Two long parallel wires placed 0.08 m apart, carry currents 3 A and 5 A in the same direction. what is the distance from the conductor carrying the larger current to the point where the resultant magnetic field is zero?

Options

A. 0.5 m
B. 0.04 m
C. 0.05 m
D. 0.4m

Answer

To solve the problem step by step, we need to find the distance from the wire carrying the larger current (5 A) to the point where the resultant magnetic field is zero.

### Step 1: Understand the setup
We have two long parallel wires, one carrying a current of 3 A and the other carrying a current of 5 A. The distance between the two wires is 0.08 m (or 8 cm). We need to find the distance from the wire carrying 5 A to the point where the magnetic fields due to both wires cancel each other out.

### Step 2: Define the variables
Let:
- \( x \) = distance from the wire carrying 5 A to the point where the magnetic field is zero.
- The distance from the wire carrying 3 A to this point will then be \( 0.08 - x \).

### Step 3: Write the formula for the magnetic field
The magnetic field \( B \) due to a long straight wire at a distance \( r \) is given by the formula:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where:
- \( \mu_0 \) = permeability of free space (a constant),
- \( I \) = current in the wire,
- \( r \) = distance from the wire.

### Step 4: Calculate the magnetic fields
1. **Magnetic field due to the 3 A wire (B1)**:
\[ B_1 = \frac{\mu_0 \cdot 3}{2\pi(0.08 - x)} \]
This field is directed into the plane.

2. **Magnetic field due to the 5 A wire (B2)**:
\[ B_2 = \frac{\mu_0 \cdot 5}{2\pi x} \]
This field is directed out of the plane.

### Step 5: Set the magnetic fields equal for cancellation
For the resultant magnetic field to be zero, the magnitudes of \( B_1 \) and \( B_2 \) must be equal:
\[ B_1 = B_2 \]
Substituting the expressions we derived:
\[ \frac{\mu_0 \cdot 3}{2\pi(0.08 - x)} = \frac{\mu_0 \cdot 5}{2\pi x} \]
The \( \mu_0 \) and \( 2\pi \) cancel out:
\[ \frac{3}{0.08 - x} = \frac{5}{x} \]

### Step 6: Cross-multiply to solve for x
Cross-multiplying gives:
\[ 3x = 5(0.08 - x) \]
Expanding the right side:
\[ 3x = 0.4 - 5x \]
Combining like terms:
\[ 3x + 5x = 0.4 \]
\[ 8x = 0.4 \]
\[ x = \frac{0.4}{8} = 0.05 \text{ m} \]

### Step 7: Conclusion
The distance from the wire carrying the larger current (5 A) to the point where the resultant magnetic field is zero is **0.05 m**.