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,
- \( m \) is the magnetic moment,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

Given:
- Magnetic moment \( m = 8 \, \text{Am}^2 \)
- Magnetic field \( B = 2 \, \text{T} \)
- Angle \( \theta = 120^\circ \)

### Step 1: Substitute the values into the formula

We substitute the given values into the potential energy formula:

\[ U = -mB \cos \theta \]

\[ U = - (8 \, \text{Am}^2)(2 \, \text{T}) \cos(120^\circ) \]

### Step 2: Calculate \( \cos(120^\circ) \)

Next, we need to find \( \cos(120^\circ) \). We know that:

\[ \cos(120^\circ) = \cos(180^\circ - 60^\circ) = -\cos(60^\circ) \]

And since \( \cos(60^\circ) = \frac{1}{2} \):

\[ \cos(120^\circ) = -\frac{1}{2} \]

### Step 3: Substitute \( \cos(120^\circ) \) back into the equation

Now we substitute \( \cos(120^\circ) \) back into our equation for potential energy:

\[ U = - (8)(2) \left(-\frac{1}{2}\right) \]

### Step 4: Simplify the expression

Now we simplify the expression:

\[ U = -16 \left(-\frac{1}{2}\right) \]

\[ U = 16 \cdot \frac{1}{2} = 8 \, \text{J} \]

### Final Answer

Thus, the potential energy of the bar magnet is:

\[ U = 8 \, \text{J} \]

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