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A cell of e.m.f E is connected to a resistance R_1 for time i and the amount of heat generated in it is H. If the resistance R_1 is replaced by another resistance R_2 and is connected to the cell for the same time i, the amount of heat generated in R_2 is 4H. Then the internal resistance of the cell is |
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Answer» `(2R_1 +R_2)/(2)` ` H=(E^2 R_1)/( (R_1 +r ^2 ) t`   `orE^2=(H (R_1+r)^2)/(R-1 t)` forsecondcaethe amountof heatgeneratedin `R_2`in thesametime t is ` 4 H = (E^2R_2 t)/( (R_2 +r)^2 )orE^2 =(4 H( R_2 +r)^2)/( R_2 t)` EQUATING(i) and (ii ) we GET ` (H (R_1+r)^2)/( r_1 t) =(4 H( R_2+r)^2)/(R_2 t)` `((R_1 +r)^2)/( R_1) =(4 (R_2 +r)^2 )/( R_2)or((R_1 +r))/(sqrt(R_1)) = (2 (R_2 +r))/(sqrt(R_2))` `impliessqrt(R_2) (R_1 +r) = 2 sqrt(R_1) (R_2 +r)` ` impliessqrt(R_2) R_1 + sqrt(R_2 ) r = 2 sqrt(R_1) R_2 + 2 sqrt( R_1)r` `IMPLIES sqrt(R_2 ) r-2sqrt(R_1 ) r=2 sqrt(R_1)R_2- sqrt(R_2)R_1` `impliesr ( sqrt(R_2) - 2 sqrt(R_1))= sqrt(R_1 R_2 )[ 2 sqrt(R_2 )- sqrt(R_1)]` `impliesr= sqrt(R_1 R_2) ( 2 sqrt(R_2)- sqrt( R_1))/( sqrt(R_2)-2 sqrt(R_1))` |
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