Electrostatics

In electrostatics, the voltage increase from point ${\displaystyle \mathbf {r} _{A}}$ to some point ${\displaystyle \mathbf {r} _{B}}$ is given by the change in electrostatic potential ${\textstyle V}$ from ${\displaystyle \mathbf {r} _{A}}$ to ${\displaystyle \mathbf {r} _{B}}$. By definition,^[17]: 78 this is:

${\displaystyle {\begin{aligned}\Delta V_{AB}&=V(\mathbf {r} _{B})-V(\mathbf {r} _{A})\\&=-\int _{\mathbf {r} _{0}}^{\mathbf {r} _{B}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}-\left(-\int _{\mathbf {r} _{0}}^{\mathbf {r} _{A}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\right)\\&=-\int _{\mathbf {r} _{A}}^{\mathbf {r} _{B}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\end{aligned}}}$

where ${\displaystyle \mathbf {E} }$ is the intensity of the electric field.

In this case, the voltage increase from point A to point B is equal to the work done per unit charge, against the electric field, to move the charge from A to B without causing any acceleration.^{[17]: 90–91} Mathematically, this is expressed as the line integral of the electric field along that path. In electrostatics, this line integral is independent of the path taken.^[17]: 91

Under this definition, any circuit where there are time-varying magnetic fields, such as AC circuits, will not have a well-defined voltage between nodes in the circuit, since the electric force is not a conservative force in those cases.^{[note 1]} However, at lower frequencies when the electric and magnetic fields are not rapidly changing, this can be neglected (see electrostatic approximation).

Electrodynamics

The electric potential can be generalized to electrodynamics, so that differences in electric potential between points are well-defined even in the presence of time-varying fields. However, unlike in electrostatics, the electric field can no longer be expressed only in terms of the electric potential.^[17]: 417 Furthermore, the potential is no longer uniquely determined up to a constant, and can take significantly different forms depending on the choice of gauge.^{[note 2][17]: 419–422}

In this general case, some authors^[18] use the word "voltage" to refer to the line integral of the electric field, rather than to differences in electric potential. In this case, the voltage rise along some path ${\displaystyle {\mathcal {P}}}$ from ${\displaystyle \mathbf {r} _{A}}$ to ${\displaystyle \mathbf {r} _{B}}$ is given by:

${\displaystyle \Delta V_{AB}=-\int _{\mathcal {P}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}$

However, in this case the "voltage" between two points depends on the path taken.

Circuit theory

In circuit analysis and electrical engineering, lumped element models are used to represent and analyze circuits. These elements are idealized and self-contained circuit elements used to model physical components.^[19]

When using a lumped element model, it is assumed that the effects of changing magnetic fields produced by the circuit are suitably contained to each element.^[19] Under these assumptions, the electric field in the region exterior to each component is conservative, and voltages between nodes in the circuit are well-defined, where^[19]

${\displaystyle \Delta V_{AB}=-\int _{\mathbf {r} _{A}}^{\mathbf {r} _{B}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}$

as long as the path of integration does not pass through the inside of any component. The above is the same formula used in electrostatics. This integral, with the path of integration being along the test leads, is what a voltmeter will actually measure.^{[20][note 3]}

If uncontained magnetic fields throughout the circuit are not negligible, then their effects can be modelled by adding mutual inductance elements. In the case of a physical inductor though, the ideal lumped representation is often accurate. This is because the external fields of inductors are generally negligible, especially if the inductor has a closed magnetic path. If external fields are negligible, we find that

${\displaystyle \Delta V_{AB}=-\int _{\mathrm {exterior} }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=L{\frac {dI}{dt}}}$

is path-independent, and there is a well-defined voltage across the inductor's terminals.^[21] This is the reason that measurements with a voltmeter across an inductor are often reasonably independent of the placement of the test leads.

Addition of voltages

The voltage between A and C is the sum of the voltage between A and B and the voltage between B and C. The various voltages in a circuit can be computed using Kirchhoff's circuit laws.

When talking about alternating current (AC) there is a difference between instantaneous voltage and average voltage. Instantaneous voltages can be added for direct current (DC) and AC, but average voltages can be meaningfully added only when they apply to signals that all have the same frequency and phase.