Two conducting rods $A$ and $B$ of same length and cross-sectional area are connected $(i)$ In series $(ii)$ In parallel as shown. In both combination a temperature difference of $100^o C$ is maintained. If thermal conductivity of $A$ is $3K$ and that of $B$ is $K$ then the ratio of heat current flowing in parallel combination to that flowing in series combination is
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(a) Heat current $H = \frac{{\Delta \theta }}{R}$ ==> $\frac{{{H_P}}}{{{H_S}}} = \frac{{{R_S}}}{{{R_P}}}$
In first case : ${R_S} = {R_1} + {R_2} = \frac{l}{{(3K)A}} + \frac{l}{{KA}} = \frac{4}{3}\frac{l}{{KA}}$
In second case :${R_P} = \frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}} = \frac{{\frac{l}{{(3K)A}} \times \frac{l}{{KA}}}}{{\left( {\frac{l}{{(3K)A}} + \frac{l}{{KA}}} \right)}} = \frac{l}{{4KA}}$
$\therefore \frac{{{H_P}}}{{{H_S}}} = \frac{{\frac{{4l}}{{3KA}}}}{{\frac{l}{{4KA}}}} = \frac{{16}}{3}$
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