Two long parallel wires `P` and `Q` are both perpendicular to the plane of the paper with distance `5m` between them. If `P` and `Q` carry current of `2.5` amp and `5` amp respectively in the same direction, then the magnetic field at a point half way between the wires is

Options

A. `(sqrt(3)mu_(0))/(2pi)`
B. `(mu_(0))/(pi)`
C. `(3mu_(0))/(2pi)`
D. `(mu_(0))/(2pi)`

Explanation

To solve the problem, we need to find the magnetic field at a point halfway between two long parallel wires carrying currents in the same direction. Let's break down the solution step-by-step.

### Step 1: Understand the Setup
We have two long parallel wires, P and Q, separated by a distance of 5 meters. Wire P carries a current \( I_1 = 2.5 \, \text{A} \) and wire Q carries a current \( I_2 = 5 \, \text{A} \). We need to find the magnetic field at a point halfway between the two wires, which is \( 2.5 \, \text{m} \) from each wire.

### Step 2: Use the Formula for Magnetic Field
The magnetic field \( B \) due to a long straight wire at a distance \( r \) from it is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)).

### Step 3: Calculate the Magnetic Field from Each Wire
1. **Magnetic Field due to Wire P**:
- Distance from point to wire P: \( r_1 = 2.5 \, \text{m} \)
- Current \( I_1 = 2.5 \, \text{A} \)
- Magnetic field \( B_1 \):
\[ B_1 = \frac{\mu_0 I_1}{2 \pi r_1} = \frac{\mu_0 \times 2.5}{2 \pi \times 2.5} = \frac{\mu_0}{2 \pi} \]

2. **Magnetic Field due to Wire Q**:
- Distance from point to wire Q: \( r_2 = 2.5 \, \text{m} \)
- Current \( I_2 = 5 \, \text{A} \)
- Magnetic field \( B_2 \):
\[ B_2 = \frac{\mu_0 I_2}{2 \pi r_2} = \frac{\mu_0 \times 5}{2 \pi \times 2.5} = \frac{2\mu_0}{2 \pi} = \frac{\mu_0}{\pi} \]

### Step 4: Determine the Direction of the Magnetic Fields
- According to the right-hand rule, for both wires carrying current in the same direction, the magnetic field at the halfway point will be directed into the plane of the paper (negative y-direction) due to wire P and out of the plane of the paper (positive y-direction) due to wire Q.

### Step 5: Calculate the Net Magnetic Field
Since both magnetic fields are in opposite directions, we can subtract their magnitudes:
\[ B_{\text{net}} = B_2 - B_1 = \frac{\mu_0}{\pi} - \frac{\mu_0}{2 \pi} = \frac{\mu_0}{\pi} - \frac{\mu_0}{2 \pi} = \frac{\mu_0}{2 \pi} \]
The direction of the net magnetic field will be out of the plane of the paper (positive y-direction).

### Final Answer
The magnitude of the magnetic field at the point halfway between the wires is:
\[ B_{\text{net}} = \frac{\mu_0}{2 \pi} \, \text{T} \]