Two separate circuits are made with resistances `r_(1)andr_(2)` connected to the same storge battery . What should be the interal resistance (r) of the storage battery for which an equal amount of heat is produced in the exteranal circuits ?

Explanation

For the first cooection , currrents `I_(1)=(E)/(r+r_(1))`
[ E= emf of the battery ]
So , the heat produced in resistance `r_(1)` in time t ., `H_(2)=(E^(2)r_(1)t)/((r+r_(2))^(2)J)`
According to the question , `H_(1)= H _(2)`
`therefore(E^(2)r_(1)t)/((r+r_(1))^(2)J)=(E^(2)r_(2)t)/((r+r_(2))^(2)J)"or",r_(1)(r+r_(2))^(2)=r_(2)(r+r_(1))^(2)`
`"or, "r_(1)(r^(2)+2rr_(2)+r_(2)^(2))=r_(2)(r^(2)+2rr_(1)+r_(1))^(2)`
`"or, "r_(1)r^(2)+r_(1)r_(2)^(2)=r_(2)r^(2)+r_(2)r_(1)^(2)`
` "or, "r^(2)(r_(1)-r_(2))=r_(1)r_(2)(r_(1)-r_(2))`
`"or, "r^(2)=r_(1)r_(2)" or, "rsqrt(r_(1)r_(2))`