The torque and magnetic potential energy of a magnetic dipole in most stable position ina uniform magnetic field `vec(B)` having magnetic moment `vec(m)` will be

Options

A. `-mB,zero`
B. `mB,zero`
C. `zero,mB`
D. `zero,-mB`

Explanation

To solve the problem regarding the torque and magnetic potential energy of a magnetic dipole in its most stable position in a uniform magnetic field, we will follow these steps:

### Step 1: Understand the Definitions
- **Torque (\(\tau\))**: The torque experienced by a magnetic dipole in a magnetic field is given by the formula:
\[ \tau = \vec{m} \times \vec{B} \]
or in magnitude form:
\[ \tau = mB \sin \theta \]
where \(m\) is the magnetic moment, \(B\) is the magnetic field strength, and \(\theta\) is the angle between \(\vec{m}\) and \(\vec{B}\).

- **Magnetic Potential Energy (\(U_m\))**: The potential energy of a magnetic dipole in a magnetic field is given by:
\[ U_m = -\vec{m} \cdot \vec{B} \]
or in magnitude form:
\[ U_m = -mB \cos \theta \]

### Step 2: Determine the Most Stable Position
- The most stable position of a magnetic dipole in a magnetic field occurs when the dipole is aligned with the field, which corresponds to \(\theta = 0^\circ\).

### Step 3: Calculate Torque at \(\theta = 0^\circ\)
- Substitute \(\theta = 0^\circ\) into the torque equation:
\[ \tau = mB \sin(0^\circ) = mB \cdot 0 = 0 \]
- Therefore, the torque in the most stable position is:
\[ \tau = 0 \]

### Step 4: Calculate Magnetic Potential Energy at \(\theta = 0^\circ\)
- Substitute \(\theta = 0^\circ\) into the potential energy equation:
\[ U_m = -mB \cos(0^\circ) = -mB \cdot 1 = -mB \]
- Therefore, the magnetic potential energy in the most stable position is:
\[ U_m = -mB \]

### Final Answer
- The torque and magnetic potential energy of a magnetic dipole in its most stable position in a uniform magnetic field are:
\[ \text{Torque} = 0, \quad \text{Magnetic Potential Energy} = -mB \]