Ratio of current flowing through the wire. If both the wire have same area and the ratio between their length is 4:1.

Options

A. `1:1`
B. `1:4`
C. `4:1`
D. `4:4`

Explanation

To solve the problem of finding the ratio of current flowing through two wires with the same area but different lengths, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Ohm's Law**: According to Ohm's law, the relationship between voltage (V), current (I), and resistance (R) is given by:
\[ V = I \times R \]
For two wires, if the potential difference across both wires is the same, we can write:
\[ V = I_1 \times R_1 = I_2 \times R_2 \]

2. **Finding the Ratio of Currents**: From the above equation, we can express the ratio of the currents as:
\[ \frac{I_1}{I_2} = \frac{R_2}{R_1} \]

3. **Calculating Resistance**: The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length of the wire, and \( A \) is the cross-sectional area.

4. **Substituting for Resistance**: For our two wires, we can write:
\[ R_1 = \frac{\rho L_1}{A} \quad \text{and} \quad R_2 = \frac{\rho L_2}{A} \]

5. **Cancelling Common Terms**: Since both wires have the same material (same resistivity) and the same area, we can cancel \( \rho \) and \( A \) from the ratio:
\[ \frac{I_1}{I_2} = \frac{R_2}{R_1} = \frac{L_2}{L_1} \]

6. **Using the Given Length Ratio**: We are given that the ratio of their lengths is \( L_1 : L_2 = 4 : 1 \). This means:
\[ L_1 = 4L \quad \text{and} \quad L_2 = L \]
Therefore, the ratio \( \frac{L_2}{L_1} \) can be expressed as:
\[ \frac{L_2}{L_1} = \frac{L}{4L} = \frac{1}{4} \]

7. **Final Ratio of Currents**: Substituting back into our earlier equation gives us:
\[ \frac{I_1}{I_2} = \frac{L_2}{L_1} = \frac{1}{4} \]
Thus, the ratio of currents \( I_1 : I_2 \) is:
\[ I_1 : I_2 = 1 : 4 \]

### Conclusion:
The ratio of the current flowing through the two wires is \( 1 : 4 \).