= Battery internal resistance
{title2=$V_L=\mathcal E-Ir$}
A useful short-interval approximation replaces a <battery> by an ideal <electromotive force> $\mathcal E$ in series with a positive internal <electrical resistance> $r$. This is the <Thévenin terminal response> with <open-circuit voltage> $\mathcal E$ and <Thévenin resistance> $r$. The model describes the terminals without asserting that there is a separate physical <resistor> hidden inside the cell. Both $\mathcal E$ and $r$ can change with composition, <temperature>, <electric current> and history.<University Physics Volume 2 by Samuel J. Ling and others>[§10.1, Internal Resistance and Terminal Voltage, current/temperature/history dependence and equation 10.1]{ref}
For a resistive <electrical load> $R>0$, take $I>0$ to mean discharge: <conventional current> leaves the positive terminal. <Kirchhoff's voltage law> and <Ohm's law> give
$$
\mathcal E-Ir-IR=0,\qquad
I=\frac{\mathcal E}{R+r},\qquad
V_L=IR=\mathcal E-Ir.
$$
For the stipulated $\mathcal E=3\,\mathrm V$, $r=1\,\Omega$, $R=4\,\Omega$,
$$
I=0.6\,\mathrm A,\qquad
Ir=0.6\,\mathrm V,\qquad
V_L=2.4\,\mathrm V.
$$
An <open-circuit voltage> of $3\,\mathrm V$ therefore does not promise $3\,\mathrm V$ at this <electrical load>. Increasing the load <electrical resistance> reduces <electric current> and brings $V_L$ closer to $\mathcal E$, as in <Thévenin load response>.
The signed relation also explains charging within this model. If an external source drives <electric current> into the positive terminal, $I<0$ and $V_L=\mathcal E+|I|r$. The terminal <electric potential difference> then exceeds $\mathcal E$. This algebra is not evidence that every <battery> is rechargeable.<University Physics Volume 2 by Samuel J. Ling and others>[§10.1, paragraph accompanying Figure 10.10: reversed current and charging terminal voltage]{ref}
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