When 6 identical cells of no internal resistance are connected in series in the secondarycircuit of a potentio meter, the balancing lengths is .l., balancing length becomes 1/3 when some cells are connected wrongly, the number of cells connected wrongly are

Options

A. 1
B. 3
C. 2
D. 4

Answer

To solve the problem step by step, we will analyze the situation with the cells connected in a potentiometer circuit.

### Step 1: Understand the Initial Setup
We have 6 identical cells connected in series, each with an electromotive force (emf) of \( e \) and no internal resistance. Therefore, the total emf when all cells are connected correctly is:
\[ E_{total} = 6e \]
The balancing length of the potentiometer in this case is given as \( l \).

**Hint:** Remember that the total emf in a series connection is the sum of the individual emfs.

### Step 2: Analyze the New Balancing Length
When some cells are connected wrongly, the balancing length becomes \( \frac{l}{3} \). This indicates that the effective emf has changed. We can denote the new effective emf as \( E_{new} \).

**Hint:** The balancing length is directly proportional to the emf in a potentiometer. A shorter balancing length means a lower effective emf.

### Step 3: Set Up the Proportionality Equation
Using the relationship between the emfs and the balancing lengths, we can write:
\[ \frac{E_{new}}{E_{total}} = \frac{l_{new}}{l_{original}} \]
Substituting the known values:
\[ \frac{E_{new}}{6e} = \frac{\frac{l}{3}}{l} \]
This simplifies to:
\[ \frac{E_{new}}{6e} = \frac{1}{3} \]

**Hint:** This equation helps us find the new effective emf based on the change in balancing length.

### Step 4: Solve for the New Effective Emf
From the equation above, we can solve for \( E_{new} \):
\[ E_{new} = 6e \cdot \frac{1}{3} = 2e \]

**Hint:** This shows that the effective emf has decreased significantly due to the wrong connections.

### Step 5: Determine the Reduction in Emf
The reduction in emf due to the wrong connections can be calculated as:
\[ \text{Reduction} = E_{total} - E_{new} = 6e - 2e = 4e \]

**Hint:** This reduction indicates how much emf is lost due to the incorrect connections.

### Step 6: Calculate the Number of Cells Connected Wrongly
Each cell contributes \( e \) to the total emf. If \( n \) cells are connected wrongly, they effectively reduce the total emf by \( n \cdot e \). We can set up the equation:
\[ n \cdot e = 4e \]
Thus, solving for \( n \):
\[ n = 4 \]

**Hint:** This means that the total reduction in emf corresponds to the number of cells connected incorrectly.

### Conclusion
The number of cells connected wrongly is \( n = 4 \).

**Final Answer:** 4 cells are connected wrongly.