Potential energy of a bar mangnet of magnetic moment M placed in a magnetic field of induction B such that it makes an angle `theta` with the direction of B is (take `theta =90^(@)` as datum)

Options

A. `-M B sin theta `
B. `-M B cos theta`
C. `MB (1- cos theta)`
D. `M B (1 + cos theta)`

Answer

To find the potential energy \( U \) of a bar magnet with magnetic moment \( M \) placed in a magnetic field \( B \) at an angle \( \theta \) with respect to the direction of the magnetic field, we can follow these steps:

### Step 1: Understand the Reference Level
We take \( \theta = 90^\circ \) as the reference level where the potential energy is defined to be zero. This means that when the magnetic moment is perpendicular to the magnetic field, the potential energy \( U = 0 \).

### Step 2: Write the Expression for Potential Energy
The potential energy \( U \) of a magnetic dipole in a magnetic field is given by the formula:
\[ U = -\vec{M} \cdot \vec{B} \]
Where \( \vec{M} \) is the magnetic moment and \( \vec{B} \) is the magnetic field.

### Step 3: Use the Dot Product
The dot product of two vectors can be expressed as:
\[ \vec{M} \cdot \vec{B} = MB \cos \theta \]
Thus, substituting this into the potential energy expression gives:
\[ U = -MB \cos \theta \]

### Step 4: Identify the Angle
Since we are considering the angle \( \theta \) with respect to the magnetic field, we can see that when \( \theta = 90^\circ \), \( \cos(90^\circ) = 0 \), which confirms that \( U = 0 \) at this angle.

### Step 5: Final Expression
Therefore, the potential energy of the bar magnet at any angle \( \theta \) is:
\[ U = -MB \cos \theta \]

### Conclusion
The potential energy of a bar magnet of magnetic moment \( M \) placed in a magnetic field \( B \) at an angle \( \theta \) is given by:
\[ U = -MB \cos \theta \]