Two wires are held perpendicular to the plane of paper and are 5 m apart. They carry currents of 2.5 A and 5 A in same direction. Then, the magnetic field strength (B) at a point midway between the wires will be

Options

A. `(mu_(0))/(4pi)T`
B. `(mu_(0))/(2pi)T`
C. `(3mu_(0))/(2pi)T`
D. `(3mu_(0))/(4pi)T`

Answer

To solve the problem of finding the magnetic field strength (B) at a point midway between two parallel wires carrying currents in the same direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Distance between the wires, \( d = 5 \, \text{m} \).
- Currents in the wires, \( I_1 = 2.5 \, \text{A} \) (for the first wire) and \( I_2 = 5 \, \text{A} \) (for the second wire).

2. **Determine the Midpoint**:
- The midpoint between the two wires is \( \frac{d}{2} = \frac{5}{2} = 2.5 \, \text{m} \) from each wire.

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

4. **Calculate Magnetic Field from Each Wire**:
- For the first wire (current \( I_1 = 2.5 \, \text{A} \)):
\[ B_1 = \frac{\mu_0 I_1}{2 \pi r_1} = \frac{\mu_0 \cdot 2.5}{2 \pi \cdot 2.5} = \frac{\mu_0}{2 \pi} \, \text{(inward direction)} \]

- For the second wire (current \( I_2 = 5 \, \text{A} \)):
\[ B_2 = \frac{\mu_0 I_2}{2 \pi r_2} = \frac{\mu_0 \cdot 5}{2 \pi \cdot 2.5} = \frac{\mu_0}{\pi} \, \text{(outward direction)} \]

5. **Determine the Direction of the Magnetic Fields**:
- According to the right-hand rule:
- The magnetic field \( B_1 \) from the first wire (2.5 A) is directed inward.
- The magnetic field \( B_2 \) from the second wire (5 A) is directed outward.

6. **Calculate the Net Magnetic Field**:
- Since \( B_1 \) is inward and \( B_2 \) is outward, we can find the net magnetic field:
\[ B_{\text{net}} = B_2 - B_1 = \frac{\mu_0}{\pi} - \frac{\mu_0}{2 \pi} \]
- Simplifying this:
\[ B_{\text{net}} = \frac{2\mu_0}{2\pi} - \frac{\mu_0}{2\pi} = \frac{\mu_0}{2\pi} \, \text{(outward direction)} \]

7. **Final Result**:
- The magnetic field strength at the midpoint between the wires is:
\[ B = \frac{\mu_0}{2\pi} \, \text{T} \, \text{(outward direction)} \]