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  4. In the circuit shown, the resistance r is a variable resistance. If for r = f R, the heat generation in r is maximum then the value of f is :

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Q. In the circuit shown, the resistance $r$ is a variable resistance. If for $r = f R$, the heat generation in $r$ is maximum then the value of $f$ is :Physics Question Image

JEE MainJEE Main 2016Current Electricity

Solution:

Heat generation in $r$ will be, $H = I _{ r }^{2} r$,
$ H=\left(\frac{ V }{ R +\frac{ Rr }{ R + r }} \times \frac{ r }{ R + r }\right)^{2} r =\left(\frac{ V }{ R ( R +2 r )}\right)^{2} r $
Putting $\frac{ dH }{ dr }=0$ gives, $r =\frac{ R }{2}$. Hence, $f =\frac{1}{2}$.