Problem 54 - A Complete Resource Book in Physics for JEE Main

Question

\(n\) identical cells, each of emf \(\varepsilon\) and internal resistance \(r\), are joined in series to form a closed circuit. The potential difference across any one cell is (A) Zero (B) \(\varepsilon\) (C) \(\frac{\varepsilon}{n}\) (D) \(\frac{n-1}{n} \varepsilon\)

Step-by-Step Solution

Answer

The potential difference across any one cell in the given closed circuit is: (A) Zero

Step 1: Calculate total EMF and resistance

As the \(n\) cells are connected in series, the total emf \(\varepsilon_{total}\) and internal resistance \(r_{total}\) can be found as follows: \[\varepsilon_{total}= n\varepsilon\] \[r_{total} = nr\]

Step 2: Calculate the current in the circuit

Ohm's law states that current (I) is the potential difference (or voltage) divided by the resistance. In this case, the potential difference is the total emf, and the resistance is the total resistance: \[I = \frac{\varepsilon_{total}}{r_{total}} = \frac{n\varepsilon}{nr}\] \[I = \frac{\varepsilon}{r}\]

Step 3: Calculate the potential difference across one cell

The potential difference across a single cell will be the sum of the cell's EMF (\(\varepsilon\)) and the voltage drop across the cell's internal resistance. To find the voltage drop across a cell's internal resistance, we use Ohm's law again, multiplying the resistance of one cell, \(r\), by the current in the circuit, \(I\). \[V_{drop} = Ir = \frac{\varepsilon}{r}r = \varepsilon\] The potential difference across one cell is the sum of its emf \(\varepsilon\) and the voltage drop across its internal resistance \(V_{drop}\): \[V_{cell} = \varepsilon + V_{drop}\] Since the voltage drop across a cell's internal resistance is equal to the cell's emf, the potential difference across one cell is given by: \[V_{cell} = \varepsilon + \varepsilon = 2\varepsilon\] However, this result appears to contradict the given options. This contradiction arises because the question implies a closed circuit formed only by the cells' internal resistances, without any external load or resistance. In this case, it's essential to realize that the actual potential difference across the entire circuit is zero due to the lack of an external load.

Step 4: Potential difference across one cell considering a closed circuit

Since the total potential difference across the circuit is zero, the potential difference across any one cell can be represented as: \[V_{cell} = \frac{V_{total}}{n} = \frac{0}{n}\] \[V_{cell} = 0\] Thus, the potential difference across any one cell is: (A) Zero