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, 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. We will denote the distance from the wire carrying the larger current as \( x \) and the distance from the wire carrying the smaller current (3 A) will then be \( 0.08 - x \) meters.

### Step-by-Step Solution:

1. **Identify the Magnetic Field Due to Each Wire:**
The magnetic field \( B \) due to a long straight wire carrying current \( I \) at a distance \( r \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where \( \mu_0 \) is the permeability of free space.

2. **Set Up the Magnetic Fields:**
- For the wire carrying 3 A (let's call it Wire 1), the magnetic field at point P (where the resultant field is zero) is:
\[ B_1 = \frac{\mu_0 \cdot 3}{2 \pi (0.08 - x)} \]
- For the wire carrying 5 A (Wire 2), the magnetic field at point P is:
\[ B_2 = \frac{\mu_0 \cdot 5}{2 \pi x} \]

3. **Determine the Direction of the Magnetic Fields:**
Since both currents are in the same direction, the magnetic field due to Wire 1 (3 A) will be directed downwards, and the magnetic field due to Wire 2 (5 A) will be directed upwards. Therefore, we can set up the equation for the point where the magnetic fields cancel each other out:
\[ B_1 = B_2 \]

4. **Set Up the Equation:**
From the above, we have:
\[ \frac{\mu_0 \cdot 3}{2 \pi (0.08 - x)} = \frac{\mu_0 \cdot 5}{2 \pi x} \]
We can cancel \( \mu_0 \) and \( 2 \pi \) from both sides:
\[ \frac{3}{0.08 - x} = \frac{5}{x} \]

5. **Cross Multiply to Solve for \( x \):**
Cross multiplying gives:
\[ 3x = 5(0.08 - x) \]
Expanding the right side:
\[ 3x = 0.4 - 5x \]

6. **Combine Like Terms:**
Adding \( 5x \) to both sides:
\[ 3x + 5x = 0.4 \]
\[ 8x = 0.4 \]

7. **Solve for \( x \):**
Dividing both sides by 8:
\[ x = \frac{0.4}{8} = 0.05 \text{ m} \]

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