Two long parallel conductors are placed at right angles to a metre scale at the `2cm` and `6cm` marks, They carry currents of `1A` and `3A` respectively. They will produce zero magnetic field at the `(` ignore the earth's magnetic field `)`

Options

A. 5 cm mark
B. 3 cm mark
C. 1 cm mark
D. 8 cm mark

Answer

To solve the problem of finding the point where the magnetic field produced by two parallel conductors carrying currents of 1A and 3A is zero, we can follow these steps:

### Step-by-step Solution:

1. **Identify the Positions of the Conductors**:
- The first conductor carrying a current of 1A is located at the 2 cm mark.
- The second conductor carrying a current of 3A is located at the 6 cm mark.

2. **Understand the Magnetic Field Direction**:
- The magnetic field produced by a long straight conductor is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi d} \]
where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( d \) is the distance from the conductor.

3. **Set Up the Problem**:
- We need to find a point \( P \) along the scale where the total magnetic field \( B \) is zero. This point will be between the two conductors since the currents are in the same direction.

4. **Define Distances**:
- Let \( x \) be the distance from the 3A conductor (located at 6 cm). Therefore, the distance from the 1A conductor (located at 2 cm) will be \( 4 - x \) cm (since the distance between the two conductors is 4 cm).

5. **Write the Magnetic Field Equations**:
- The magnetic field \( B_1 \) due to the 1A conductor at point \( P \) is:
\[ B_1 = \frac{\mu_0 \cdot 1}{2 \pi (4 - x)} \]
- The magnetic field \( B_2 \) due to the 3A conductor at point \( P \) is:
\[ B_2 = \frac{\mu_0 \cdot 3}{2 \pi x} \]

6. **Set the Magnetic Fields Equal**:
- For the net magnetic field to be zero:
\[ B_1 = B_2 \]
This leads to:
\[ \frac{\mu_0 \cdot 1}{2 \pi (4 - x)} = \frac{\mu_0 \cdot 3}{2 \pi x} \]

7. **Cancel Common Terms**:
- Cancel \( \mu_0 \) and \( 2 \pi \) from both sides:
\[ \frac{1}{4 - x} = \frac{3}{x} \]

8. **Cross Multiply**:
- Cross multiplying gives:
\[ x = 3(4 - x) \]
Simplifying this:
\[ x = 12 - 3x \]
\[ 4x = 12 \]
\[ x = 3 \]

9. **Determine the Position**:
- Since \( x = 3 \) cm is the distance from the 3A conductor, the position of point \( P \) from the start of the scale (0 cm) is:
\[ 6 \text{ cm} - 3 \text{ cm} = 3 \text{ cm} \]

### Final Answer:
The point where the magnetic field is zero is at the **3 cm mark** on the scale.

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