Two long parallel wires carry currents `i_(1)` and `i_(2)` such that `i_(1) gt i_(2)`. When the currents are in the same direction, the magnetic field at a point midway between the wires is `6xx10^(-6)T`. If the direction of `i_(2)` is reversed, the field becomes `3xx10^(-5)T`. The ratio `(i_(1))/(i_(2))` is

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(3)/(2)`
D. `(1)/(5)`

Answer

To solve the problem, we need to analyze the magnetic fields produced by two long parallel wires carrying currents \( I_1 \) and \( I_2 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two long parallel wires carrying currents \( I_1 \) and \( I_2 \) such that \( I_1 > I_2 \).
- When the currents are in the same direction, the magnetic field at a point midway between the wires is \( 6 \times 10^{-6} \, T \).
- When the direction of \( I_2 \) is reversed, the magnetic field becomes \( 3 \times 10^{-5} \, T \).

2. **Magnetic Field Calculation**:
- The magnetic field \( B \) at a distance \( D \) from a long straight wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2 \pi D} \]
- For the first scenario (currents in the same direction):
\[ B = B_1 - B_2 = \frac{\mu_0 I_1}{2 \pi D} - \frac{\mu_0 I_2}{2 \pi D} = \frac{\mu_0 (I_1 - I_2)}{2 \pi D} \]
- Setting this equal to the given magnetic field:
\[ \frac{\mu_0 (I_1 - I_2)}{2 \pi D} = 6 \times 10^{-6} \quad \text{(Equation 1)} \]

3. **Second Scenario**:
- When the direction of \( I_2 \) is reversed, the magnetic fields add up:
\[ B = B_1 + B_2 = \frac{\mu_0 I_1}{2 \pi D} + \frac{\mu_0 I_2}{2 \pi D} = \frac{\mu_0 (I_1 + I_2)}{2 \pi D} \]
- Setting this equal to the new magnetic field:
\[ \frac{\mu_0 (I_1 + I_2)}{2 \pi D} = 3 \times 10^{-5} \quad \text{(Equation 2)} \]

4. **Dividing the Equations**:
- Divide Equation 2 by Equation 1:
\[ \frac{\frac{\mu_0 (I_1 + I_2)}{2 \pi D}}{\frac{\mu_0 (I_1 - I_2)}{2 \pi D}} = \frac{3 \times 10^{-5}}{6 \times 10^{-6}} \]
- Simplifying gives:
\[ \frac{I_1 + I_2}{I_1 - I_2} = 5 \]

5. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ I_1 + I_2 = 5(I_1 - I_2) \]
- Expanding this:
\[ I_1 + I_2 = 5I_1 - 5I_2 \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ 5I_2 + I_2 = 5I_1 - I_1 \]
- This simplifies to:
\[ 6I_2 = 4I_1 \]

7. **Finding the Ratio**:
- Dividing both sides by \( I_2 \):
\[ \frac{I_1}{I_2} = \frac{6}{4} = \frac{3}{2} \]

### Final Answer:
The ratio \( \frac{I_1}{I_2} \) is \( \frac{3}{2} \).