Question
Explain combination of cells in parallel.
✓
Answer
$i.$ Consider two cells which are connected in parallel. Here, positive terminals of all the cells are connected together and the negative terminals of all the cells are connected together.
$ii.$ In parallel connection, the current is divided among the branches i.e. $I_1$ and $I_2$ as shown in figure.
$iii$. Consider points $A$ and $B$ having potentials $V_A$ and $V_B$, respectively.
$iv.$ For the first cell the potential difference across its terminals is, $V=V_A-V_B=E_1-I_1 r_1$
$\therefore I _1=\frac{E_1 V}{r_1}.........(1)$
$v.$ Point $A$ and $B$ are connected exactly similarly to the second cell.
Hence, considering the second cell,
$V = V _{ A }- V _{ B }= E _2- I _2 r _2$
$\therefore I _2=\frac{E_2 V}{r_2} \ldots \ldots \ldots \ldots(2)$
$vi.$ Since, $I=I_1+I_2$
Combining equations $(1), (2)$ and $(3),$
$\therefore I=\frac{E_1}{r_1}-\frac{V}{r_1}+\frac{E_2}{r_2}-\frac{V}{r_2}=\left(\frac{E_1}{r_1}+\frac{E_2}{r_2}\right)-V\left(\frac{1}{r_1}+\frac{1}{r_2}\right)$
$vii.$ Thus, $V\left(\frac{1}{r_1}+\frac{1}{r_2}\right)=\left(\frac{E_1}{r_1}+\frac{E_2}{r_2}\right)-I$
$\therefore V\left(\frac{r_1+r_2}{r_1 r_2}\right)=\frac{E_1 r_2+E_2 r_1}{r_1 r_2}-I$
$ V=\frac{E_1 r_2+E_2 r_1}{r_1+r_2}-I \frac{r_1 r_2}{r_1+r_2}$
$viii.$ If we replace the cells by a single cell connected between points $A$ and $B$ with the emf $E_{e q}$ and the internal resistance $r_{e q}$ then,
$V = E _{ eq }- Ir _{ eq }$
From equations $(4)$ and $(5),$
$E_{e q}=\frac{E_1 r_2+E_2 r_1}{r_1+r_2}$
$r_{e q}=\frac{r_1 r_2}{r_1+r_2}$
$\text { i.e. } \frac{1}{r_{ rq }}=\frac{1}{r_1}+\frac{1}{r_2}$
$\frac{E_{ eq }}{r_{ rq }}=\frac{E_1}{r_1}+\frac{E_2}{r_2}$
$ix.$ For $n$ number of cells connected in parallel with emf $E_1, E_2, E_3, \ldots \ldots \ldots \ldots ., E_n$ and internal
resistance $r _1, r _2, r _3, \ldots \ldots \ldots ., r _{ n }$
$\frac{1}{ r _{ rq }}=\frac{1}{ r _1}+\frac{1}{ r _2}+\frac{1}{ r _3}+\ldots \ldots \ldots+\frac{1}{ r _{ n }}$
and $\frac{ E _{ eq }}{ r _{ rq }}=\frac{ E _1}{ r _1}+\frac{ E _2}{ r _2}+\ldots \ldots \ldots+\frac{ E _{ n }}{ r _{ n }}$
$ii.$ In parallel connection, the current is divided among the branches i.e. $I_1$ and $I_2$ as shown in figure.
$iii$. Consider points $A$ and $B$ having potentials $V_A$ and $V_B$, respectively.
$iv.$ For the first cell the potential difference across its terminals is, $V=V_A-V_B=E_1-I_1 r_1$
$\therefore I _1=\frac{E_1 V}{r_1}.........(1)$
$v.$ Point $A$ and $B$ are connected exactly similarly to the second cell.
Hence, considering the second cell,
$V = V _{ A }- V _{ B }= E _2- I _2 r _2$
$\therefore I _2=\frac{E_2 V}{r_2} \ldots \ldots \ldots \ldots(2)$
$vi.$ Since, $I=I_1+I_2$
Combining equations $(1), (2)$ and $(3),$
$\therefore I=\frac{E_1}{r_1}-\frac{V}{r_1}+\frac{E_2}{r_2}-\frac{V}{r_2}=\left(\frac{E_1}{r_1}+\frac{E_2}{r_2}\right)-V\left(\frac{1}{r_1}+\frac{1}{r_2}\right)$
$vii.$ Thus, $V\left(\frac{1}{r_1}+\frac{1}{r_2}\right)=\left(\frac{E_1}{r_1}+\frac{E_2}{r_2}\right)-I$
$\therefore V\left(\frac{r_1+r_2}{r_1 r_2}\right)=\frac{E_1 r_2+E_2 r_1}{r_1 r_2}-I$
$ V=\frac{E_1 r_2+E_2 r_1}{r_1+r_2}-I \frac{r_1 r_2}{r_1+r_2}$
$viii.$ If we replace the cells by a single cell connected between points $A$ and $B$ with the emf $E_{e q}$ and the internal resistance $r_{e q}$ then,
$V = E _{ eq }- Ir _{ eq }$
From equations $(4)$ and $(5),$
$E_{e q}=\frac{E_1 r_2+E_2 r_1}{r_1+r_2}$
$r_{e q}=\frac{r_1 r_2}{r_1+r_2}$
$\text { i.e. } \frac{1}{r_{ rq }}=\frac{1}{r_1}+\frac{1}{r_2}$
$\frac{E_{ eq }}{r_{ rq }}=\frac{E_1}{r_1}+\frac{E_2}{r_2}$
$ix.$ For $n$ number of cells connected in parallel with emf $E_1, E_2, E_3, \ldots \ldots \ldots \ldots ., E_n$ and internal
resistance $r _1, r _2, r _3, \ldots \ldots \ldots ., r _{ n }$
$\frac{1}{ r _{ rq }}=\frac{1}{ r _1}+\frac{1}{ r _2}+\frac{1}{ r _3}+\ldots \ldots \ldots+\frac{1}{ r _{ n }}$
and $\frac{ E _{ eq }}{ r _{ rq }}=\frac{ E _1}{ r _1}+\frac{ E _2}{ r _2}+\ldots \ldots \ldots+\frac{ E _{ n }}{ r _{ n }}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
State and prove parallelogram law of vector addition and determine magnitude and direction of resultant vector.
Explain the use of dimensional analysis to check the correctness of a physical equation.
What is the effect of humidity of air on velocity of sound?
State advantages and disadvantages of amplitude modulation.
As i was standing on a weighing machine inside a lift it recorded $50 kg$ wt. Suddenly for few seconds it recorded $45 kg$ wt. What must have happened during that time? Explain with complete numerical analysis. [Ans: Lift must be coming down with acceleration $\frac{ g }{10}=$ $\left.1 ms ^{-2}\right]$
→‘The distances travelled by an object starting from rest and having a positive uniform acceleration in successive seconds are in the ratio $1:3:5:7….$’ Prove it.
→Explain characteristics and structure of silicon using a neat labelled diagram.
Define absolute error, mean absolute error, relative error and percentage error.
State some important features of the depletion region.
Distinguish between uniform rectilinear motion and non-uniform rectilinear motion.