Exploring Voltage: Electric Potential & 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.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[APhysicist]

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