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 will follow a systematic approach to find the ratio \( \frac{I_1}{I_2} \).

### Step 1: Understanding the Magnetic Field due to Long Parallel Wires

When two long parallel wires carry currents \( I_1 \) and \( I_2 \), the magnetic field at a point between them can be calculated using the formula:
\[ B = \frac{\mu_0}{2\pi d} I \]
where \( \mu_0 \) is the permeability of free space, \( d \) is the distance from the wire to the point of interest, and \( I \) is the current in the wire.

### Step 2: Magnetic Field When Currents are in the Same Direction

Given that the currents \( I_1 \) and \( I_2 \) are in the same direction, the magnetic fields at the midpoint between the two wires will be:
- \( B_1 \) due to wire 1 (downward)
- \( B_2 \) due to wire 2 (upward)

Since \( I_1 > I_2 \), we can express the net magnetic field \( B \) at the midpoint as:
\[ B = B_1 - B_2 = \frac{\mu_0}{2\pi d} I_1 - \frac{\mu_0}{2\pi d} I_2 \]
This simplifies to:
\[ B = \frac{\mu_0}{2\pi d} (I_1 - I_2) \]
According to the problem, this magnetic field is given as \( 10 \mu T \):
\[ \frac{\mu_0}{2\pi d} (I_1 - I_2) = 10 \mu T \quad \text{(Equation 1)} \]

### Step 3: Magnetic Field When the Direction of \( I_2 \) is Reversed

When the direction of \( I_2 \) is reversed, both currents now flow in the same direction (downward). The magnetic fields will both be downward:
\[ B = B_1 + B_2 = \frac{\mu_0}{2\pi d} I_1 + \frac{\mu_0}{2\pi d} I_2 \]
This simplifies to:
\[ B = \frac{\mu_0}{2\pi d} (I_1 + I_2) \]
According to the problem, this magnetic field is given as \( 30 \mu T \):
\[ \frac{\mu_0}{2\pi d} (I_1 + I_2) = 30 \mu T \quad \text{(Equation 2)} \]

### Step 4: Forming a Ratio from the Two Equations

Now we have two equations:
1. \( \frac{\mu_0}{2\pi d} (I_1 - I_2) = 10 \)
2. \( \frac{\mu_0}{2\pi d} (I_1 + I_2) = 30 \)

We can divide Equation 1 by Equation 2:
\[ \frac{10}{30} = \frac{I_1 - I_2}{I_1 + I_2} \]
This simplifies to:
\[ \frac{1}{3} = \frac{I_1 - I_2}{I_1 + I_2} \]

### Step 5: Cross-Multiplying to Solve for \( I_1 \) and \( I_2 \)

Cross-multiplying gives:
\[ I_1 + I_2 = 3(I_1 - I_2) \]
Expanding this:
\[ I_1 + I_2 = 3I_1 - 3I_2 \]
Rearranging terms:
\[ I_1 + I_2 + 3I_2 = 3I_1 \]
This simplifies to:
\[ 4I_2 = 2I_1 \]

### Step 6: Finding the Ratio \( \frac{I_1}{I_2} \)

Now, dividing both sides by \( 2I_2 \):
\[ \frac{I_1}{I_2} = \frac{4}{2} = 2 \]

### Conclusion

Thus, the ratio \( \frac{I_1}{I_2} \) is:
\[ \boxed{2} \]