Question

# A cell of E.M.F. E and internal resistance r supplies currents for the same time t through external resistanceR1 and R2 respectively. If the heat produced in both cases is the same then the internal resistance is

A
1/r = 1/R1 + 1/R2
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B
r = (R1 + R2)/2
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C
r = (R1R2)
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D
r = R1+R2
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Solution

## The correct option is C r = (R1R2)If R1 is thereheat released =l2R1t →(i)i=εR1+r →(2)Put (2) in (1)Heat released =ε2(R1+r)2R1t →(3)If R2 is there heat released =i2R2t→(5)i=εR2+r→(6)=ε2(R2+r)2R2t→(4)Given (4) and (3) are sameε2R1t(R1+r)2=ε2R2t(R2+r)R1(R1+r)2=R2(R2+r)2→R1(R22+r2+2R2r) = R2(R21+r2+2R1r)→R1R22+R1r2+2R1R2r=R2R21+R2r2+2R1R2r→R1R22+R1r2=R2R21+R2R2→R1R22−R2R21=r2(R2−R1)→R1R2(R2−R1)=r2(R2−R1)→r2=R1R2

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