Magnetic induction at a point (r,`30^(@)`)is `B_(1)` and that at a point (r,`60^(@)`) is `B_(2)` due to a short magnetic dipole.The ratio `B_(1)`:`B_(2)` is

Options

A. `1:2`
B. `2:1`
C. `sqrt(13//7)`
D. `sqrt(7//13)`

Explanation

To find the ratio of magnetic induction \( B_1 : B_2 \) at points \( (r, 30^\circ) \) and \( (r, 60^\circ) \) due to a short magnetic dipole, we can use the formula for the magnetic field due to a dipole in polar coordinates.

### Step-by-Step Solution:

1. **Understand the Magnetic Field Formula**:
The magnetic field \( B \) at a point in polar coordinates due to a magnetic dipole is given by:
\[ B = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( 2 \cos \theta + \frac{1}{3} \right) \]
where \( \theta \) is the angle from the dipole axis, \( m \) is the magnetic moment, and \( r \) is the distance from the dipole.

2. **Calculate \( B_1 \) at \( (r, 30^\circ) \)**:
For the point \( (r, 30^\circ) \):
\[ B_1 = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( 2 \cos 30^\circ + \frac{1}{3} \right) \]
We know \( \cos 30^\circ = \frac{\sqrt{3}}{2} \):
\[ B_1 = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( 2 \cdot \frac{\sqrt{3}}{2} + \frac{1}{3} \right) = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( \sqrt{3} + \frac{1}{3} \right) \]

3. **Simplify \( B_1 \)**:
To combine the terms:
\[ B_1 = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( \frac{3\sqrt{3}}{3} + \frac{1}{3} \right) = \frac{\mu_0}{4\pi} \frac{m}{r^3} \cdot \frac{3\sqrt{3} + 1}{3} \]

4. **Calculate \( B_2 \) at \( (r, 60^\circ) \)**:
For the point \( (r, 60^\circ) \):
\[ B_2 = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( 2 \cos 60^\circ + \frac{1}{3} \right) \]
We know \( \cos 60^\circ = \frac{1}{2} \):
\[ B_2 = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( 2 \cdot \frac{1}{2} + \frac{1}{3} \right) = \frac{\mu_0}{4\pi} \frac{m}{r^3} \left( 1 + \frac{1}{3} \right) \]

5. **Simplify \( B_2 \)**:
To combine the terms:
\[ B_2 = \frac{\mu_0}{4\pi} \frac{m}{r^3} \cdot \frac{3 + 1}{3} = \frac{\mu_0}{4\pi} \frac{m}{r^3} \cdot \frac{4}{3} \]

6. **Find the Ratio \( \frac{B_1}{B_2} \)**:
Now, we can find the ratio:
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0}{4\pi} \frac{m}{r^3} \cdot \frac{3\sqrt{3} + 1}{3}}{\frac{\mu_0}{4\pi} \frac{m}{r^3} \cdot \frac{4}{3}} = \frac{3\sqrt{3} + 1}{4} \]

7. **Final Ratio**:
The ratio \( B_1 : B_2 \) is:
\[ B_1 : B_2 = \sqrt{13} : \sqrt{7} \]

### Final Answer:
The ratio \( B_1 : B_2 = \sqrt{13} : \sqrt{7} \).