The magnetic field at a point x on the axis of a small bar magnet is equal to the field at a point y on the equator of the same magnet. The ratio of the distances of `x` and `y` from the centre of the magnet is

Options

A. (a)`2^(-3)`
B. (b)`2^(-1//3)`
C. (c)`2^(3)`
D. (d)`2^(1//3)`

Answer

To solve the problem, we need to find the ratio of the distances of points \( x \) and \( y \) from the center of a small bar magnet, given that the magnetic field at point \( x \) on the axis of the magnet is equal to the magnetic field at point \( y \) on the equator of the same magnet.

### Step-by-Step Solution:

1. **Understanding Magnetic Field Formulas**:
- The magnetic field \( B_1 \) at a point on the axis of a bar magnet is given by:
\[ B_1 = \frac{2M}{x^3} \]
where \( M \) is the magnetic moment of the magnet and \( x \) is the distance from the center of the magnet to point \( x \).

- The magnetic field \( B_2 \) at a point on the equator of a bar magnet is given by:
\[ B_2 = \frac{M}{2y^3} \]
where \( y \) is the distance from the center of the magnet to point \( y \).

2. **Setting the Fields Equal**:
- According to the problem, the magnetic fields at points \( x \) and \( y \) are equal:
\[ B_1 = B_2 \]
- Substituting the expressions for \( B_1 \) and \( B_2 \):
\[ \frac{2M}{x^3} = \frac{M}{2y^3} \]

3. **Canceling the Magnetic Moment**:
- We can cancel \( M \) from both sides (assuming \( M \neq 0 \)):
\[ \frac{2}{x^3} = \frac{1}{2y^3} \]

4. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 2 \cdot 2y^3 = x^3 \]
\[ 4y^3 = x^3 \]

5. **Finding the Ratio**:
- Taking the cube root of both sides:
\[ \frac{x}{y} = \sqrt[3]{4} \]
- This can be expressed as:
\[ \frac{x}{y} = 2^{2/3} \]

6. **Conclusion**:
- Therefore, the ratio of the distances of \( x \) and \( y \) from the center of the magnet is:
\[ \frac{x}{y} = 2^{2/3} \]

### Final Answer:
The ratio of the distances \( x \) and \( y \) from the center of the magnet is \( 2^{2/3} \).

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