Question

10 identical cells each of potential 'E' and internal resistance 'r' are connected in series , so formed a closed circuit. An ideal voltmeter is connected across three cells will read

• A. 13 E
• B. 10 E
• C. 7 E
• D. 3 E

Answer 1

The Correct Option is D

We are given 10 identical cells, each with electromotive force (EMF) 'E' and internal resistance 'r'.

These cells are connected in series to form a closed circuit. This means the positive terminal of the last cell is connected to the negative terminal of the first cell, forming a loop.

1. Calculate the total EMF and total internal resistance for the closed circuit: 

Since the 10 cells are in series:

• Total EMF: \( E_{total} = 10 \times E = 10E \)
• Total internal resistance: \( r_{total} = 10 \times r = 10r \)

2. Calculate the current flowing in the closed circuit:

Using Ohm's law for the complete circuit (where the external resistance is zero):

\[ I = \frac{\text{Total EMF}}{\text{Total Resistance}} = \frac{E_{total}}{r_{total}} \]

\[ I = \frac{10E}{10r} = \frac{E}{r} \]

This is the current flowing through each cell in the closed loop.

3. Determine the voltmeter reading across three cells:

An ideal voltmeter measures the potential difference (terminal voltage) between the two points it is connected across. It is connected across three cells in series.

The combined EMF of these three cells is \( 3E \).

The combined internal resistance of these three cells is \( 3r \).

The potential difference \( V \) across the terminals of these three cells while the current \( I \) is flowing is given by:

\[ V = (\text{EMF of 3 cells}) - (\text{Voltage drop across internal resistance of 3 cells}) \]

\[ V = (3E) - (I \times 3r) \]

Now, substitute the value of the current \( I = \frac{E}{r} \) we calculated for the closed circuit:

\[ V = 3E - \left(\frac{E}{r} \times 3r\right) \]

\[ V = 3E - 3E \]

\[ V = 0 \]

According to this calculation, based on the description "formed a closed circuit", the reading of the ideal voltmeter should be 0.

4. Reconsidering the options:

The calculated value of 0 is not among the given options (13E, 10E, 7E, 3E). This suggests a possible ambiguity or error in the question statement or the options provided.

Often in such scenarios, if the circuit were open (meaning no current flows, \( I=0 \)), the voltmeter would measure the total EMF of the cells it spans.

If we assume the circuit is open or that the question implicitly asks for the combined EMF of the three cells (despite connecting a voltmeter which measures potential difference), then:

\[ V (\text{open circuit}) = 3E - I \times (3r) \] \[ V (\text{open circuit}) = 3E - (0) \times (3r) \] \[ V (\text{open circuit}) = 3E \]

This value (3E) matches one of the options.

Conclusion based on options:

Given the discrepancy between the calculation for a "closed circuit" (V=0) and the provided options, the most likely intended answer, corresponding to option 3E, assumes either an open circuit condition (\(I=0\)) or that the question is simply asking for the total EMF of the three cells.

Answer 2

In this case, there are 10 identical cells connected in series, so the total potential difference across the circuit is \( 10E \).

The internal resistance of the circuit is also 10 times the internal resistance of a single cell, i.e., \( 10r \). 

When an ideal voltmeter is connected across three cells, it has infinite resistance and draws negligible current. As a result, the potential difference across the voltmeter will be the same as the potential difference across the three cells.

Since the three cells are connected in series, the potential difference across them is \( 3E \). Therefore, the voltmeter will read \( 3E \).

Thus, the correct answer is: (D) \( 3E \).