Two parallel wires carrying equal currents `i_(1)` and `i_(2)` with `i_(1)gti_(2)`. When the current are in the same direction, the `10 mT`. If the direction of `i_(2)` is reversed, the field becomes `30 mT`. The ratio `i_(1)//i_(2)` is

Explanation

To solve the problem, we need to analyze the magnetic fields produced by two parallel wires carrying currents \( i_1 \) and \( i_2 \).

### Step 1: Understand the Magnetic Field from Each Wire
The magnetic field \( B \) at a distance \( D \) from a long straight wire carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi D} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Calculate the Magnetic Field When Currents are in the Same Direction
When both currents \( i_1 \) and \( i_2 \) are in the same direction, the magnetic fields \( B_1 \) and \( B_2 \) at the midpoint (distance \( D \) from each wire) are:
\[ B_1 = \frac{\mu_0 i_1}{2 \pi D} \]
\[ B_2 = \frac{\mu_0 i_2}{2 \pi D} \]
Since \( i_1 > i_2 \), the net magnetic field \( B \) is:
\[ B = B_1 - B_2 = \frac{\mu_0 i_1}{2 \pi D} - \frac{\mu_0 i_2}{2 \pi D} = \frac{\mu_0 (i_1 - i_2)}{2 \pi D} \]
Given that this net magnetic field is \( 10 \, \text{mT} \):
\[ \frac{\mu_0 (i_1 - i_2)}{2 \pi D} = 10 \times 10^{-3} \, \text{T} \quad \text{(Equation 1)} \]

### Step 3: Calculate the Magnetic Field When Currents are in Opposite Directions
When the direction of \( i_2 \) is reversed, the magnetic fields at the midpoint are:
\[ B_1 = \frac{\mu_0 i_1}{2 \pi D} \]
\[ B_2 = \frac{\mu_0 i_2}{2 \pi D} \]
Now, since the currents are in opposite directions, the net magnetic field \( B \) is:
\[ B = B_1 + B_2 = \frac{\mu_0 i_1}{2 \pi D} + \frac{\mu_0 i_2}{2 \pi D} = \frac{\mu_0 (i_1 + i_2)}{2 \pi D} \]
Given that this net magnetic field is \( 30 \, \text{mT} \):
\[ \frac{\mu_0 (i_1 + i_2)}{2 \pi D} = 30 \times 10^{-3} \, \text{T} \quad \text{(Equation 2)} \]

### Step 4: Set Up the Ratio of the Two Equations
From Equation 1:
\[ \mu_0 (i_1 - i_2) = 20 \pi D \times 10^{-3} \]
From Equation 2:
\[ \mu_0 (i_1 + i_2) = 60 \pi D \times 10^{-3} \]

### Step 5: Divide the Two Equations
Dividing Equation 2 by Equation 1:
\[ \frac{i_1 + i_2}{i_1 - i_2} = \frac{60}{20} = 3 \]
Cross-multiplying gives:
\[ i_1 + i_2 = 3(i_1 - i_2) \]
Expanding this:
\[ i_1 + i_2 = 3i_1 - 3i_2 \]
Rearranging terms:
\[ 3i_2 + i_2 = 3i_1 - i_1 \]
\[ 4i_2 = 2i_1 \]
Thus:
\[ \frac{i_1}{i_2} = 2 \]

### Final Answer
The ratio \( \frac{i_1}{i_2} \) is \( 2 \).