Question

Two identical cells of emf 1.5V each joined in parallel, supply energy to an external circuit consisting of two resistances of $7\Omega $each joined in parallel. A very high resistance voltmeter reads the terminal voltage of cells to be 1.4V . Calculate the internal resistance of each cell.

Answer 1

Hint: Assume some value of internal resistance of each cell (let's say r) and then solve for the whole circuit using ohm’s law you will get the expression of current once you know the value of current then put the value of current in the formula of terminal voltage across a cell to find the value of internal resistance.

Complete step by step solution:
Let say each cell has an internal resistance r. Then total resistance of the circuit will be sum of all the resistances (internal resistances of the cells and external resistances) i.e; total resistance $R = r + r + 7 + 7 = 2(r + 7)\Omega $
There are two cells so total emf will be the sum of the emf of each cell i.e; total emf .
${E_{total}} = {E_1} + {E_2} \\
\Rightarrow{E_{total}} = 1.5 + 1.5 \\
\Rightarrow{E_{total}} = 3V$

Using ohm's law ,current in the circuit,
$i = \dfrac{{{E_{total}}}}{R} \\
 \Rightarrow i = \dfrac{3}{{2(r + 7)}}$
Now we will find terminal voltage across each cell. We know that, terminal voltage $V = E - ir$, putting values we have,
$1.4 = 1.5 - ir$
$ \Rightarrow 1.4 = 1.5 - \dfrac{3}{{2(r + 7)}}r$
$ \Rightarrow 1.5 - 1.4 = \dfrac{{3r}}{{2r + 14}}$
$ \therefore r = \dfrac{{0.1 \times 14}}{{3 - 0.2}} = 0.5\Omega $

Hence the correct answer is $0.5\Omega $.

Additional information:
Internal resistance: Internal resistance refers to the opposition to the current flow generated by the cells and batteries themselves, which contributes to heat generation. Sulfation and degradation of the grid are the major contributors to the increase in lead acid internal resistance. The resistance is also influenced by temperature; heat decreases it and cold elevates it. To have additional runtime, heating the battery will temporarily lower the internal resistance.

Terminal voltage: The voltage or potential difference between the terminals of a cell or battery is the terminal voltage. If the cell or battery is not attached to the circuit, the voltage or potential difference of the terminal is equal to the cell or battery's e.m.f.

Note:Here we have to know what internal resistance is along with its formula to properly calculate the value. Also the internal resistance should not be confused with resistance. Internal resistance is the resistance inside a battery, or another source of voltage, that when there is a current, causes a drop in the source voltage. Opposition to the movement of current provided by any load is commonly referred to as outward resistance or simply called resistance.