What is the relationship between voltage, electric potential, and power?

[Greg Bernhardt]

Definition/Summary

Voltage is electric potential difference, which is potential energy difference per charge: V\ =\ U/q

Energy per charge equals energy per time divided by charge per time, which is power divided by current (watts per amp): V\ =\ U/q\ =\ P/I

Since potential energy is just another name for work done (by a conservative force), voltage is also electric force "dot" displacement per charge, ie electric field "dot" displacement:V\ =\ \int{E}\cdot d{x}

The unit of voltage is the volt, V, also equal to the joule per coulomb, J/C.

Equations

Equations for DC and instantaneous equations for AC:

V\ =\ IR

V\ =\ P/I\ =\ \sqrt{PR}

P\ =\ V^2/R\ =\ I^2R\ =\ VI

Average equations for AC:

P_{average}\ =\ V_{rms}^2/R

P_{average}\ =\ V_{rms}I_{rms}cos\phi

P_{apparent} \ =\ V_{rms}I_{rms} \ =\ |P_{complex}|\ =\ \sqrt{P_{average}^2 + Q_{average}^2}

P_{average}\ =\ V_{rms}^2\cos\phi/|Z|

V_{average}\ =\ (2\sqrt{2}/\pi)V_{rms}\ =\ (2/\pi)V_{peak}

where \phi is the phase difference between voltage and current, Z is the (complex) impedance, Q is the reactive or imaginary power (involving no net transfer of energy), and V_{rms}\text{ and }I_{rms} are the root-mean-square voltage and current, V_{peak}/\sqrt{2}\text{ and }I_{peak}/\sqrt{2}.

Extended explanation

Two ways of defining voltage:

voltage = energy/charge = work/charge = force"dot"distance/charge = (from the Lorentz force) electric field"dot"distance, or dV = E.dr

but also voltage = energy/charge = (energy/time)/(charge/time) = power/current, or V = P/I

Volt:

The volt is defined as the potential difference across a conductor when a current of one amp dissipates one watt of power.

Kirchhoff's second rule: (syn. Kirchhoff's Law, KVL)

The sum of potential differences around any loop is zero.

So potential difference is "additive" for components in series: the total potential difference is the sum of the individual potential differences.

Across a DC or AC resistance, V\ =\ IR. Across an AC capacitor or inductor, V\ =\ IX, where X is the reactance.

For a general AC load, V_{rms}\ =\ I_{rms}|Z|, where the complex number Z\ =\ R+jX is the impedance (purely real for a resistance and purely imaginary for a capacitor or inductor). If phase is important, we use V\ =\ IZ, where V and I are complex numbers also.

Alternating current (AC):

The "official" voltage delivered by electricity generators and marked on electrical equipment (such as 240V or 100V) is the root mean square voltage, V_{rms}, which is the peak voltage (amplitude) divided by √2.

Voltage may be out of phase with current, by a phase difference (phase angle), \phi.

Instantaneous power equals instantaneous voltage times instantaneous current: P\ =\ VI, but average power is V_{rms}I_{rms}\cos\phi, or the apparent power times the phase factor.

AC power:

AC power, P, usually means the power (true power, or real power) which transfers net energy (does net work), as opposed to the reactive power (imaginary power), Q, which transfers no net energy.

Complex power is S\ =\ P\ +\ jQ.​

Electromotive force (emf):

Electromotive force has different meanings for different authors (and is not a force anyway): see http://en.wikipedia.org/wiki/Electromotive_force#Terminology. Sometimes it means voltage.

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[APhysicist]

Thanks for the explanation! I found it really helpful in understanding the concept of voltage.