Voltage

The voltage increase from some point ${\textstyle x_{A}}$ to some point ${\textstyle x_{B}}$ is given by

${\displaystyle {\begin{aligned}\Delta V_{AB}&=V(x_{B})-V(x_{A})\\&=-\int _{r_{0}}^{x_{B}}{\vec {E}}\cdot d{\vec {l}}-\left(-\int _{r_{0}}^{x_{A}}{\vec {E}}\cdot d{\vec {l}}\right)\\&=-\int _{x_{A}}^{x_{B}}{\vec {E}}\cdot d{\vec {l}}\end{aligned}}}$

The electric field around the rod exerts a force on the charged pith ball, in an electroscope

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. Mathematically, this is expressed as the line integral of the electric field along that path. Under this definition, the voltage difference between two points is not uniquely defined when there are time-varying magnetic fields since the electric force is not a conservative force in such cases.

In a static field, the work is independent of the path

If this definition of voltage is used, any circuit where there are time-varying magnetic fields,[note 1] such as circuits containing inductors, will not have a well-defined voltage between nodes in the circuit. However, if magnetic fields are suitably contained to each component, then the electric field is conservative in the region exterior[note 2] to the components, and voltages are well-defined in that region.[6] In this case, the voltage across an inductor, viewed externally, turns out to be

${\displaystyle U=\Delta V=-L{\frac {dI}{dt}}}$

despite the fact that, internally, the electric field in the coil is zero[6] (assuming it is a perfect conductor).

Using the above definition, the electric potential is not defined whenever magnetic fields change with time. In physics, it's sometimes useful to generalize the electric potential by only considering the conservative part of the electric field. This is done by the following decomposition used in electrodynamics:

${\displaystyle {\vec {E}}=-\nabla V-{\frac {\partial {\vec {A}}}{\partial t}}}$

where ${\textstyle {\vec {A}}}$ is the magnetic vector potential. The above decomposition is justified by Helmholtz's theorem.

In this case, the voltage increase from ${\textstyle x_{A}}$ to ${\textstyle x_{B}}$ is given by

${\displaystyle {\begin{aligned}\Delta V_{AB}&=-\int _{x_{A}}^{x_{B}}{\vec {E}}_{\mathrm {conservative} }\cdot d{\vec {l}}\\&=-\int _{x_{A}}^{x_{B}}\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot d{\vec {l}}\\&=-\int _{x_{A}}^{x_{B}}({\vec {E}}-{\vec {E}}_{\mathrm {induced} })\cdot d{\vec {l}}\end{aligned}}}$

where ${\textstyle {\vec {E}}_{\mathrm {induced} }}$ is the rotational electric field due to time-varying magnetic fields. In this case, the voltage between points is always uniquely defined.

In circuit analysis and electrical engineering, the voltage across an inductor is not considered to be zero or undefined, as the standard definition would suggest. This is because electrical engineers use a lumped element model to represent and analyze circuits.

When using a lumped element model, it is assumed that there are no magnetic fields in the region surrounding the circuit and that the effects of these are contained in 'lumped elements', which are idealized and self-contained circuit elements used to model physical components.[7] If the assumption of negligible leaked fields is too inaccurate, their effects can be modelled by parasitic components.

In the case of a physical inductor though, the ideal lumped representation is often accurate. This is because the leaked fields of the inductor are generally negligible, especially if the inductor is a toroid. If leaked fields are negligible, we find that

${\displaystyle \int _{\mathrm {exterior} }{\vec {E}}\cdot d{\vec {l}}=-L{\frac {dI}{dt}}}$

is path-independent, and there is a well-defined voltage across the inductor's terminals.[6] This is the reason that measurements with a voltmeter across an inductor are often reasonably independent of the placement of the test leads.

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.