Voltage: What is it? (Definition, Formula And How To Measure Potential Difference)

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Key learnings:

Voltage Definition: Voltage is defined as the potential energy difference per unit charge between two points in an electrical field.
Understanding Through Analogy: Voltage can be likened to water pressure in a hydraulic system, where higher pressure pushes water through pipes, similar to voltage pushing electrons through a circuit.
Measurement Units: The standard unit for voltage is the volt, represented by the symbol V, quantifying the energy per unit charge.
Ohm’s Law Application: Voltage is crucial in Ohm’s Law, calculated as the product of current and resistance, helping determine the potential difference in circuits.
Practical Measurement: To measure voltage, instruments like voltmeters or multimeters are used, connected in parallel with the circuit element to assess the potential difference accurately.

What is Voltage?

Voltage is the electric potential difference between two points: the work per unit charge to move a test charge between those points in an electric field. In circuit work the usual symbols are V and, in older texts, E. Calling voltage “potential difference per unit charge” would divide volts by coulombs. EMF is the open-circuit source voltage, or the line integral of the non-electrostatic field in a source.

For a water-pressure picture, skip to this section.

The formal definition continues below.

In a static electric field, voltage between two points is the work per coulomb to move a small positive test charge between them. In symbols,

$\begin{align*} Voltage = \frac{Work\,\,Done\ (W)}{Charge\ (Q)} \end{align*}$

Work is in joules. Charge is in coulombs. One volt is one joule per coulomb.

$\begin{align*} Thus, Voltage = \frac{joule}{coulomb} \end{align*}$

In a circuit, that is the difference in electric potential energy per coulomb between two nodes, not the stored energy of the whole circuit.

One node sits at a higher potential than another. Voltage is that potential difference, not a difference in how much charge sits on the two nodes.

A potential difference drives current in a closed circuit. Conventional current is drawn from the higher-potential terminal toward the lower. Free electrons in a metal drift the other way, from lower potential toward higher.

For a given resistance, a larger voltage means a larger current (Ohm’s law). With zero potential difference between two nodes, no net current flows between them. Charges still move thermally; there is no directed drift.

Older British cable language also used “electric tension.” A 1 kV class is low voltage in IEC terms (up to 1 kV). 11 kV and 33 kV are medium-voltage distribution classes in many grids. “Super tension” is dated UK jargon, not a universal IEC name for 33 kV.

Definition of Potential Difference as Potential of Electric Field

The field definition uses line integrals of E, as in the stored pair below.

Take two points A and B.

V_A relative to V_B is the work per coulomb associated with moving a positive test charge between A and B in field E. Watch the path limits on the stored integral: texts differ on whether they write the integral from A to B or B to A, and that choice sets the minus sign.

The stored form is

$\begin{align*} V_A_B = \frac{W}{Q} = -\int_B^A E^- * dl^-\end{align*}$

With B as the reference, that difference is also written

$\begin{align*} V_A_B = V_A - V_B \end{align*}$

A pressure analogy is often used next.

The usual picture is water in pipes (the hydraulic analogy), used only as a teaching aid.

Understanding Voltage By Analogy

In that analogy:

Map the quantities as follows.

Electric potential (voltage) maps to hydraulic pressure
Electric current is equivalent to hydraulic water flow rate
Charge maps to a quantity of water
An electrical conductor is equivalent to a pipe

Analogy 1

Figure (a) shows two tanks at the same water level. With no pressure difference, no flow runs from one tank to the other.

In Figure (b) the levels differ, so water flows until the levels match.

If you join two cells of different open-circuit voltage with a conductor, current flows. The lower-voltage cell is charged only if polarity allows it and the higher source can source current. They do not always “equalize” like two open tanks: internal chemistry sets each cell voltage.

Analogy 2

A tank sits above ground with a hose at the bottom, as in the figure.

Hose pressure stands in for voltage. The water quantity stands in for charge. More water (a taller head) raises the pressure at the hose.

Drain the tank and the hose pressure falls. A discharging storage battery is a rough parallel: terminal voltage sags and a lamp on that battery runs dimmer, if the lamp is a simple resistive load.

Analogy 3

Work in the electric circuit maps to work in a pumped water loop, as in the figure below.

Working Analogy of Hydraulic and Electric Circuit

A pump drives water in a pipe. The pipe stands in for a conducting wire.

The pump’s pressure difference can drive a turbine, so the water does work.

A battery’s potential difference can drive current through a lamp, and that current does electrical work in the lamp.

What is Voltage Measured In (Voltage Units)?

SI Unit of Voltage

The SI unit of voltage is the volt (symbol V), a derived unit named for Alessandro Volta (1745-1827), who built the voltaic pile, an early battery.

Volt in SI Base Units

One volt is the potential difference between two points when one joule is exchanged per coulomb that passes. The stored line (including the “chrage” spelling) is

$\begin{align*} 1\,\,Volt = \frac{potential \ energy} {chrage} = \frac{1\,\, joule}{1\,\,coulomb} = \frac{kg\,\, m^2}{A\,\,s^3} \end{align*}$

In SI base units that is $\frac{kg\,\,m^2}{A\,\,s^3}$ or $kg\,\,m^2\,\,s^-^3\,\,A^-^1$ .

Equivalents include W/A and A·Ω, from P = VI and V = IR for the same operating point.

Voltage Formula

A circuit form of voltage is shown below. These Ohm’s-law and power forms hold for a resistor (or another element that obeys V = IR at that instant). The field definition remains V = W/Q.

Voltage Formula 1 (Ohm’s Law)

From Ohm’s law,

$\begin{align*} Voltage = Current * Resistance \end{align*}$

$\begin{align*} V = I * R \end{align*}$

Example 1

A current of 4 A flows in a 15 Ω resistance. Find the voltage drop.

Solution:

Given: $I = 4\,\,A$ , $R=15\,\,\Omega$

Ohm’s law:

$\begin{align*} & V = I * R \\ & = 4 * 15 \\ & V = 60\,\,Volts \end{align*}$

The drop is 60 V.

Voltage Formula 2 (Power And Current)

Instantaneous electrical power into a two-terminal element is v·i. For DC (or RMS on a resistor) P = V I.

$\begin{align*} P = V * I \end{align*}$

Substitute $I=\frac{V}{R}$ and you also get P = V²/R, so

(1) $\begin{equation*} P = V * I = \frac{V^2}{R} \end{equation*}$

So V = P/I when that power formula applies:

$\begin{align*} V = \frac{P}{I} \,\,Volts \end{align*}$

Example 2

A 48 W lamp draws 2 A. Find the supply voltage, treating P = V I as valid for that lamp at this operating point.

Solution:

Given: $I = 2\,\,A$ , $P = 48 \,\,W$

From V = P/I,

$\begin{align*} & V = \frac{P}{I} \\ & = \frac{48}{2} \\ & V = 24 \,\,Volts \end{align*}$

The supply voltage is 24 V.

Voltage Formula 3 (Power And Resistance)

From P = V²/R you also have V = √(P R) for a positive DC (or RMS) value:

$\begin{align*} V = \sqrt{P*R} \end{align*}$

Example 3

Find the voltage that puts 5 W in a 2 Ω lamp, using V = √(P R). (The stored wording “resistance of current” is the 2 Ω resistance.)

Solution:

Given: $P = 5 \,\, W$ , $R = 2 \,\, \Omega$

Then

$\begin{align*} & V = \sqrt{P*R} \\ & = \sqrt{5*2} \\ & = \sqrt{10} \\ & V = 3.16 \,\,Volts \end{align*}$

√10 ≈ 3.16, so about 3.16 V across that $5 W, 2\Omega$ lamp.

Voltage Circuit Symbol (AC And DC)

AC Voltage Symbol

A common schematic mark for AC (alternating current) voltage is below.

DC Voltage Symbol

A common schematic mark for DC (direct current) voltage is below.

Dimensions of Voltage

Voltage is electric potential energy per unit charge, so its SI dimensions follow from joule/coulomb.

In M, L, T and current A that is $M L^2 T^-^3 A^-^1$ .

$\begin{align*} V = \frac{W}{Q} = \frac{M L^2 T^-^2}{A T} = M L^2 T^-^3 A^-^1 \end{align*}$

Some texts use I for the current dimension instead of A, and then write $M L^2 T^-^3 I^-^1$ .

How to Measure Voltage

You can measure voltage between two nodes, or between a node and a chosen 0 V reference (chassis or earth, if that is the reference).

On a three-phase system, phase-to-neutral voltage is line-to-neutral (phase voltage). Call it line-to-ground only when that neutral is earthed and you are actually measuring to earth.

Voltage between two phases is line-to-line voltage.

Common instruments:

Voltmeter Method

A voltmeter reads the potential difference between its two leads. Connect it in parallel with the nodes you want.

Put one lead on each node. Do not insert a voltmeter in series as if it were an ammeter: its high resistance would starve the circuit, and you would not be measuring the original voltage.

The same meter can read a drop across one part, or across a string, depending on where the leads sit. The figure shows a resistor measurement.