A short bar magnet has a magnetic moment of `0.39` J `"T"^(-1)`. The magnitude and direction of the magnetic field produced by the magnet at a distance of 20 cm from the centre of the magnet on the equatorial line of the magnet is

Options

A. 0.049 G, N-S direction
B. 4.95 G, S-N direction
C. 0.0195 G, S-N direction
D. 19.5 G, N-S direction

Explanation

To find the magnitude and direction of the magnetic field produced by a short bar magnet at a distance of 20 cm from its center on the equatorial line, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Magnetic moment (M) = 0.39 J/T
- Distance (d) = 20 cm = 0.2 m

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

3. **Neglect the Length of the Magnet:**
Since the magnet is short, we can neglect its length (L) in our calculations. Thus, we can simplify the formula to:
\[ B_E = \frac{\mu_0}{4\pi} \cdot \frac{M}{d^3} \]

4. **Substitute the Values into the Formula:**
- First, calculate \(\mu_0 / 4\pi\):
\[ \frac{\mu_0}{4\pi} = 10^{-7} \, \text{T m/A} \]
- Now substitute M and d into the formula:
\[ B_E = 10^{-7} \cdot \frac{0.39}{(0.2)^3} \]

5. **Calculate \(d^3\):**
\[ (0.2)^3 = 0.008 \, \text{m}^3 \]

6. **Calculate the Magnetic Field:**
\[ B_E = 10^{-7} \cdot \frac{0.39}{0.008} = 10^{-7} \cdot 48.75 = 0.049 \times 10^{-4} \, \text{T} \]

7. **Convert Tesla to Gauss:**
- Since \(1 \, \text{T} = 10^4 \, \text{Gauss}\):
\[ B_E = 0.049 \times 10^{-4} \, \text{T} = 0.049 \, \text{Gauss} \]

8. **Determine the Direction of the Magnetic Field:**
- The magnetic field direction on the equatorial line of a bar magnet is from the north pole to the south pole of the magnet.

### Final Answer:
- The magnitude of the magnetic field at a distance of 20 cm from the center of the magnet on the equatorial line is **0.049 Gauss**.
- The direction of the magnetic field is from the **north pole to the south pole** of the magnet.