In the circuit shown in the figure , , and have constant values but R can be varied. We have to find an expression for R that results in the maximum heating in that resistor.

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We will find the current flowing through R. We have a multi-loop circuit with one junction. Equations that follow from Kirchoff’s laws are

Also,

Rate of Joule heating in the resistance R is

We find the extremum solution of by requiring

This condition gives the equation

As

we have