The magnetic potential at a point along the axis of a short magnetic dipole is

Options

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

Answer

To find the magnetic potential at a point along the axis of a short magnetic dipole, we can follow these steps:

### Step 1: Understand the Magnetic Dipole
A magnetic dipole consists of two equal and opposite magnetic poles, denoted as +m (north pole) and -m (south pole), separated by a distance of 2d (where d is half the length of the dipole).

### Step 2: Define the Point of Interest
Let’s consider a point P located at a distance r from the center of the dipole along the axis of the dipole. The distance from the north pole to point P is (r - d) and from the south pole to point P is (r + d).

### Step 3: Calculate the Potential Due to Each Pole
The magnetic potential \( V \) at a point due to a magnetic pole is given by the formula:
\[ V = \frac{\mu_0 m}{4 \pi r} \]
For the north pole at distance \( r - d \):
\[ V_N = \frac{\mu_0 m}{4 \pi (r - d)} \]
For the south pole at distance \( r + d \):
\[ V_S = -\frac{\mu_0 m}{4 \pi (r + d)} \]
(Note the negative sign for the south pole.)

### Step 4: Find the Net Potential
The total magnetic potential \( V \) at point P is the sum of the potentials due to both poles:
\[ V = V_N + V_S = \frac{\mu_0 m}{4 \pi (r - d)} - \frac{\mu_0 m}{4 \pi (r + d)} \]

### Step 5: Combine the Potentials
To combine the potentials, we need a common denominator:
\[ V = \frac{\mu_0 m}{4 \pi} \left( \frac{(r + d) - (r - d)}{(r - d)(r + d)} \right) \]
This simplifies to:
\[ V = \frac{\mu_0 m}{4 \pi} \left( \frac{2d}{r^2 - d^2} \right) \]

### Step 6: Simplify Under the Short Dipole Approximation
For a short magnetic dipole, we assume that \( r \) is much greater than \( d \) (i.e., \( r \gg d \)). Therefore, we can neglect \( d^2 \) in the denominator:
\[ V \approx \frac{\mu_0 m}{4 \pi} \left( \frac{2d}{r^2} \right) \]
Since \( 2d \) is the magnetic length \( l \) of the dipole, we can express it as:
\[ V \approx \frac{\mu_0 m l}{4 \pi r^2} \]

### Step 7: Final Expression
Thus, the magnetic potential at a point along the axis of a short magnetic dipole is:
\[ V = \frac{\mu_0 m}{4 \pi r^2} \]
where \( m \) is the magnetic moment of the dipole.