A resistance of R Ω draws current from a potentiometer as shown in the figure. The potentiometer has a total resistance R_{o} Ω. A voltage V is supplied to the potentiometer. Derive an expression for the voltage across R when the sliding contact is in the middle of the potentiometer.

#### Solution

From the figure given in the question, current *I* through *R _{o} *is

`I=V/R_o`

Now, the sliding contact is in the middle of the wire. From the relation `R=rhol/A`

R ∝ l (where* ρ* and *A* are constants)

where *l* is the length of the wire, *ρ* is the resistivity and *A* is the area of the wire.

∴ Resistance of half of the wire of the potentiometer is

`R'=R_o/2`

Now *R' *and *R* are in parallel; therefore, voltage across *R *and *R' *is equal. Therefore, voltage across R is

`:.V'=I.R'=V/R_o"."R_o/2`

`=>V'=V/2`