When two parallel wires carrying current l and 2l in same direction are placed at a distance d apart, the ratio of force per unit length acting upon the wire due to other is

Options

A. `1 : 1`
B. `1 : 2`
C. `2 : 1`
D. None of these

Answer

To solve the problem of finding the ratio of the force per unit length acting on two parallel wires carrying currents \( I \) and \( 2I \) in the same direction, we can follow these steps:

### Step 1: Understand the formula for the force between two parallel currents
The force per unit length \( F \) between two parallel wires carrying currents \( I_1 \) and \( I_2 \) separated by a distance \( d \) is given by the formula:
\[ F = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Assign values to the currents
Let:
- \( I_1 = I \)
- \( I_2 = 2I \)

### Step 3: Calculate the force per unit length on wire 1 due to wire 2
Using the formula, the force per unit length \( F_{21} \) (force on wire 2 due to wire 1) is:
\[ F_{21} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d} = \frac{\mu_0}{2\pi} \frac{I \cdot 2I}{d} = \frac{\mu_0 I^2}{\pi d} \]

### Step 4: Calculate the force per unit length on wire 2 due to wire 1
Now, the force per unit length \( F_{12} \) (force on wire 1 due to wire 2) is:
\[ F_{12} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d} = \frac{\mu_0}{2\pi} \frac{2I \cdot I}{d} = \frac{\mu_0 I^2}{\pi d} \]

### Step 5: Find the ratio of the forces
Now we can find the ratio of the forces per unit length:
\[ \text{Ratio} = \frac{F_{12}}{F_{21}} = \frac{\frac{\mu_0 I^2}{\pi d}}{\frac{\mu_0 I^2}{\pi d}} = 1 \]

### Conclusion
Thus, the ratio of the force per unit length acting upon one wire due to the other is:
\[ \text{Ratio} = 1 : 1 \]