A cell of constant $e.m.f.$ first connected to a resistance ${R_1}$ and then connected to a resistance ${R_2}$. If power delivered in both cases is then the internal resistance of the cell is
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(a) Power dissipated$ = {i^2}R = {\left( {\frac{E}{{R + r}}} \right)^2}R$
${\left( {\frac{E}{{{R_1} + r}}} \right)^2}{R_1} = {\left( {\frac{E}{{{R_2} + r}}} \right)^2}{R_2}$
$==>$ ${R_1}(R_2^2 + {r_2} + 2{R_2}r) = {R_2}(R_1^2 + {r^2} + 2{R_1}r)$
$==>$ $R_2^2{R_1} + {R_1}{r^2} + 2{R_2}r = R_1^2{R_2} + {R_2}{r^2} + 2{R_1}{R_2}r$
$==>$ $({R_1} - {R_2}){r^2} = ({R_1} - {R_2}){r^2} = ({R_1} - {R_2}){R_1}{R_2}$
$==>$ $r = \sqrt {{R_1}{R_2}} $
${\left( {\frac{E}{{{R_1} + r}}} \right)^2}{R_1} = {\left( {\frac{E}{{{R_2} + r}}} \right)^2}{R_2}$
$==>$ ${R_1}(R_2^2 + {r_2} + 2{R_2}r) = {R_2}(R_1^2 + {r^2} + 2{R_1}r)$
$==>$ $R_2^2{R_1} + {R_1}{r^2} + 2{R_2}r = R_1^2{R_2} + {R_2}{r^2} + 2{R_1}{R_2}r$
$==>$ $({R_1} - {R_2}){r^2} = ({R_1} - {R_2}){r^2} = ({R_1} - {R_2}){R_1}{R_2}$
$==>$ $r = \sqrt {{R_1}{R_2}} $
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