Two parallel conductors are separated by 5cm. They carry 6A and 2A in the same direction. Between them the location from first at which magnetic needdle lies along `B_(H)` is

Options

A. `1.25cm`
B. 3.75 cm
C. 2.5 cm
D. no point exist

Answer

To solve the problem of finding the point between two parallel conductors where the magnetic needle lies along the line of the magnetic field, we can follow these steps:

### Step-by-step Solution:

1. **Identify the Given Information:**
- Current in the first conductor, \( I_1 = 6 \, \text{A} \)
- Current in the second conductor, \( I_2 = 2 \, \text{A} \)
- Distance between the two conductors, \( d = 5 \, \text{cm} \)

2. **Define the Point of Interest:**
- Let point P be at a distance \( x \) from the first conductor. Therefore, the distance from the second conductor to point P will be \( d - x = 5 - x \).

3. **Write the Expression for Magnetic Field:**
- The magnetic field \( B_1 \) at point P due to the first conductor is given by:
\[ B_1 = \frac{\mu_0 I_1}{2 \pi x} \]
- The magnetic field \( B_2 \) at point P due to the second conductor is given by:
\[ B_2 = \frac{\mu_0 I_2}{2 \pi (5 - x)} \]

4. **Set the Magnetic Fields Equal:**
- Since the magnetic needle lies along the line of the magnetic field, we set \( B_1 = B_2 \):
\[ \frac{\mu_0 I_1}{2 \pi x} = \frac{\mu_0 I_2}{2 \pi (5 - x)} \]
- The \( \mu_0 \) and \( 2 \pi \) terms cancel out:
\[ \frac{I_1}{x} = \frac{I_2}{5 - x} \]

5. **Substitute the Values of Currents:**
- Substitute \( I_1 = 6 \) A and \( I_2 = 2 \) A into the equation:
\[ \frac{6}{x} = \frac{2}{5 - x} \]

6. **Cross Multiply to Solve for x:**
- Cross multiplying gives:
\[ 6(5 - x) = 2x \]
- Expanding this results in:
\[ 30 - 6x = 2x \]

7. **Combine Like Terms:**
- Rearranging the equation gives:
\[ 30 = 8x \]

8. **Solve for x:**
- Divide both sides by 8:
\[ x = \frac{30}{8} = 3.75 \, \text{cm} \]

### Conclusion:
The magnetic needle lies at a distance of **3.75 cm** from the first conductor.