal references, v=L di/dt. Here terminal voltage accompanies a rate of change of current. A finite voltage cannot instantaneously change this ideal inductor’s current. At constant current its ideal terminal voltage is zero.

That statement does not make every real coil a zero-resistance wire. Windings have resistance, cores may behave nonlinearly or saturate, and losses depend on conditions. Likewise, a real capacitor has losses and limits. Constant C and L are deliberately restricted models that make state and energy accounting tractable.

Read the square-law energy relationships

The stored energies in these linear models are EC=½Cv² and EL=½Li². Their dependence on a square means doubling the capacitor voltage or inductor current quadruples stored energy when C or L stays constant. Joules describe the stored state; watts describe the rate of transfer into or out of it.

The figure plots original hypothetical values rather than material or component data. A resistor dissipates electrical energy within its model, whereas an ideal storage element can return previously stored energy to a circuit. A real route may combine storage, heat loss and other transfers. Opening one supply connection does not, by itself, account for energy already in storage.

Compare two specified stored states

Take an ideal 100 μF capacitor at 4 V. Since 100 μF=100×10⁻⁶ F, separated charge is 0.0004 C=400 μC. Stored energy is ½×100×10⁻⁶×4²=0.0008 J, or 0.8 mJ.

For an ideal 20 mH inductor carrying 0.2 A, L=0.020 H and stored energy is ½×0.020×0.2²=0.0004 J=0.4 mJ. These values compare two constructed states; they do not rank all capacitors against all inductors.

At 8 V the same capacitor holds 3.2 mJ. At 0.4 A the same ideal inductor holds 1.6 mJ. Each energy quadruples. Actual voltage/current limits, stored-energy access and the rest of a system require information beyond these equations.

Try the reasoning

1. Can the ideal capacitor carry current while its voltage is changing?

   Show the answer and reasoning

   Yes. The model relation is i=C dv/dt. A constant voltage, rather than all voltage, gives zero ideal current.

2. Why can a real coil heat while a constant-L model says zero steady terminal voltage?

   Show the answer and reasoning

   The ideal inductance omits winding resistance and other losses. Add the relevant elements and material behaviour to model the real coil.

3. Does doubling voltage double stored capacitor energy?

   Show the answer and reasoning

   At constant C, it quadruples the energy because EC depends on voltage squared.

What to carry forward

Capacitor voltage and inductor current are distinct stored states. Use their element laws and energy equations within declared conditions.