The ratio of magnitude of magnetic induction (B)at any point due to short magnetic dipole and the magnetic potential (V) at that point is given by

Options

A. `(B)/(V)=(sqrt(3cos^(2)theta+1))/(rcostheta)`
B. `(B)/(V)=(r costheta)/(sqrt(3cos^(2)theta+1))`
C. `(B)/(V)=(sqrt(2cos^(2)theta+1))/(rcostheta)`
D. `(B)/(V)=(sqrt(cos^(2)theta+1))/(rcostheta)`

Answer

To find the ratio of the magnitude of magnetic induction (B) at any point due to a short magnetic dipole and the magnetic potential (V) at that point, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Magnetic induction (B) at a point due to a magnetic dipole is analogous to the electric field due to an electric dipole.
- Magnetic potential (V) at a point due to a magnetic dipole is analogous to the electric potential due to an electric dipole.

2. **Use the Formulas for Electric Dipole**:
- The electric field (E) at a point due to an electric dipole is given by:
\[ E = \frac{P}{4 \pi \epsilon_0 r^3} (3 \cos^2 \theta + 1) \]
- The electric potential (V) at a point due to an electric dipole is given by:
\[ V = \frac{P \cos \theta}{4 \pi \epsilon_0 r^2} \]

3. **Analogous Relationships**:
- For a magnetic dipole, we can replace the electric dipole moment (P) with the magnetic dipole moment (M).
- The constant \( \frac{1}{4 \pi \epsilon_0} \) is replaced by \( \frac{\mu_0}{4 \pi} \) for magnetic dipoles.

4. **Write the Formulas for Magnetic Dipole**:
- The magnetic induction (B) at a point due to a magnetic dipole:
\[ B = \frac{M}{4 \pi r^3} (3 \cos^2 \theta + 1) \]
- The magnetic potential (V) at a point due to a magnetic dipole:
\[ V = \frac{M \cos \theta}{4 \pi r^2} \]

5. **Calculate the Ratio \( \frac{B}{V} \)**:
- Now, we can find the ratio of B to V:
\[ \frac{B}{V} = \frac{\frac{M}{4 \pi r^3} (3 \cos^2 \theta + 1)}{\frac{M \cos \theta}{4 \pi r^2}} \]
- Simplifying this expression:
\[ \frac{B}{V} = \frac{(3 \cos^2 \theta + 1)}{\cos \theta \cdot r} \]
- This gives us:
\[ \frac{B}{V} = \frac{3 \cos^2 \theta + 1}{r \cos \theta} \]

6. **Final Result**:
- The ratio of the magnitude of magnetic induction (B) at any point due to a short magnetic dipole to the magnetic potential (V) at that point is:
\[ \frac{B}{V} = \frac{3 \cos^2 \theta + 1}{r \cos \theta} \]