If `B_(E)` represents equatorial magnetic field and `B_(A)` represents axial magnetic field due to a bar magnet. Which of the following relationships between `B_(E)` and `B_(A)` is correct ?

Options

A. `B_(E)=2B _(A)`
B. `B_(A)= 2B_(E)`
C. `B _(E)=4B_(A)`
D. `B_(A)=4B_(E)`

Explanation

To solve the problem, we need to compare the expressions for the equatorial magnetic field \( B_E \) and the axial magnetic field \( B_A \) of a bar magnet.

### Step 1: Write the formula for the equatorial magnetic field \( B_E \)
The equatorial magnetic field \( B_E \) due to a bar magnet at a distance \( r \) from its center is given by the formula:
\[ B_E = \frac{\mu_0 m}{4 \pi r^3} \]
where \( \mu_0 \) is the permeability of free space, and \( m \) is the magnetic moment of the bar magnet.

### Step 2: Write the formula for the axial magnetic field \( B_A \)
The axial magnetic field \( B_A \) due to a bar magnet at a distance \( r \) from its center is given by the formula:
\[ B_A = \frac{\mu_0 m \cdot 2}{4 \pi r^3} \]

### Step 3: Relate \( B_E \) and \( B_A \)
Now we can relate \( B_E \) and \( B_A \) using the formulas derived in the previous steps. We can express \( B_A \) in terms of \( B_E \):
\[ B_A = 2 \cdot B_E \]

### Conclusion
From the relationship we derived, we can conclude that:
\[ B_A = 2 B_E \]
Thus, the correct relationship between \( B_E \) and \( B_A \) is that the axial magnetic field \( B_A \) is twice the equatorial magnetic field \( B_E \).

### Final Answer
The correct relationship is:
\[ B_A = 2 B_E \]