Question

Two long parallel wires carry currents \( I_1 \) and \( I_2 \) (\( I_1 > I_2 \)). When currents are flowing in the same direction, the magnetic field at a point midway between the wires is \( 6 \times 10^{-6} \, \text{T} \). If the direction of \( I_2 \) is reversed, the field at the midpoint becomes \( 3 \times 10^{-5} \, \text{T} \). The ratio \( I_1 : I_2 \) is

• A. 3 : 2
• B. 2 : 3
• C. 3 : 5
• D. 6 : 7

Hint:

When two currents flow in the same direction, their magnetic fields at a midpoint add up. When the currents flow in opposite directions, their magnetic fields subtract from each other.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The net magnetic field at the midpoint is the vector sum of fields from individual wires. Directions depend on current directions.
Step 2: Key Formula or Approach:
Field from a long wire at distance \( r \): \( B = \frac{\mu_0 I}{2\pi r} \).
Let midpoint be at distance \( d \) from each wire. \( B_1 = k I_1, B_2 = k I_2 \).
Step 3: Detailed Explanation:
1. Same direction: Fields are in opposite directions at midpoint.
\[ B_{net} = B_1 - B_2 = k(I_1 - I_2) = 6 \times 10^{-6} \]
2. Opposite direction: Fields are in the same direction at midpoint.
\[ B'_{net} = B_1 + B_2 = k(I_1 + I_2) = 3 \times 10^{-5} = 30 \times 10^{-6} \]
Dividing the two equations:
\[ \frac{I_1 + I_2}{I_1 - I_2} = \frac{30 \times 10^{-6}}{6 \times 10^{-6}} = 5 \]
\[ I_1 + I_2 = 5 I_1 - 5 I_2 \]
\[ 6 I_2 = 4 I_1 \]
\[ \frac{I_1}{I_2} = \frac{6}{4} = \frac{3}{2} \]
Step 4: Final Answer:
The ratio \( I_1 : I_2 \) is 3 : 2.