Question

Consider N = n1n2 identical cells, each of emf ε and internal resistance r. Suppose n1 cells are joined in series to form a line and n2 such lines are connected in parallel.
The combination drives a current in an external resistance R. (a) Find the current in the external resistance. (b) Assuming that n1 and n2 can be continuously varied, find the relation between n1, n2, R and r for which the current in R is maximum.

Answer 1

Answer:

Given:

Emf of one cell = E

∴ Total e.m.f. of n1 cells in one row = n1E

Total emf of one row will be equal to the net emf across all the n2 rows because of parallel connection.

Total resistance in one row = n1r

Total resistance of n2 rows in parallel =n1r/n2

Net resistance of the circuit = R + n1r/n2

∴Current, I = [n1E/R+(n1r/n2)]

= n1n2E/n2R+n1r

(b) From (a),

I = n1n2E/n2R+n1r

For I to be maximum, (n1r + n2R) should be minimum

⇒(√n1r-√n2R)²+2√n1R n2r = min

It is minimum when

√n1r = √n2R

n1r = 2R

∴ I is maximum when n1r = n2R .

Answer 2

(a) Current (I) = $\frac{n_{1} E}{n_{1} / n_{2} r+R}=\frac{n_{1} n_{2} E}{n_{1} r+n_{2} R}$

(b) Current (i) is maximum when n1r=n2R

Explanation:

Step 1:

Emf of one cell = E

Total e.m.f f one row will be equal to the net emf across all the n2 rows because of parallel connection.

$\text { Total resistance of } \mathrm{n} 2=\mathrm{n} 1 \mathrm{r} / \mathrm{n} 2$

$\text { Total resistance in all rows }=\frac{\mathrm{n}_{1} \mathrm{r}}{\mathrm{n}_{2}}$

$\text { Net resistance }=\frac{n_{1} \mathrm{r}}{\mathrm{n}_{2}}+\mathrm{R}$

$\text { Current }=\frac{n_{1} E}{n_{1} / n_{2} r+R}=\frac{n_{1} n_{2} E}{n_{1} r+n_{2} R}$

Step 2:

$\text { b) } 1=\frac{n_{1} n_{2} E}{n_{1} r+n_{2} R}$

$\text { for } \mathrm{I}=\mathrm{max}$

$\mathrm{n}_{1} \mathrm{r}+\mathrm{n}_{2} \mathrm{R}=\mathrm{min}$

$\Rightarrow(\sqrt{n_{1} r}-\sqrt{n_{2} R})^{2}+2 \sqrt{n_{1} r n_{2} R}=\min$

It is min when,

$\sqrt{n_{1} r}=\sqrt{n_{2} R}$

$\begin{aligned}&\Rightarrow \mathrm{n}_{1} \mathrm{r}=\mathrm{n}_{2} \mathrm{R}\\&\text { I is max when } \mathrm{n}_{1} \mathrm{r}=\mathrm{n}_{2} \mathrm{R} \text { . }\end{aligned}$