EMF, Terminal Voltage & Internal Resistance

PHYCells & Circuits

This subtopic distinguishes a cell's EMF (the energy it gives per coulomb on open circuit) from its terminal voltage (what appears across its terminals when current flows), the difference being the internal $Ir$ drop. The central relation $V=E-Ir$ underlies almost every cell problem.

ISC asks students to find terminal voltage from EMF and load, or to back out the internal resistance from a measured current and external resistance.

Terminal voltage (discharging)

V = E - I r

V

terminal voltage (V),

E

EMF (V),

I

current drawn (A),

r

internal resistance (

\Omega

V<E

whenever the cell supplies current.

Derivation

Apply Kirchhoff's loop rule to the single-loop circuit (cell of emf $E$ , internal resistance $r$ , external resistor $R$ ): the emf rise equals the sum of the $IR$ drops across $r$ and $R$ :
$E = Ir + IR$
The terminal voltage $V$ is the potential difference measured across the cell's terminals, which equals the drop across the external resistor:
$V = IR = E - Ir$
So $V<E$ while the cell discharges (current leaves the $+$ terminal); $V\to E$ only as $I\to0$ , i.e. on open circuit.

Circuit current with internal resistance

I = \dfrac{E}{R + r}

R

external load (

\Omega

r

internal resistance (

\Omega

). The total circuit resistance is

R+r

, not just

R

Internal resistance from measurements

r = \dfrac{E - V}{I} = R \left( \dfrac{E - V}{V} \right)

V=IR

is the terminal voltage across the load. The lost volts

E-V

equal the internal drop

Ir

Terminal voltage across the load

V = I R

V

terminal voltage equals the PD across the external resistance for a single-loop circuit. Combine with

I=E/(R+r)

to eliminate

I

EMF is the maximum voltage available on open circuit (when $I=0$ ); terminal voltage falls below it as soon as current flows.
The 'lost volts' $E-V$ are dropped across the internal resistance and equal $Ir$ .
While charging the cell the relation flips to $V=E+Ir$ , because current is forced in against the EMF.
The total resistance limiting the current is $R+r$ ; ignoring $r$ overestimates the current.
Terminal voltage equals EMF only when no current flows (open circuit or potentiometer balance).
A higher internal resistance causes a larger drop in terminal voltage as load current increases.
For a single loop the load PD $V=IR$ and the cell relation $V=E-Ir$ describe the same terminal voltage.

Where the marks go

Confusing EMF with terminal voltage — they are equal only at zero current.
Omitting $r$ from the total resistance and using $I=E/R$ instead of $I=E/(R+r)$ .
Using $V=E+Ir$ for a discharging cell; the plus sign applies only during charging.
Forgetting that the lost volts $E-V$ equal $Ir$ , leading to a wrong value of $r$ .

How the board asks it

Numerical $V=E-Ir$ with $I=E/(R+r)$
A cell of EMF $2.1\,V$ and internal resistance $0.2\,\Omega$ is connected across an external resistance of $10\,\Omega$ . Calculate the current drawn from the cell and its terminal voltage.
Numericalinternal resistance from lost volts $E-V=Ir$
A cell of EMF $2.0\,V$ gives a current of $0.5\,A$ when connected to an external resistance of $3.6\,\Omega$ . Calculate the terminal voltage of the cell and its internal resistance.
Define / stateemf as energy per coulomb on open circuit
Define the EMF of a cell and state the condition under which the terminal voltage of a cell becomes equal to its EMF.
Distinguishemf versus terminal voltage
Distinguish between the EMF and the terminal voltage of a cell, stating one condition under which the two are equal.
Derive / prove $V=E-Ir$ for a single loop
A cell of EMF $E$ and internal resistance $r$ is connected to an external resistance $R$ . Obtain an expression for the terminal voltage of the cell in terms of $E$ , $R$ and $r$ .
Give reasonsinternal $Ir$ drop while discharging
Explain why the terminal voltage of a cell is less than its EMF when the cell is supplying current to an external circuit.

Practise this topic

PreviousElectrical Power, Energy & Heating Effect NextSeries & Parallel Combinations

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.