A planar coil having 12 turns carries 15 A current. The coil is oriented with respect to the uniform magnetic field `B= 0.2hati T` such that its directed area is `A=0.04 hatim^(2).` the potential energy of the coil in the given orientation is

Options

A. 0
B. `+0.72 J`
C. `+1.44 J`
D. `- 1.44 J`

Answer

To find the potential energy of a planar coil in a magnetic field, we can use the formula for the potential energy \( U \) of a magnetic moment \( \mathbf{m} \) in a magnetic field \( \mathbf{B} \):

\[ U = -\mathbf{m} \cdot \mathbf{B} \]

Where:
- \( \mathbf{m} \) is the magnetic moment of the coil, given by \( \mathbf{m} = n \cdot I \cdot \mathbf{A} \)
- \( n \) is the number of turns in the coil
- \( I \) is the current flowing through the coil
- \( \mathbf{A} \) is the area vector of the coil
- \( \mathbf{B} \) is the magnetic field vector

### Step-by-Step Solution:

1. **Identify the given values:**
- Number of turns \( n = 12 \)
- Current \( I = 15 \, \text{A} \)
- Area \( A = 0.04 \, \text{m}^2 \)
- Magnetic field \( B = 0.2 \, \text{T} \)

2. **Calculate the magnetic moment \( \mathbf{m} \):**
\[ \mathbf{m} = n \cdot I \cdot A \]
Substituting the values:
\[ \mathbf{m} = 12 \cdot 15 \cdot 0.04 \]

3. **Perform the multiplication:**
\[ \mathbf{m} = 12 \cdot 15 = 180 \]
\[ \mathbf{m} = 180 \cdot 0.04 = 7.2 \, \text{A m}^2 \]

4. **Calculate the potential energy \( U \):**
Since the area vector \( \mathbf{A} \) and the magnetic field \( \mathbf{B} \) are in the same direction, the angle \( \theta = 0 \) degrees. Therefore, \( \cos(0) = 1 \).

Now, substituting into the potential energy formula:
\[ U = -\mathbf{m} \cdot \mathbf{B} = -mB \cos(0) \]
\[ U = -7.2 \cdot 0.2 \cdot 1 \]

5. **Perform the multiplication:**
\[ U = -1.44 \, \text{J} \]

### Final Answer:
The potential energy of the coil in the given orientation is \( U = -1.44 \, \text{J} \).

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