Name: Applications of Potentiometer
Uploaded: 2019-07-19
Duration: 4 min 30 s
Description: Since potentiometers are very versatile and handy to use, it is vital to understand their working principle and applications fully. In this video, learn abou...

Question

Two cells of emf

ε_{1}

and

ε_{2}

have internal resistance

r_{1}

and

r_{2}

. Deduce an expression for equivalent emf of their parallel combination.

Accepted Answer

Let $ε_{1}$ and $ε_{2}$ be the emfs of the two cells connected in parallel.

Let $r_{2}$ and $r_{2}$ be their internal resistance respectively.

$I_{1}$ and $I_{2}$ are the currents leaving the positive electrodes of the cells.

At $B_{2}$ , currents $I_{1}$ and $I_{2}$ flow in whereas the current $I$ flows out.

Therefore, $I = I_{1} + I_{2} $

Let $V_{1}$ and $V_{2}$ be the potentials at $B_{1}$ and $B_{2}$ respectively.

Potential difference across the first cell is $V_{1} - V_{2} = V = ε_{1} - I_{1} r_{1} $

Potential difference across the second cell is also $V_{1} - V_{2} = V = ε_{2} - I_{2} r_{2} $

Hence, $I_{1} = \frac{ε_{1} - V}{r_{1}} and I_{2} = \frac{ε_{2} - V}{r_{2}}$

$I = I_{1} + I_{2} = \frac{ε_{1} - V}{r_{1}} + \frac{ε_{2} - V}{r_{2}}$
$I = ( \frac{ε_{1}}{r_{1}} + \frac{ ε_{2}}{r_{2}} ) - V (\frac{1}{r_{1}} + \frac{1}{r_{2}})$

Rearranging

$V (\frac{r_{1} + r_{2}}{r_{1} r_{2}}) = \frac{ε_{1} r_{2} + ε_{2} r_{1}}{r_{1} r_{2}} - I$

$V (r_{1} + r_{2}) = ε_{1} r_{2} + ε_{2} r_{1} - I r_{1} r_{2}$

$V = \frac{ε_{1} r_{2} + ε_{2} r_{1}}{(r_{1} + r_{2})} - \frac{I r_{1} r_{2}}{(r_{1} + r_{2})}$

If the parallel combination of cell is replaced by a single cell between $B_{1}$ and $B_{2}$ of emf $ε_{P}$ and internal resistance $r_{p}$ then, $V = ε_{P} - I r_{p}$

Comparing the above equations

$ε_{P} = \frac{ε_{1} r_{2} + ε_{2} r_{1}}{(r_{1} + r_{2})}$

Two cells of emf $ε_{1}$ and $ε_{2}$ have internal resistance $r_{1}$ and $r_{2}$ . Deduce an expression for equivalent emf of their parallel combination.

Important Questions on Direct Current Circuits

Learn and Prepare for any exam you want !

Two cells of emf ε1 and ε2 have internal resistance r1 and r2. Deduce an expression for equivalent emf of their parallel combination.

Important Questions on Direct Current Circuits

Learn and Prepare for any exam you want !

Two cells of emf $ε_{1}$ and $ε_{2}$ have internal resistance $r_{1}$ and $r_{2}$ . Deduce an expression for equivalent emf of their parallel combination.