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 the coil in the given magnetic field, we can follow these steps:

### Step 1: Identify the given values
- Number of turns (N) = 12
- Current (I) = 15 A
- Area (A) = 0.04 m²
- Magnetic field (B) = 0.2 T

### Step 2: Calculate the magnetic moment (m)
The magnetic moment (m) of the coil can be calculated using the formula:
\[ m = N \cdot I \cdot A \]
Substituting the values:
\[ m = 12 \cdot 15 \cdot 0.04 \]

### Step 3: Perform the multiplication
Calculating the above expression:
\[ m = 12 \cdot 15 = 180 \]
\[ m = 180 \cdot 0.04 = 7.2 \, \text{A m}^2 \]

### Step 4: Calculate the potential energy (U)
The potential energy (U) of the coil in the magnetic field is given by the formula:
\[ U = -m \cdot B \cdot \cos(\theta) \]
Since the coil is oriented such that the area vector is in the same direction as the magnetic field, the angle \( \theta = 0^\circ \). Thus, \( \cos(0) = 1 \).
So, the formula simplifies to:
\[ U = -m \cdot B \]

### Step 5: Substitute the values into the potential energy formula
Substituting the values we have:
\[ U = -7.2 \cdot 0.2 \]

### Step 6: Calculate the potential energy
Calculating the above expression:
\[ U = -1.44 \, \text{J} \]

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

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