Two long straight wires are set parallel to each other Each carries a current i in the same direction and the separation between them is 2r. The intensity of the magnetic field midway between them is

Options

A. `mu_(0)i//r`
B. `4mu_(0)i//r`
C. zero
D. `mu_(0)i//4r`

Answer

To find the intensity of the magnetic field midway between two long straight parallel wires carrying the same current in the same direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Setup**:
- We have two long straight wires, Wire 1 and Wire 2, carrying a current \( I \) in the same direction.
- The distance between the two wires is \( 2r \), which means the distance from each wire to the midpoint is \( r \).

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

3. **Calculate the Magnetic Field at the Midpoint**:
- For Wire 1, the magnetic field at the midpoint (due to Wire 1) will be directed into the page (inward).
- For Wire 2, the magnetic field at the midpoint (due to Wire 2) will be directed out of the page (outward).

4. **Direction of Magnetic Fields**:
- Using the right-hand rule, we can determine the direction of the magnetic fields:
- For Wire 1 (current flowing in the same direction), the magnetic field at the midpoint is inward.
- For Wire 2 (also with current flowing in the same direction), the magnetic field at the midpoint is outward.

5. **Magnitude of the Magnetic Fields**:
- The magnitudes of the magnetic fields due to both wires at the midpoint are the same:
\[ B_1 = B_2 = \frac{\mu_0 I}{2\pi r} \]

6. **Net Magnetic Field Calculation**:
- Since the magnetic fields are equal in magnitude but opposite in direction, they will cancel each other out:
\[ B_{\text{net}} = B_1 - B_2 = \frac{\mu_0 I}{2\pi r} - \frac{\mu_0 I}{2\pi r} = 0 \]

### Conclusion:
The intensity of the magnetic field midway between the two wires is:
\[ B_{\text{net}} = 0 \]