Question

A constant voltage `V_0` is applied to a potentiometer of resistance `R` connected to an ammeter. A constant resistor `r` is connected to the sliding contact of the potentiometer and the fixed end of the potentiometer. How will the reading of ammeter vary as the sliding contact is moved from one end of the potentiometer to the other ? The resistance of ammeter is assumed to be negligible.

Answer 1

Let `x` : resistance of potentiometer between point `a` and sliding contact
Equivalent resistance between `a` and sliding contact `= (rx)/( r+ x)`
Total resistance of circuit `= (rx)/(r + x) + (R - x)`

`V_1 = ((rx)/(r + x) V_0)/((rx)/(r + x) + (R - x)) = (rx V_0)/(rR + Rx - x^2)`
Current in ammeter
`i_1 = (V_1)/(x) = (V_0 r)/(Rx - x^2 + Rr)`
For `i_1` to be (maximum, minimum or constant)
`(di_1)/(dx) = 0 rArr R - 2x = 0 rArr x = R//2`
At `x = R//2, i_1` is minimum.