Three equal resistance, each of R ohm, are connected as shown. A cell of 10 volts and internal resistance 0.5 Ω is connected across the circuit. Calculate the value of R for which heat generated in the circuit is maximum.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1.5 Ω
The power delivered by a cell to an external resistance \( R_{\mathrm{ext}} \) is maximum when \( R_{\mathrm{ext}} \) equals the internal resistance r of the cell (maximum power transfer).
\( \displaystyle P = I^{2}R_{\mathrm{ext}} = \frac{E^{2}R_{\mathrm{ext}}}{(R_{\mathrm{ext}} + r)^{2}} \)
Setting \( dP/dR_{\mathrm{ext}} = 0 \) gives \( (R_{\mathrm{ext}} + r) - 2R_{\mathrm{ext}} = 0 \), that is \( R_{\mathrm{ext}} = r \).
From the figure, the three resistors of R each are connected in parallel across the cell, so
\( \displaystyle R_{\mathrm{ext}} = \frac{R}{3} \)
\( \displaystyle \frac{R}{3} = r = 0.5\,\Omega \Rightarrow R = 1.5\,\Omega \)