question_answer

A resistance of \[R\,\Omega \] draws current from a potentiometer. The potentiometer has a total resistance of \[{{R}_{0}}\,\Omega .\] A voltage V is supplied to the potentiometer. Derive an expression for the voltage fed into the circuit when the slide is in the middle of the potentiometer.

Answer:

The equivalent resistance of the circuit             \[{{R}_{aq}}=\frac{{{R}_{0}}}{2}+\frac{R\cdot ({{R}_{0}}/2)}{R+({{R}_{0}}/2)}\] \[\therefore \]      Current in the circuit, \[I=\frac{N}{{{R}_{aq}}}\] \[V=I{{R}_{aq}}=I\left[ \frac{{{R}_{0}}}{2}+\frac{R\cdot ({{R}_{0}}/2)}{R+({{R}_{0}}/2)} \right]\]             \[=I\left( \frac{{{R}_{0}}}{2}+\frac{R{{R}_{0}}}{2R+{{R}_{0}}} \right)\] \[\therefore \]      \[V=\frac{I{{R}_{0}}}{2}\left( 1+\frac{2R}{2R+{{R}_{0}}} \right)\]