Two parallel straight long conducting wires, which are placed at a distance r from each other, are carrying equal currents I in opposite directions. The value of magnetic induction at a point situated at a point situated x from one wire in between the wires will be:

Options

A. `(mu_(0)i)/(2pi){(1)/(r-x)-(1)/(x)}`
B. `(mu_(0)i)/(2pi){(1)/(r-x)+(1)/(x)}`
C. `(mu_(0)i)/(2pi(r-x))`
D. `(mu_(0)i)/(2pix)`

Answer

To solve the problem of finding the magnetic induction at a point situated at a distance \( x \) from one of the two parallel wires carrying equal currents \( I \) in opposite directions, we can follow these steps:

### Step 1: Understand the Configuration
We have two long parallel wires separated by a distance \( R \). One wire carries current \( I \) upwards, and the other wire carries current \( I \) downwards. We need to find the magnetic field at a point located \( x \) from one wire.

### Step 2: Determine the Distances
- The distance from the first wire (the one carrying current \( I \) upwards) to the point is \( x \).
- The distance from the second wire (the one carrying current \( I \) downwards) to the point is \( R - x \).

### Step 3: Calculate the Magnetic Field due to Each Wire
The magnetic field \( B \) created by a long straight wire at a distance \( d \) from it is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi d} \]

**Magnetic Field due to Wire 1 (upwards current):**
\[ B_1 = \frac{\mu_0 I}{2 \pi x} \]
The direction of \( B_1 \) is into the page (using the right-hand rule).

**Magnetic Field due to Wire 2 (downwards current):**
\[ B_2 = \frac{\mu_0 I}{2 \pi (R - x)} \]
The direction of \( B_2 \) is also into the page.

### Step 4: Find the Net Magnetic Field
Since both magnetic fields \( B_1 \) and \( B_2 \) are in the same direction (into the page), we can add them:
\[ B_{\text{net}} = B_1 + B_2 = \frac{\mu_0 I}{2 \pi x} + \frac{\mu_0 I}{2 \pi (R - x)} \]

### Step 5: Simplify the Expression
Combining the two terms gives:
\[ B_{\text{net}} = \frac{\mu_0 I}{2 \pi} \left( \frac{1}{x} + \frac{1}{R - x} \right) \]

### Final Result
Thus, the value of the magnetic induction at the point situated \( x \) from one wire in between the wires is:
\[ B_{\text{net}} = \frac{\mu_0 I}{2 \pi} \left( \frac{1}{x} + \frac{1}{R - x} \right) \]