Electromotive Force Formula: ε = I(R + r), Unit and Examples

The electromotive force formula for a cell in a closed circuit is $\varepsilon = I(R + r)$ : the emf equals the current times the external resistance plus the internal resistance. At its root, emf is the energy a source hands to each coulomb of charge it pushes round, $\varepsilon = \frac{W}{q}$ , so one volt of emf means one joule given to every coulomb.

Despite the name, emf is not a force at all. It is measured in volts, like potential difference. The difference is where the energy goes. Emf is energy supplied by the cell, chemical energy in a battery or mechanical energy in a dynamo. Terminal voltage is the energy each coulomb delivers to the outside circuit after some is lost inside the cell. This topic sits beside the voltage formula and Ohm's law in the electricity formulas section.

Quick facts

Formula
$\varepsilon = I(R + r)$
Definition
$\varepsilon = \frac{W}{q}$ , work done per unit charge
SI unit
volt (V), equal to J/C
Dimensional formula
[M¹ L² T⁻³ I⁻¹]
Type
Scalar
Terminal voltage
$V = \varepsilon - Ir$ while the cell supplies current
Motional emf
$\varepsilon = Blv$ for a rod moving through a field

The electromotive force formula, symbol by symbol

Quantities in ε = I(R + r)
Symbol	Quantity it represents	SI unit
$\varepsilon$	Emf of the cell or generator	volt (V)
$I$	Current through the whole loop	ampere (A)
$R$	Resistance of the outside circuit, the load	ohm (Ω)
$r$	Internal resistance of the source itself	ohm (Ω)

Read the formula as an energy budget. The cell supplies $\varepsilon$ joules per coulomb. Of that, $IR$ joules per coulomb are used in the load and $Ir$ joules per coulomb are wasted as heat inside the cell. Rearranged, this gives terminal voltage $V = \varepsilon - Ir$ . Some US courses write the emf as $\mathcal{E}$ instead of $\varepsilon$ ; both mean the same thing.

Steps for a cell and load problem

Draw the loop. Mark the cell, its internal resistance r in series, and the load R.
Combine the load first. If the outside circuit has several resistors, reduce them to one equivalent R using series and parallel rules.
Apply the formula. Use $I = \frac{\varepsilon}{R + r}$ to find the current.
Find what is asked. Terminal voltage is $IR$ or $\varepsilon - Ir$ ; lost volts are $Ir$ .
Sanity check. Terminal voltage must come out less than the emf when the cell is discharging.

The calculator above solves the same relation for E, I, R or r. Type three values in volts, amperes and ohms and it returns the fourth.

Solved examples on emf

Example 1: a torch cell

A cell drives 0.5 A through a 2.6 Ω bulb. Its internal resistance is 0.4 Ω. Find the emf.

Given: I = 0.5 A, R = 2.6 Ω, r = 0.4 Ω.

$\varepsilon = 0.5 \times (2.6 + 0.4) = 0.5 \times 3.0 = 1.5\ \text{V}$

The bulb receives 0.5 × 2.6 = 1.3 V; the other 0.2 V is used up inside the cell.

Example 2: starting a car engine

A 12.0 V battery with internal resistance 0.05 Ω supplies 100 A to the starter motor. What is the terminal voltage?

$V = \varepsilon - Ir = 12.0 - (100)(0.05) = 12.0 - 5.0 = 7.0\ \text{V}$

This is why headlights dim while the engine cranks.

Example 3: emf from energy, with a unit change

A cell does 0.375 J of work shifting 250 mC through the loop. What is its emf?

Convert: 250 mC = 0.250 C.

$\varepsilon = \frac{W}{q} = \frac{0.375}{0.250} = 1.5\ \text{V}$

Emf versus terminal potential difference

How emf and terminal voltage differ
Point	Emf $\varepsilon$	Terminal voltage V
Meaning	Energy supplied per coulomb	Energy delivered to the load per coulomb
Depends on current	No, fixed by the cell	Yes, falls as I rises
Measured when	No current flows (open circuit)	Current is flowing
Relation	$V = \varepsilon - Ir$ while discharging, $V = \varepsilon + Ir$ while charging

A plot of V against I for a cell is a straight line. Its intercept on the V axis is the emf and its slope is $-r$ . This is the standard practical for measuring internal resistance in Class 12.

Other sources of emf and the unit check

Emf is not only a battery idea. A metal rod of length l moving at speed v across a field B produces motional emf $\varepsilon = Blv$ . A changing magnetic flux through a coil induces $\varepsilon = -N\frac{d\Phi}{dt}$ , which is Faraday's law; see the magnetic flux formula for $\Phi$ . A coil opposing its own changing current gives $\varepsilon = -L\frac{dI}{dt}$ , covered on the self inductance page.

Every form must end in volts. From $\varepsilon = W/q$ , the dimensions are [M L² T⁻²] ÷ [I T] = [M¹ L² T⁻³ I⁻¹], the same as any voltage, as on the dimensional formula of voltage page.

Mistakes students make with emf

Leaving out r. Writing $\varepsilon = IR$ treats the cell as perfect. Only an ideal cell has r = 0.
Calling emf a force. It has units of volts, not newtons. Say "emf" in answers to avoid the trap.
Wrong sign when charging. A cell being charged has terminal voltage above its emf: $V = \varepsilon + Ir$ .
Cells in parallel. Identical cells in parallel keep the same emf but divide the internal resistance; cells in series add both emf and r.

An ideal voltmeter across an open cell reads the emf. Once a load is connected, the reading drops to the terminal voltage.

Frequently asked questions

What is the formula of electromotive force?

The emf of a source is the work done per unit charge, ε = W/q. In a circuit with load R and internal resistance r, it is ε = I(R + r).

Which SI unit is emf measured in?

Emf is measured in volts, and one volt means one joule handed to each coulomb. Broken into base quantities, the volt gives [M¹ L² T⁻³ I⁻¹].

Is electromotive force really a force?

No. The name is historical. Emf is energy per unit charge and is measured in volts, so it is a scalar, not a force measured in newtons.

What is the difference between emf and potential difference?

Emf is the energy the source supplies to each coulomb, and it does not change with current. Potential difference across the load is what each coulomb gives up outside the cell, and it equals ε − Ir while current flows.

Can terminal voltage be more than emf?

Yes, while the cell is being charged by an outside source. Current then flows into the positive terminal and V = ε + Ir.