Two long parallel wires are at a distance ` 2d` apart. They carry steady equal currents flowing out of the plane of the paper , as shown. The variation of the magnetic field `B` along the line `XX` is given by

Options

A.
B.
C.
D.

Explanation

To solve the problem of finding the variation of the magnetic field \( B \) along the line \( XX \) due to two long parallel wires carrying equal currents, we can follow these steps:

### Step 1: Understand the Setup
We have two long parallel wires separated by a distance of \( 2d \), with both wires carrying equal currents flowing out of the plane of the paper. We need to analyze the magnetic field at various points along the line \( XX \).

### Step 2: Define the Position
Let’s denote the left wire as wire 1 and the right wire as wire 2. We can define a point \( P \) located at a distance \( x \) from wire 1. Consequently, the distance from wire 2 to point \( P \) will be \( 2d - x \).

### Step 3: Calculate the Magnetic Field from Each Wire
Using the formula for the magnetic field \( B \) due to a long straight wire, we have:
\[ B_1 = \frac{\mu_0 I}{2 \pi x} \quad \text{(magnetic field due to wire 1)} \]
\[ B_2 = \frac{\mu_0 I}{2 \pi (2d - x)} \quad \text{(magnetic field due to wire 2)} \]

### Step 4: Determine the Directions of the Magnetic Fields
Using the right-hand rule:
- For wire 1 (left wire), the magnetic field \( B_1 \) at point \( P \) will be directed upwards (positive \( j \)-direction).
- For wire 2 (right wire), the magnetic field \( B_2 \) at point \( P \) will be directed downwards (negative \( j \)-direction).

### Step 5: Write the Net Magnetic Field
The net magnetic field \( B_{\text{net}} \) at point \( P \) can be expressed as:
\[ B_{\text{net}} = B_1 - B_2 = \frac{\mu_0 I}{2 \pi x} - \frac{\mu_0 I}{2 \pi (2d - x)} \]

### Step 6: Simplify the Expression
Factoring out the common terms:
\[ B_{\text{net}} = \frac{\mu_0 I}{2 \pi} \left( \frac{1}{x} - \frac{1}{2d - x} \right) \]

### Step 7: Analyze Special Cases
1. **At \( x = d \)**:
\[ B_{\text{net}} = \frac{\mu_0 I}{2 \pi} \left( \frac{1}{d} - \frac{1}{d} \right) = 0 \]
The magnetic field is zero at the midpoint between the two wires.

2. **For \( x < d \)**:
The term \( \frac{1}{x} \) dominates, making \( B_{\text{net}} > 0 \) (upward direction).

3. **For \( x > d \)**:
The term \( \frac{1}{2d - x} \) dominates, making \( B_{\text{net}} < 0 \) (downward direction).

### Step 8: Conclusion
The variation of the magnetic field \( B \) along the line \( XX \) can be summarized as:
- \( B > 0 \) for \( x < d \) (upward)
- \( B = 0 \) for \( x = d \)
- \( B < 0 \) for \( x > d \) (downward)