A short bar magnet is placed along N-S direction with N pole pointing north. The neutral points are located 20 cm away from the bar magnet. If `B_(H)` is the horizontal component of earth's magnetic field then the magnetic field due to the bar magnet at distance of 40 cm along its axis is

Options

A. `(B_(H))/(2)`
B. `(B_(H))/(4)`
C. `(B_(H))/(8)`
D. `(B_(H))/(16)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the magnetic field due to a bar magnet
The magnetic field \( B \) due to a short bar magnet at a distance \( r \) along its axis is given by the formula:
\[ B \propto \frac{1}{r^3} \]
This means that the magnetic field strength decreases with the cube of the distance from the magnet.

### Step 2: Identify the distances
We have two distances:
- \( R_1 = 20 \) cm (the distance to the neutral point where the magnetic field is equal to the horizontal component of Earth's magnetic field \( B_H \))
- \( R_2 = 40 \) cm (the distance at which we want to find the magnetic field due to the bar magnet)

### Step 3: Set up the ratio of magnetic fields
Using the inverse cube relationship, we can write:
\[ \frac{B_1}{B_2} = \frac{R_2^3}{R_1^3} \]
Where:
- \( B_1 = B_H \) (the magnetic field at the neutral point)
- \( B_2 \) is the magnetic field at 40 cm

### Step 4: Substitute the values
Substituting the values of \( R_1 \) and \( R_2 \):
\[ \frac{B_H}{B_2} = \frac{40^3}{20^3} \]
Calculating the cubes:
\[ 40^3 = 64000 \quad \text{and} \quad 20^3 = 8000 \]
Thus:
\[ \frac{B_H}{B_2} = \frac{64000}{8000} = 8 \]

### Step 5: Solve for \( B_2 \)
Rearranging the equation gives:
\[ B_2 = \frac{B_H}{8} \]

### Conclusion
The magnetic field due to the bar magnet at a distance of 40 cm along its axis is:
\[ B_2 = \frac{B_H}{8} \]