Read the following paragraph and answer the question:
A magnetic field can be produced by moving charges or electric currents. The basic equation governing the magnetic field due to a current distribution is the Biot-Savart law.
Finding the magnetic field resulting from a current distribution involves the vector product and is inherently a calculus problem when the distance from the current to the field point is continuously changing.
According to this law, the magnetic field at a point due to a current element of length `vec(dl)` carrying current I, at a distance r from the element is dB = `(mu_0)/(4pi) (I(dvec1 xx vecr))/(r^3)`.
Biot-Savart law has certain similarities as well as difference with Coulomb's law for electrostatic field e.g., there is an angle dependence in Biot-Savart law which is not present in electrostatic case.
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. `prop_0 i/r`
B. `4prop_0 i/r`
C. zero
D. `prop_0 i/(4r)`

Answer

To solve the problem of finding 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. **Understanding the Setup**:
- We have two long straight wires, each 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. **Using the Biot-Savart Law**:
- According to the Biot-Savart law, the magnetic field \( dB \) at a point due to a current element \( dL \) is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \times \hat{r}}{r^2} \]
- For long straight wires, the magnetic field \( B \) at a distance \( r \) from a wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]

3. **Calculating the Magnetic Field from Each Wire**:
- For Wire 1 (let's denote the magnetic field at the midpoint due to Wire 1 as \( B_1 \)):
\[ B_1 = \frac{\mu_0 I}{2\pi r} \]
- For Wire 2 (denote the magnetic field at the midpoint due to Wire 2 as \( B_2 \)):
\[ B_2 = \frac{\mu_0 I}{2\pi r} \]

4. **Determining the Direction of the Magnetic Fields**:
- Using the right-hand rule, the magnetic field direction due to Wire 1 at the midpoint will be into the page (or screen).
- The magnetic field direction due to Wire 2 at the midpoint will be out of the page (or screen).

5. **Finding the Net Magnetic Field**:
- Since both magnetic fields \( B_1 \) and \( B_2 \) have the same magnitude but opposite directions, we can express the net magnetic field \( B \) at the midpoint as:
\[ B = B_1 - B_2 \]
- Substituting the values:
\[ B = \frac{\mu_0 I}{2\pi r} - \frac{\mu_0 I}{2\pi r} = 0 \]

6. **Conclusion**:
- The intensity of the magnetic field midway between the two wires is:
\[ B = 0 \]