Question

Heat generated in the given circuit will be maximum when r is

• A. $2\Omega$
• B. $4\Omega$
• C. $\frac{22}{7}\Omega$
• D. $\frac{30}{7}\Omega$

Answer 1

The correct option is D $\frac{30}{7}\Omega$

For maximum heat, $R_{eq}=2\Omega$ [internal resistance of battery]
$\epsilon =iReq=(i_1+2i_2+i_3)Req-\left(1\right)$
$\epsilon =i_{1}r-\left(2\right)$

By K.V.L
$i_2+i_2+\frac{i_3}{2}=i_1-\left(3\right)$
$\&i_3+\frac{i_3}{2}=i_2-\left(4\right)$

By (3) & (4)
$\frac{3i_3}{2}=i_2\&i_3=\frac{2i_1}{7}$

By (1)
$\epsilon =(i_1+\frac{6}{7}{i}_1+\frac{2}{7}{i}_1)R_{eq}\Rightarrow R_{eq}=\frac{7r}{15}$
$\therefore \frac{7}{15}r=2\Omega \Rightarrow r=\frac{30}{7}\Omega$