The magnetic potential at a point due to pole of strength m at a distance r is

Options

A. `V=(mu_(0))/(4pi) (m)/(r)`
B. `V=(mu_(0))/(pir) (m)/(r^(2))`
C. `V=(mu_(0))/(4pi)(m)/(r^(3))`
D. `V=(mu_(0))/(4pi) (2m)/(r)`

Answer

To find the magnetic potential \( V \) at a point due to a magnetic pole of strength \( m \) at a distance \( r \), we can use the analogy between electrostatics and magnetism. The formula for electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:

\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]

Where:
- \( \epsilon_0 \) is the permittivity of free space.

In magnetism, we can draw an analogy where:
- The magnetic pole strength \( m \) corresponds to the electric charge \( Q \).
- The constant \( \frac{1}{4 \pi \epsilon_0} \) corresponds to \( \frac{\mu_0}{4 \pi} \), where \( \mu_0 \) is the permeability of free space.

Thus, the magnetic potential \( V_m \) at a distance \( r \) from a magnetic pole of strength \( m \) can be expressed as:

\[ V_m = \frac{\mu_0}{4 \pi} \frac{m}{r} \]

### Step-by-step Solution:
1. **Identify the analogous quantities**: Recognize that the magnetic pole strength \( m \) corresponds to the electric charge \( Q \), and the constants in the formulas are also analogous.

2. **Write the electric potential formula**: Recall the formula for electric potential due to a point charge:
\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]

3. **Replace quantities with their magnetic counterparts**: Substitute \( Q \) with \( m \) and \( \frac{1}{4 \pi \epsilon_0} \) with \( \frac{\mu_0}{4 \pi} \):
\[ V_m = \frac{\mu_0}{4 \pi} \frac{m}{r} \]

4. **Final expression**: The magnetic potential at a distance \( r \) from a magnetic pole of strength \( m \) is:
\[ V_m = \frac{\mu_0}{4 \pi} \frac{m}{r} \]