Two short bar magnets of length `1cm` each have magnetic moments `1.20 Am^(2) and 1.00 Am^(2)` respectively. They are placed on a horizontal table parallel to each other with their `N` poles pointing towards the south. They have a common magnetic equator and are separted by a distance of `20.0 cm`. The value of the resultant horizontal magnetic induction at the mid - point `O` of the line joining their centres is close to (Horizontal component of earths magnetic induction is `3.6 xx 10.5 Wh//m^(2)`

Options

A. ` 3.6 xx 10.5 Wh//m^(2)`
B. ` 2.56 xx 10.4 Wh//m^(2)`
C. ` 3.50 xx 10.4 Wh//m^(2)`
D. ` 5.80 xx 10.4 Wh//m^(2)`

Explanation

To solve the problem, we need to calculate the resultant magnetic induction at the midpoint O between the two bar magnets. Here are the steps to find the solution:

### Step 1: Understand the Configuration
We have two bar magnets with magnetic moments \( M_1 = 1.20 \, \text{Am}^2 \) and \( M_2 = 1.00 \, \text{Am}^2 \). They are placed parallel to each other with their north poles pointing towards the south. The distance between the magnets is \( 20.0 \, \text{cm} \).

### Step 2: Determine the Midpoint
The midpoint O is located halfway between the two magnets. Therefore, the distance from each magnet to the midpoint is:
\[ R = \frac{20.0 \, \text{cm}}{2} = 10.0 \, \text{cm} = 0.1 \, \text{m} \]

### Step 3: Calculate the Magnetic Field Due to Each Magnet
The magnetic field \( B \) at a distance \( R \) from a magnetic dipole is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{R^3} \]
where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \).

#### For Magnet 1:
Using \( M_1 = 1.20 \, \text{Am}^2 \):
\[ B_1 = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times 1.20}{(0.1)^3} = 10^{-7} \cdot \frac{2.4}{0.001} = 2.4 \times 10^{-4} \, \text{T} \]

#### For Magnet 2:
Using \( M_2 = 1.00 \, \text{Am}^2 \):
\[ B_2 = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times 1.00}{(0.1)^3} = 10^{-7} \cdot \frac{2.0}{0.001} = 2.0 \times 10^{-4} \, \text{T} \]

### Step 4: Determine the Direction of the Magnetic Fields
Since both magnets have their north poles pointing towards the south, the magnetic field \( B_1 \) from magnet 1 at point O will point towards the north (upward), and the magnetic field \( B_2 \) from magnet 2 will also point towards the north (upward).

### Step 5: Calculate the Resultant Magnetic Field
The resultant magnetic field \( B_{\text{net}} \) at point O is the sum of the individual magnetic fields:
\[ B_{\text{net}} = B_1 + B_2 = 2.4 \times 10^{-4} + 2.0 \times 10^{-4} = 4.4 \times 10^{-4} \, \text{T} \]

### Step 6: Include the Earth's Magnetic Field
The horizontal component of the Earth's magnetic field is given as \( B_H = 3.6 \times 10^{-5} \, \text{T} \). Since the Earth's magnetic field is also directed towards the north, we add it to the resultant magnetic field:
\[ B_{\text{total}} = B_{\text{net}} + B_H = 4.4 \times 10^{-4} + 3.6 \times 10^{-5} = 4.76 \times 10^{-4} \, \text{T} \]

### Step 7: Final Result
The resultant horizontal magnetic induction at the midpoint O is approximately:
\[ B_{\text{total}} \approx 4.76 \times 10^{-4} \, \text{T} \text{ or } 4.76 \times 10^{-4} \, \text{Wb/m}^2 \]