Question

The minimum number of cells in mixed grouping required to produce a maximum current of \[1.5{\text{ A}}\]through an external resistance of $30{\text{ }}\Omega $, given the emf of each cell is $1.5{\text{ V}}$and internal resistance is $1\Omega $is
A. $30$
B. $120$
C. $40$
D. $60$

Answer 1

Hint: To answer this question students should know what is ohm’s law and equation of ohm’s law. Ohm’s law gives the relation between current passing and voltage applied across any circuit. Then what is resistance? How is it related to ohm’s law?

Complete step by step answer:
Ohm’s Law states that the current flowing through a conductor is directly proportional to the potential difference applied across its ends, provided the temperature and other physical conditions remain unchanged. As per ohm’s law Current is directly proportional to voltage difference through a resistor. That is, if the current doubles, then so does the voltage. To make a current flow through a resistance there must be a voltage across that resistance. Ohm's Law shows the relationship between the voltage (V), current (I) and resistance (R).
$I\alpha V$ or $I\alpha V$
$ \Rightarrow V = IR$

Where $R$ is a constant called resistance of the conductor. The value of this constant depends on the nature, length, area of cross section and temperature of the conductor.
$V = IR$
$ \Rightarrow 1.5 \times 30 = 45{\text{ v}}$
Now, the effective resistance in a mixed group of cells for the same resistance is given as,
${R_{eff}} = \dfrac{{nr}}{m}$ where ‘n’ is series and ‘m’ is parallel path
Substituting the given values,
$30 = \dfrac{{n \times 1}}{m}$
$n = 30m - - - - (1)$

Now, the maximum current is given as ${I_{\max }} = 1.5{\text{ A}}$
Substituting in the equation of the current,
${I_{\max }} = \dfrac{{nE}}{{R + \dfrac{{nr}}{m}}} = 1.5$
$\dfrac{{n \times 1.5}}{{30 + \dfrac{n}{m}}} = 1.5$
Further simplifying we get,
$\dfrac{n}{{\dfrac{n}{m} + 30}} = 1 - - - - (a)$
From equation (1) we have $n = 30m$ . Substitute it in equation (a) and find the values of m and n respectively. Therefore
$m = 2{\text{ and n = 60}}$
Total number of cells $ = mn$
$ \therefore 60 \times 2 = 120$

Therefore, the correct answer is option B.

Additional information: Ohm's law helps us in determining the values of resistance, the current flowing through a circuit, and the voltage applied. Hence with the help of these values, we can find the values of other factors like drift speed, resistivity and many more. It also allows us to calculate the value of power consumption.

Note: For ohm’s law it is a necessary condition that the temperature must be constant. It’s true only for linear electrical elements. For non-linear electrical elements with parameters like capacitance, resistance etc, the voltage and current won’t be constant with respect to time making it difficult to use Ohm’s law. Ohm’s law is not applicable for unilateral electrical elements like diodes and transistors as they allow the current to flow through in one direction only.