A dc source with internal resistance `R_(0)` is loaded with three identical resistance R as shown in the figure. At what value of R will the thermal power generated in this circuit be the highest?

Explanation

The equivalent circuit may be drawn as in the figure.
Resistance of the network `= R_(0) + (R//3)`
Let, assume that `e.m.f` of the cell in `xi`, then current
`i = (xi)/(R_(0) + (R//3))`
Now, thermal power, generated in the circuit
`P = i^(2) R//3 = (xi^(2))/((R_(0) + (R//3))^(2)) (R//3)`
For `P` to be maximum, `(dP)/(dR) = 0`, which yields
`R = 3 R_(0)`