A cell when balanced with potentiometer gave a balance length of50 cm. `4.5 Omega` external resistance is introduced in the circuit, now it is balanced on 45 cm. The internal resistance of cell is

Options

A. `0.25 Omega`
B. `0.5 Omega`
C. `1.0 Omega`
D. `1.5 Omega`

Answer

To find the internal resistance of the cell, we can follow these steps:

### Step 1: Understand the Initial Conditions
Initially, the balance length with the potentiometer was 50 cm. This means that the electromotive force (EMF) of the cell (E) can be expressed as:
\[ E = \lambda \times 50 \]
where \( \lambda \) is the potential gradient (voltage per unit length) of the potentiometer wire.

### Step 2: Introduce External Resistance
When a 4.5 ohm external resistance is introduced, the new balance length is 45 cm. The potential drop across the external resistance can be expressed as:
\[ V_{ext} = \lambda \times 45 \]

### Step 3: Relate Current to Potential Drop
Let \( I \) be the current flowing through the circuit. The potential drop across the external resistance is also given by Ohm's Law:
\[ V_{ext} = I \times 4.5 \]
Thus, we can equate the two expressions for the potential drop:
\[ \lambda \times 45 = I \times 4.5 \]

### Step 4: Solve for Current
From the equation above, we can express the current \( I \):
\[ I = \frac{\lambda \times 45}{4.5} \]
This simplifies to:
\[ I = 10 \lambda \]

### Step 5: Apply Kirchhoff's Loop Law
According to Kirchhoff's Loop Law, the EMF of the cell is equal to the sum of the potential drops across the internal resistance and the external resistance:
\[ E = I \times r + I \times 4.5 \]
Substituting \( E \) and \( I \) into this equation gives:
\[ 50 \lambda = (10 \lambda) \times r + (10 \lambda) \times 4.5 \]

### Step 6: Simplify the Equation
We can factor out \( 10 \lambda \):
\[ 50 \lambda = 10 \lambda (r + 4.5) \]
Dividing both sides by \( 10 \lambda \) (assuming \( \lambda \neq 0 \)):
\[ 5 = r + 4.5 \]

### Step 7: Solve for Internal Resistance
Now, we can solve for the internal resistance \( r \):
\[ r = 5 - 4.5 \]
\[ r = 0.5 \, \Omega \]

### Conclusion
The internal resistance of the cell is \( 0.5 \, \Omega \).

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