Force per unit length acting at one end of each of the two long parallel wires, carrying current I each, kept distance r apart is

Options

A. `(mu_(0))/(4pi)(i^(2))/(r)`
B. `(mu_(0))/(4pi)(2i^(2))/(r)`
C. `(mu_(0))/(4pi)((2i)^(2))/(r)`
D. `(mu_(0))/(4pi)(i^(2))/(4r)`

Answer

To find the force per unit length acting at one end of two long parallel wires carrying current \( I \) each and separated by a distance \( r \), we can follow these steps:

### Step 1: Understand the Concept
When two parallel wires carry current, they exert a force on each other. The direction of this force depends on the direction of the currents. If the currents are in the same direction, the force is attractive; if they are in opposite directions, the force is repulsive.

### Step 2: Use the Formula for Force per Unit Length
The formula for the force per unit length \( \frac{F}{L} \) between two parallel wires carrying currents \( I_1 \) and \( I_2 \) separated by a distance \( r \) is given by:

\[ \frac{F}{L} = \frac{\mu_0}{2\pi} \cdot \frac{I_1 I_2}{r} \]

Where:
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)).
- \( I_1 \) and \( I_2 \) are the currents in the wires.
- \( r \) is the distance between the wires.

### Step 3: Substitute the Values
Since both wires carry the same current \( I \), we can set \( I_1 = I_2 = I \). Thus, the formula simplifies to:

\[ \frac{F}{L} = \frac{\mu_0}{2\pi} \cdot \frac{I^2}{r} \]

### Step 4: Final Expression
Now we can express the force per unit length acting on each wire:

\[ \frac{F}{L} = \frac{\mu_0 I^2}{2\pi r} \]

### Step 5: Simplify the Expression
To match the format often used in physics problems, we can multiply and divide by 2:

\[ \frac{F}{L} = \frac{\mu_0}{4\pi} \cdot \frac{2I^2}{r} \]

### Conclusion
Thus, the final expression for the force per unit length acting on one end of each of the two long parallel wires is:

\[ \frac{F}{L} = \frac{\mu_0}{4\pi} \cdot \frac{2I^2}{r} \]