Two long parallel wires carry equal current I flowing in the same direction are at a distance 2d apart. The magnetic field B at a point lying on the perpendicular line joining the wires and at a distance x from the midpoint is

Options

A. `(mu_(0)id)/(pi(d^(2)+x^(2)))`
B. `(mu_(0)ix)/(pi(d^(2)-x^(2)))`
C. `(mu_(0)ix)/((d^(2)+x^(2)))`
D. `(mu_(0)id)/((d^(2)+x^(2)))`

Answer

To find the magnetic field \( B \) at a point \( P \) lying on the perpendicular line joining two long parallel wires carrying equal currents \( I \) in the same direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
- We have two long parallel wires separated by a distance \( 2d \).
- The point \( P \) is located at a distance \( x \) from the midpoint of the line joining the two wires.

2. **Magnetic Field Due to a Single Wire**:
- The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- Here, \( \mu_0 \) is the permeability of free space.

3. **Calculate the Distance from Each Wire**:
- The distance from the first wire to point \( P \) is \( d - x \).
- The distance from the second wire to point \( P \) is \( d + x \).

4. **Calculate the Magnetic Field at Point \( P \)**:
- The magnetic field due to the first wire at point \( P \) (directed into the plane) is:
\[ B_1 = \frac{\mu_0 I}{2 \pi (d - x)} \]
- The magnetic field due to the second wire at point \( P \) (directed out of the plane) is:
\[ B_2 = \frac{\mu_0 I}{2 \pi (d + x)} \]

5. **Determine the Net Magnetic Field**:
- Since the currents are in the same direction, the magnetic fields will oppose each other. Thus, the net magnetic field \( B \) at point \( P \) is:
\[ B = B_1 - B_2 = \frac{\mu_0 I}{2 \pi (d - x)} - \frac{\mu_0 I}{2 \pi (d + x)} \]

6. **Combine the Terms**:
- Factor out \( \frac{\mu_0 I}{2 \pi} \):
\[ B = \frac{\mu_0 I}{2 \pi} \left( \frac{1}{d - x} - \frac{1}{d + x} \right) \]

7. **Simplify the Expression**:
- To combine the fractions:
\[ B = \frac{\mu_0 I}{2 \pi} \left( \frac{(d + x) - (d - x)}{(d - x)(d + x)} \right) \]
- This simplifies to:
\[ B = \frac{\mu_0 I}{2 \pi} \left( \frac{2x}{d^2 - x^2} \right) \]
- Therefore, the final expression for the magnetic field \( B \) at point \( P \) is:
\[ B = \frac{\mu_0 I x}{\pi (d^2 - x^2)} \]