The magnetic potential at any point on equator of a short magnetic dipole is

Options

A. `V=(mu_(0))/(4pi)(Mcostheta)/(r^(2))`
B. `V=(mu_(0))/(4pi)(M)/(r^(2))`
C. V=0
D. V=M

Explanation

To find the magnetic potential at any point on the equator of a short magnetic dipole, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Magnetic Dipole**:
A short magnetic dipole can be represented by a bar magnet with a North pole (N) and a South pole (S). The magnetic potential is defined around these poles.

2. **Identifying the Equatorial Plane**:
The equatorial plane is the plane that is perpendicular to the axis of the dipole and bisects it. For our bar magnet, this means that at any point on the equator, the distances from the North and South poles are equal.

3. **Distance from the Poles**:
Let’s denote the distance from the North pole to a point on the equator as \( r \). Since the equatorial point is symmetrically placed, the distance from the South pole to the same point is also \( r \).

4. **Calculating Magnetic Potential**:
The magnetic potential \( V \) due to a magnetic dipole at a distance \( r \) from the pole is given by the formula:
\[ V = \frac{\mu_0}{4\pi} \frac{m}{r} \]
where \( m \) is the magnetic moment of the dipole and \( \mu_0 \) is the permeability of free space.

5. **Potential from Each Pole**:
- The potential due to the North pole at the equatorial point is:
\[ V_N = \frac{\mu_0}{4\pi} \frac{m}{r} \]
- The potential due to the South pole at the same point is:
\[ V_S = -\frac{\mu_0}{4\pi} \frac{m}{r} \]
(The negative sign indicates that the potential due to the South pole is in the opposite direction.)

6. **Net Magnetic Potential**:
To find the total magnetic potential at the equatorial point, we add the potentials from both poles:
\[ V_{total} = V_N + V_S = \frac{\mu_0}{4\pi} \frac{m}{r} - \frac{\mu_0}{4\pi} \frac{m}{r} = 0 \]

7. **Conclusion**:
Therefore, the magnetic potential at any point on the equator of a short magnetic dipole is:
\[ V_{total} = 0 \]

### Final Answer:
The magnetic potential at any point on the equator of a short magnetic dipole is **zero**.

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