= Resistor power under fixed drive
{title2=$P_I=I_0^2R,\quad P_V=V_0^2/R$}
Whether increasing <electrical resistance> increases <Joule heating> depends on what the supply holds fixed. Consider separate ideal <resistor> experiments with $R=5\,\Omega$ and $R=10\,\Omega$, each maintained at its specified <temperature>. At fixed <electric current> $I_0=1\,\mathrm A$, doubling $R$ changes the required <electric potential difference> from $5$ to $10\,\mathrm V$ and doubles <power> from $5$ to $10\,\mathrm W$. At fixed <electric potential difference> $V_0=5\,\mathrm V$, doubling $R$ halves <electric current> from $1$ to $0.5\,\mathrm A$ and halves <power> from $5$ to $2.5\,\mathrm W$. These are stipulated model values, not measured data.
\Image[/joule-heating-fixed-current-fixed-voltage.png]
{title=Increasing <electrical resistance> has opposite effects on <Joule heating> under fixed <electric current> and fixed <electric potential difference>}
{id=joule-heating-fixed-current-fixed-voltage}
{description=Original calculated comparisons, with $I_0=1\,\mathrm A$ on the left and $V_0=5\,\mathrm V$ on the right. The marked $5$ and $10\,\Omega$ cases are separate ideal experiments. Each lower panel shows the companion quantity that changes when the supply holds the other quantity fixed.}
The comparison also predicts feedback when a particular <resistor>'s <electrical resistance> rises with <temperature>: a supply maintaining <electric current> increases its heating <power>, whereas one maintaining terminal <electric potential difference> reduces it. Predicting the actual <temperature> still needs the material response $R(T)$ and its <heat transfer> to the surroundings; a nominal supply setting does not establish which drive is actually constant at the <resistor>.
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