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 $I^2$ 
 A: =$^{2}$ and also $=$. So my question is- Should we say that  is proportional to  or 2
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.
A: 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!      
A: 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. 
