Two bar magnets of length 0.1m and poles strength 75m each, are placed on the same line. The distance between their centres is 0.2 m. what is the reultant force due to one on the other when (i) the north pole of one faces the south pole of the other, (ii) the north pole of one faces the north pole of the other ?

Answer

To solve the problem of finding the resultant force between two bar magnets, we will use the formula for the magnetic force between two magnetic poles. The steps are as follows:

### Given Data:
- Length of each magnet (l) = 0.1 m
- Pole strength of each magnet (m) = 75 A·m
- Distance between the centers of the magnets (d) = 0.2 m

### Step 1: Calculate the Magnetic Moment (m) of Each Magnet
The magnetic moment (m) of a bar magnet is given by the product of its pole strength and length:
\[ m = \text{pole strength} \times \text{length} = 75 \, \text{A·m} \times 0.1 \, \text{m} = 7.5 \, \text{A·m}^2 \]

### Step 2: Calculate the Force Between the Magnets
The force \( F \) between two magnetic poles can be calculated using the formula:
\[ F = \frac{\mu_0}{4\pi} \cdot \frac{m_1 \cdot m_2}{r^4} \]
where:
- \( \mu_0 \) (permeability of free space) = \( 4\pi \times 10^{-7} \, \text{T·m/A} \)
- \( m_1 = m_2 = 7.5 \, \text{A·m}^2 \)
- \( r = d = 0.2 \, \text{m} \)

Substituting the values:
\[ F = \frac{(4\pi \times 10^{-7})}{4\pi} \cdot \frac{7.5 \cdot 7.5}{(0.2)^4} \]
\[ F = 10^{-7} \cdot \frac{56.25}{0.0016} \]
\[ F = 10^{-7} \cdot 35156.25 \]
\[ F = 3.515625 \times 10^{-3} \, \text{N} \approx 3.52 \times 10^{-3} \, \text{N} \]

### Step 3: Determine the Direction of the Force
1. **When the north pole of one magnet faces the south pole of the other**:
- The force will be attractive. Thus, the resultant force is \( 3.52 \times 10^{-3} \, \text{N} \) directed towards each other.

2. **When the north pole of one magnet faces the north pole of the other**:
- The force will be repulsive. The magnitude remains the same, \( 3.52 \times 10^{-3} \, \text{N} \), but the direction is away from each other.

### Final Results:
- (i) Resultant force when north pole faces south pole: \( 3.52 \times 10^{-3} \, \text{N} \) (attractive)
- (ii) Resultant force when north pole faces north pole: \( 3.52 \times 10^{-3} \, \text{N} \) (repulsive)