Question
Two identical batteries each of emf $$2\,V$$ and internal resistance $$1\,\Omega $$ are available to produce heat in an external resistance by passing a current through it. The maximum power that can be developed across $$R$$ using these batteries is
A.
$$3.2\,W$$
B.
$$2\,W$$
C.
$$1.28\,W$$
D.
$$\frac{8}{9}W$$
Answer :
$$2\,W$$
Solution :
To receive maximum current, the two batteries should be connected in series.
Given, $$R = 1\,\Omega + 1\,\Omega = 2\,\Omega .$$
Hence, power developed across the resistance $$R$$ with the batteries in series is
$$\eqalign{
& P = {i^2}R = {\left( {\frac{{2E}}{{R + 2r}}} \right)^2}R\,\,\,\left[ {I = \frac{E}{{{R_{{\text{eq}}}}}}} \right] \cr
& = \left( {\frac{{2 \times 2}}{{2 + 2}}} \right) \times 2 = 2\,W\,\,\left[ {\because r = 1\,\Omega } \right] \cr} $$
NOTE
In case of batteries connected in series, equivalent emf is given by $${E_{{\text{eq}}}} = {E_1} + {E_2}$$