Two parallel long wires carry currents 18A and 3A. When the currents are in the same direction, the magnetic field at a point midway between the wire is `B_(1)`. If the direction of `i_(2)` is reversed, the field becomes `B_(2)`. Then the value of `B_(1)//B_(2)` is

Options

A. `5:7`
B. `7:5`
C. `3:5`
D. `5:3`

Answer

To solve the problem, we need to calculate the magnetic fields \( B_1 \) and \( B_2 \) at a point midway between two parallel wires carrying currents \( I_1 = 18 \, \text{A} \) and \( I_2 = 3 \, \text{A} \).

### Step-by-Step Solution:

1. **Understanding the Magnetic Field due to a Current-Carrying Wire**:
The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0}{2 \pi r} I \]
where \( \mu_0 \) is the permeability of free space.

2. **Calculating \( B_1 \) when Currents are in the Same Direction**:
When both currents are in the same direction, the magnetic fields due to both wires at the midpoint will add up. Therefore, the magnetic field \( B_1 \) at the midpoint is:
\[ B_1 = \frac{\mu_0}{2 \pi r} I_1 - \frac{\mu_0}{2 \pi r} I_2 \]
Substituting the values:
\[ B_1 = \frac{\mu_0}{2 \pi r} (I_1 - I_2) = \frac{\mu_0}{2 \pi r} (18 - 3) = \frac{\mu_0}{2 \pi r} \cdot 15 \]

3. **Calculating \( B_2 \) when Currents are in Opposite Directions**:
When the direction of \( I_2 \) is reversed, the magnetic fields will oppose each other. Therefore, the magnetic field \( B_2 \) at the midpoint is:
\[ B_2 = \frac{\mu_0}{2 \pi r} (I_1 + I_2) \]
Substituting the values:
\[ B_2 = \frac{\mu_0}{2 \pi r} (18 + 3) = \frac{\mu_0}{2 \pi r} \cdot 21 \]

4. **Finding the Ratio \( \frac{B_1}{B_2} \)**:
Now we can find the ratio of \( B_1 \) to \( B_2 \):
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0}{2 \pi r} \cdot 15}{\frac{\mu_0}{2 \pi r} \cdot 21} \]
The \( \frac{\mu_0}{2 \pi r} \) terms cancel out:
\[ \frac{B_1}{B_2} = \frac{15}{21} = \frac{5}{7} \]

5. **Conclusion**:
Therefore, the value of \( \frac{B_1}{B_2} \) is \( \frac{5}{7} \).

### Final Answer:
\[ \frac{B_1}{B_2} = \frac{5}{7} \]