A bar magnet having centre `O` has a length of `4 cm`. Point `P_(1)` is in the broad side-on and `P_(2)` is in the end side-on position with `OP_(1)=OP_(2)=10 metres`. The ratio of magnetic intensities `H` at `P_(1)` and `P_(2)` is

Options

A. `H_(1):H_(2)=16:100`
B. `H_(1):H_(2)=2:1`
C. `H_(1):H_(2)=1:2`
D. `H_(1):H_(2)=100:16`

Answer

To find the ratio of magnetic intensities \( H \) at points \( P_1 \) (broad side-on position) and \( P_2 \) (end side-on position) of a bar magnet, we can use the formulas for magnetic field intensity in both positions.

### Step-by-Step Solution:

1. **Identify the Positions**:
- Point \( P_1 \) is in the broad side-on position (equatorial position).
- Point \( P_2 \) is in the end side-on position (axial position).
- The distance from the center of the magnet \( O \) to both points is \( OP_1 = OP_2 = 10 \, \text{m} \).

2. **Length of the Magnet**:
- The length of the bar magnet is given as \( 4 \, \text{cm} \) which is \( 0.04 \, \text{m} \).

3. **Magnetic Field Intensity Formulas**:
- For a bar magnet, the magnetic field intensity \( H \) at a point in the axial position (end side) is given by:
\[ H_{end} = \frac{\mu_0 \cdot 2 \cdot m}{r^3} \]
- For a point in the equatorial position (broad side), the magnetic field intensity is:
\[ H_{broad} = \frac{\mu_0 \cdot m}{r^3} \]
Where \( \mu_0 \) is the permeability of free space and \( m \) is the pole strength of the magnet.

4. **Calculate the Magnetic Intensities**:
- For \( P_1 \) (broad side):
\[ H_{P_1} = \frac{\mu_0 \cdot m}{(10)^3} = \frac{\mu_0 \cdot m}{1000} \]
- For \( P_2 \) (end side):
\[ H_{P_2} = \frac{\mu_0 \cdot 2 \cdot m}{(10)^3} = \frac{2 \mu_0 \cdot m}{1000} \]

5. **Find the Ratio**:
- The ratio of magnetic intensities \( \frac{H_{P_1}}{H_{P_2}} \) is:
\[ \frac{H_{P_1}}{H_{P_2}} = \frac{\frac{\mu_0 \cdot m}{1000}}{\frac{2 \mu_0 \cdot m}{1000}} = \frac{1}{2} \]

6. **Final Result**:
- Therefore, the ratio of magnetic intensities at points \( P_1 \) and \( P_2 \) is:
\[ \frac{H_{P_1}}{H_{P_2}} = \frac{1}{2} \]

### Conclusion:
The ratio of magnetic intensities \( H \) at \( P_1 \) and \( P_2 \) is \( \frac{1}{2} \).