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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 $I^2$

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    $\begingroup$ 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 $I^2$. $\endgroup$ – Eagle May 6 at 17:01
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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!

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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.

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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.

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