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 lying on the perpendicular line joining two long parallel wires carrying equal current \( I \) in the same direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Setup**:
- Let the two long parallel wires be separated by a distance of \( 2d \).
- The midpoint between the wires is denoted as point \( M \).
- The point where we want to calculate the magnetic field is at a distance \( x \) from the midpoint \( M \).

2. **Determine Distances**:
- The distance from the first wire to point \( A \) (the point where we are calculating the magnetic field) is \( d + x \).
- The distance from the second wire to point \( A \) is \( d - x \).

3. **Calculate Magnetic Fields**:
- The magnetic field \( B_1 \) due to the first wire at point \( A \) is given by the formula:
\[ B_1 = \frac{\mu_0 I}{2 \pi (d + x)} \]
- The magnetic field \( B_2 \) due to the second wire at point \( A \) is given by:
\[ B_2 = \frac{\mu_0 I}{2 \pi (d - x)} \]

4. **Determine the Direction of the Magnetic Fields**:
- Using the right-hand rule, the magnetic field due to the first wire (which is going into the page) is negative, and the magnetic field due to the second wire (which is coming out of the page) is positive.

5. **Calculate the Net Magnetic Field**:
- The net magnetic field \( B \) at point \( A \) is the difference between \( B_2 \) and \( B_1 \):
\[ B = B_2 - B_1 = \frac{\mu_0 I}{2 \pi (d - x)} - \frac{\mu_0 I}{2 \pi (d + x)} \]

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, find a common denominator:
\[ B = \frac{\mu_0 I}{2 \pi} \left( \frac{(d + x) - (d - x)}{(d - x)(d + x)} \right) = \frac{\mu_0 I}{2 \pi} \left( \frac{2x}{d^2 - x^2} \right) \]

8. **Final Result**:
- Thus, the magnetic field \( B \) at point \( A \) is:
\[ B = \frac{\mu_0 I x}{\pi (d^2 - x^2)} \]

### Final Answer:
\[ B = \frac{\mu_0 I x}{\pi (d^2 - x^2)} \]