Maximum Power Transfer

The power delivered to an external load by a source with internal resistance $r$ is maximised when the load resistance equals the internal resistance, a result that follows from analysing $P=I^{2}R={\left(\frac{\epsilon }{R+r}\right)}^{2}R$ and building on the source concepts in NCERT Class 12, Chapter 3 (Current Electricity). Differentiating this expression with respect to $R$ and setting it to zero yields the maximum power transfer condition $R=r$. Since $r=3$ ohm, the optimal load is $R=3$ ohm. The option 6 ohm is wrong because although larger loads give higher terminal voltage, the current falls, reducing delivered power beyond the matched point. The option 0 ohm gives maximum current but zero power in the load, as all power dissipates internally. The option 12 ohm, matching the emf number, has no physical basis for power matching. A plausibility check confirms the answer: at $R=r=3$ ohm the current is $\frac{12}{6}=2$ A and the load power is $2^2\times 3=12$ W, and testing nearby values like 2 ohm or 4 ohm gives slightly less than 12 W, confirming the maximum.