The potential enargy of bar magnet of magnetic moment `8 Am ^(2)` placed in a uniform magnetic field of 2 T at an angle of `120^(@)` is equal to

Options

A. `-16J`
B. `16 J`
C. `-8J`
D. `8 J`

Answer

To find the potential energy of a bar magnet in a uniform magnetic field, we can use the formula:

\[ U = -\vec{m} \cdot \vec{B} = -mB \cos(\theta) \]

where:
- \( U \) is the potential energy,
- \( \vec{m} \) is the magnetic moment,
- \( \vec{B} \) is the magnetic field,
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

### Step 1: Identify the given values
- Magnetic moment, \( m = 8 \, \text{Am}^2 \)
- Magnetic field, \( B = 2 \, \text{T} \)
- Angle, \( \theta = 120^\circ \)

### Step 2: Calculate \( \cos(120^\circ) \)
The cosine of \( 120^\circ \) is:
\[ \cos(120^\circ) = -\frac{1}{2} \]

### Step 3: Substitute the values into the potential energy formula
Now we can substitute the values into the potential energy formula:
\[ U = -mB \cos(\theta) = - (8 \, \text{Am}^2)(2 \, \text{T}) \left(-\frac{1}{2}\right) \]

### Step 4: Simplify the expression
Calculating the expression step by step:
\[ U = - (8 \times 2) \left(-\frac{1}{2}\right) = -16 \left(-\frac{1}{2}\right) = 8 \, \text{J} \]

### Final Result
The potential energy of the bar magnet in the magnetic field is:
\[ U = 8 \, \text{J} \]