= Battery load power and efficiency
{title2=$P_L=\mathcal E^2R/(R+r)^2$}
In the constant-parameter <battery internal resistance> model, the source's available electrical <power> splits between the <electrical load> and internal dissipation:
$$
\mathcal E I=\underbrace{V_LI}_{P_L}+\underbrace{I^2r}_{P_{\rm int}},\qquad
P_L=\frac{\mathcal E^2R}{(R+r)^2},\qquad
P_{\rm int}=\frac{\mathcal E^2r}{(R+r)^2}.
$$
The stipulated $3\,\mathrm V$, $1\,\Omega$, $4\,\Omega$ calculation gives $P_L=1.44\,\mathrm W$, $P_{\rm int}=0.36\,\mathrm W$, and $\mathcal E I=1.80\,\mathrm W$. A resistive <electrical load> converts its share to <Joule heating>; another <electrical load> might convert some of it to mechanical <work> or <light>. The source term is available electrical <work> per time, not a complete cell <heat> budget: reaction <entropy> can also matter.
\Image[/battery-load-voltage-power-efficiency.png]
{title=Calculated battery terminal voltage, power and delivery efficiency versus load resistance}
{description=Original predictions for constant $\mathcal E=3\,\mathrm V$ and $r=1\,\Omega$. Circles mark $R=4\,\Omega$; dashed vertical lines mark $R=r$. The three panels have distinct vertical units. Delivery efficiency is $\eta=P_L/(\mathcal E I)$ for finite discharge <electric current>; its open-circuit limit is one although both input and output <power> vanish. The $R=0$ endpoint is a mathematical limiting model, not an experimental instruction. No measured data or copied source figure is used.}
Define the delivery efficiency relative to the source's available electrical <work>, rather than to all chemical <energy> or to a complete charging cycle:
$$
\eta=\frac{P_L}{\mathcal E I}=\frac{V_L}{\mathcal E}=\frac{R}{R+r}.
$$
Here $\eta=0.80$. The largest load <power> occurs at $R=r$: since $(R+r)^2\geq4Rr$,
$$
P_L\leq\frac{\mathcal E^2}{4r},\qquad
P_L\big|_{R=r}=2.25\,\mathrm W,\qquad
\eta\big|_{R=r}=\frac12.
$$
Thus maximum load <power> and maximum delivery efficiency ask different questions. As $R\to\infty$, $V_L\to\mathcal E$ and $\eta\to1$, but $I\to0$ and $P_L\to0$: a high nearly unloaded voltage supplies almost no <power>. As $R\to0$, the model gives $V_L\to0$, $P_L\to0$ and internal <power> tending to $\mathcal E^2/r$. Real high-current operation can invalidate constant $r$ and $\mathcal E$; the mathematical maximum does not establish a permissible operating point for an actual <battery>.
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