# Thermal Energy in a Conductor

We know that thermal energy developed in a current-carrying resistor is given by

$$U=I^2Rt$$ and also $$U=VIt$$. So my question is- Should we say that $$U$$ is proportional to $$I$$ or $$I^2$$

• Depends which other terms like V, and R are same in the comparison. For example, if you are talking about two scenarios where V is same, then you say it's proportional to I, and if R is same in the two cases, then say that it's proportional to $I^2$. – Eagle May 6 at 17:01

When looking for a proportionality – or indeed any other relationship – you must decide what is to be kept constant.

In this case, if we keep R and t constant, we can say from $$U=I^{2}Rt$$ that U is proportional to $$I^2.$$

If we keep V and t constant, the equation $$U=VIt$$ suggests that U is proportional to I. This is true, but how can I change at all, if $$V$$ is kept constant? Only by our changing the resistance, R, since $$I=V/R.$$ So U is proportional to I if V and t are constant and R is varied.

But you were probably regarding R as a constant, in which case the second paragraph is the interpretation that makes sense.

The moral: be clear as to what is to be kept constant before deciding on how two variables are related!

𝑈=$$𝐼^{2}𝑅𝑡$$ and also $$𝑈=𝑉𝐼𝑡$$. So my question is- Should we say that 𝑈 is proportional to 𝐼 or 𝐼2

It is proportional to $$I$$ if the proportionality constant is $$V$$ and proportional to $$I^{2}$$ if the proportionality constant is $$R$$. But since $$V=IR$$ and $$R=\frac{V}{I}$$ per ohms law, they are basically the same thing.

$$P=VI=(IR)I=I^{2}R$$

or

$$P=I^{2}R=I^{2}\frac{V}{I}=VI$$

Hope this helps.

The problem is the that the other terms in the expression are not constants. So, let's say the voltage is constant. So on varying I, you find by second expression (=VI), that U is proportional to I. Note that in this case, R has to vary such that IR is constant. Similarly, if let's say R is constant, then, V is proportional to square of I. Note that in this case, V has to vary proportional to I as V=IR should always hold. In fact, these are the situations where we need to use the two forms of expressions interchangeably as per the required conditions.