Two parallel, long wires carry currents `i_1 and i_2` with `i_1gti_2`. When the currents are in the same direction, the magnetic field at a point midway between the wires is 10mu T. If the direction of `i_2` is reversed, the field becomes 30mu T . The ration` i_1/i_2` is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the problem, we need to analyze the magnetic field produced by two parallel wires carrying currents \( I_1 \) and \( I_2 \). We are given two scenarios:

1. When both currents are in the same direction, the magnetic field at the midpoint is \( 10 \, \mu T \).
2. When the direction of \( I_2 \) is reversed, the magnetic field at the midpoint becomes \( 30 \, \mu T \).

Let's denote the distance between the two wires as \( 2D \), so the distance from each wire to the midpoint is \( D \).

### Step 1: Write the expression for the magnetic field in the first case
When the currents are in the same direction, the magnetic fields due to both wires at the midpoint will oppose each other. The magnetic field \( B \) at the midpoint can be expressed as:

\[ B_1 = \frac{\mu_0 I_1}{2\pi D} - \frac{\mu_0 I_2}{2\pi D} \]

This simplifies to:

\[ B_1 = \frac{\mu_0}{2\pi D} (I_1 - I_2) \]

Given that \( B_1 = 10 \, \mu T \), we can write:

\[ 10 \times 10^{-6} = \frac{\mu_0}{2\pi D} (I_1 - I_2) \quad \text{(1)} \]

### Step 2: Write the expression for the magnetic field in the second case
When the direction of \( I_2 \) is reversed, the magnetic fields due to both wires at the midpoint will add up. The magnetic field \( B \) at the midpoint now becomes:

\[ B_2 = \frac{\mu_0 I_1}{2\pi D} + \frac{\mu_0 I_2}{2\pi D} \]

This simplifies to:

\[ B_2 = \frac{\mu_0}{2\pi D} (I_1 + I_2) \]

Given that \( B_2 = 30 \, \mu T \), we can write:

\[ 30 \times 10^{-6} = \frac{\mu_0}{2\pi D} (I_1 + I_2) \quad \text{(2)} \]

### Step 3: Set up the equations
From equations (1) and (2), we have:

1. \( 10 \times 10^{-6} = \frac{\mu_0}{2\pi D} (I_1 - I_2) \)
2. \( 30 \times 10^{-6} = \frac{\mu_0}{2\pi D} (I_1 + I_2) \)

### Step 4: Divide the two equations
Dividing equation (2) by equation (1):

\[ \frac{30 \times 10^{-6}}{10 \times 10^{-6}} = \frac{I_1 + I_2}{I_1 - I_2} \]

This simplifies to:

\[ 3 = \frac{I_1 + I_2}{I_1 - I_2} \]

### Step 5: Cross-multiply to solve for the ratio
Cross-multiplying gives:

\[ 3(I_1 - I_2) = I_1 + I_2 \]

Expanding this:

\[ 3I_1 - 3I_2 = I_1 + I_2 \]

Rearranging terms:

\[ 3I_1 - I_1 = 3I_2 + I_2 \]

This simplifies to:

\[ 2I_1 = 4I_2 \]

### Step 6: Solve for the ratio
Dividing both sides by \( 2I_2 \):

\[ \frac{I_1}{I_2} = \frac{4}{2} = 2 \]

### Conclusion
The ratio \( \frac{I_1}{I_2} \) is \( 2 \).