For the circuit shown here, would tha balancing length increase, decrease or remain the same, if
(i) `R_(1)` is decreased (ii) `R_(2)` is increased
without any other change, (in each case) in the rest of the circuit? Justify your answers in each case.

Explanation

The resistance of potentiometer wire be `R_(AB)`,
`rArr R_(AB) = rho xx (AB)/(Lamda_(0))`
Current in the circuit in the balanced condition is
`I= (E )/(R_(AB) + R)`
Let `R_(AJ)` be the resistance of the length AJ.
`:. (R_(AJ))/(R_(AB)) = (AJ)/(AB) rArr R_(AJ) = R_(AB) xx (AJ)/(AB)`
Voltage across length AJ
`V_(AJ) = epsi - i_(g) r, :. i_(g) = 0`
`epsi = ((E (AJ))/(R_(AB) + R)) xx ((R_(AB))/(AB))`
Balancing length: `AJ = (epsi)/(E ) ((R_(AB) + R) (AB))/(R_(AB))`
`AJ = (epsi)/(E ) [AB + (A_(0)R)/(rho)]`
(i) Balancing length is directly proportional to resistance `R_(1)`
`AJ prop R_(1)`
(ii) It is independent of resistance `R_(2)` on balancing length.