Heat Energy $propto 1/R$ or $propto R$?

Question

I know that heat energy can be calculated by $I^2 R t$. Therefore, it's directly proportional to resistance.
However,
$$I^2  R  t = frac{V^2}{R}t$$
In this case, won't the heat energy be inversely proportional to the resistance? Won't these 2 statements contradict each other?

Tony · Accepted Answer

You are implicitly assuming that $I$ and $V$ are constant quantities.  If one says that $x$ is proportional to $y$, it means $x = ay$ where $a$ is a constant.
But consider what happens when you change $R$: either $V$ or $I$, or both, change.  They are not constant.
I think it would be better to say that power is proportional to $I^2$, with coefficient of proportionality $R$, and power is also proportional to $V^2$, with coefficient of proportionality $1/R$.
Changing $I$ or $V$ does not affect the value of $R$ (not in simple analyses anyway).

Heat Energy $propto 1/R$ or $propto R$?

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