A galvanometer connected with an unknown resistor and two identical cells in series each of emf 2 V shows a current of 1 A. If the cells are connected in parallel, it shows 0.8 A. Then the internal resistance of the cell is

Options

A. `1Omega`
B. `0.5Omega`
C. `0.25Omega`
D. `0.33Omega`

Explanation

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

### Step 1: Analyze the Series Circuit
When the two identical cells are connected in series, the total EMF (E) is the sum of the individual EMFs:
\[ E = E_1 + E_2 = 2V + 2V = 4V \]
The total current (I) flowing through the circuit is given as 1 A.

Using Ohm's law, we can express the total voltage in terms of the total resistance (R_total):
\[ V = I \cdot R_{total} \]
Where \( R_{total} = R + 2r \) (R is the unknown resistor and r is the internal resistance of each cell).

Thus, we have:
\[ 4V = 1A \cdot (R + 2r) \]
This simplifies to:
\[ R + 2r = 4 \quad \text{(Equation 1)} \]

### Step 2: Analyze the Parallel Circuit
When the two identical cells are connected in parallel, the EMF remains the same (2V), but the internal resistance changes. The equivalent internal resistance (r_eq) for two identical resistances in parallel is:
\[ r_{eq} = \frac{r}{2} \]

The current flowing through the circuit in this case is given as 0.8 A. Using Ohm's law again:
\[ V = I \cdot R_{total} \]
Where \( R_{total} = R + r_{eq} = R + \frac{r}{2} \).

Thus, we have:
\[ 2V = 0.8A \cdot (R + \frac{r}{2}) \]
This simplifies to:
\[ 2 = 0.8(R + \frac{r}{2}) \]
\[ R + \frac{r}{2} = \frac{2}{0.8} = 2.5 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( R + 2r = 4 \) (Equation 1)
2. \( R + \frac{r}{2} = 2.5 \) (Equation 2)

From Equation 2, we can express R in terms of r:
\[ R = 2.5 - \frac{r}{2} \]

Substituting this expression for R into Equation 1:
\[ (2.5 - \frac{r}{2}) + 2r = 4 \]
Combining like terms:
\[ 2.5 + \frac{3r}{2} = 4 \]
Subtracting 2.5 from both sides:
\[ \frac{3r}{2} = 1.5 \]
Multiplying both sides by \( \frac{2}{3} \):
\[ r = 1 \, \Omega \]

### Conclusion
The internal resistance of the cell is \( r = 1 \, \Omega \).

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