The earth's magnetic field lines resemble that of a dipole at the centre of the earth . If the magnetic moment of this dipole is close to `8xx10^(22) Am^(2)` , the value of earth's magnetic field near the equator is closed to (radius of the earth `=6.4xx10^(6)m)`

Options

A. `0.6` Gauss
B. `1.2` Gauss
C. `1.8` Gauss
D. `0.32` Gauss

Explanation

To solve the problem of finding the Earth's magnetic field near the equator given the magnetic moment and the radius of the Earth, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Values**:
- Magnetic moment \( M = 8 \times 10^{-22} \, \text{Am}^2 \)
- Radius of the Earth \( R = 6.4 \times 10^6 \, \text{m} \)

2. **Use the Formula for the Magnetic Field of a Dipole**:
The magnetic field \( B \) at a distance \( d \) from a magnetic dipole is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

3. **Substituting the Values**:
Here, we will take \( d \) as the radius of the Earth \( R \):
\[ B = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times (8 \times 10^{-22})}{(6.4 \times 10^6)^3} \]
Simplifying this gives:
\[ B = 10^{-7} \cdot \frac{16 \times 10^{-22}}{(6.4 \times 10^6)^3} \]

4. **Calculating \( (6.4 \times 10^6)^3 \)**:
\[ (6.4 \times 10^6)^3 = 6.4^3 \times 10^{18} \approx 262.144 \times 10^{18} \approx 2.62144 \times 10^{20} \]

5. **Substituting Back into the Equation**:
Now substituting this back into the equation for \( B \):
\[ B = 10^{-7} \cdot \frac{16 \times 10^{-22}}{2.62144 \times 10^{20}} \]

6. **Calculating the Magnetic Field**:
\[ B = 10^{-7} \cdot \frac{16 \times 10^{-22}}{2.62144 \times 10^{20}} = 10^{-7} \cdot \frac{16}{2.62144} \times 10^{-2} \]
\[ B \approx 10^{-7} \cdot 6.09 \times 10^{-2} \approx 6.09 \times 10^{-9} \, \text{T} \]
Converting to Gauss (1 T = 10^4 Gauss):
\[ B \approx 0.609 \, \text{Gauss} \]

7. **Final Answer**:
The value of the Earth's magnetic field near the equator is approximately \( 0.6 \, \text{Gauss} \).