A ring of mass 'm' and radius 'r' carrying current `i_(0)` is lying in the X-Y plane with centre at origin. A uniform magnetic field of strength `B=B_(0) (2hat(i)-3hatj+5hatk)T` is applied in the region. If the ring rotates about the axid initial magnetic energy stored in the ring will be. (In joules)

Options

A. `-pi^(2)i_(0)B_(0)`
B. `-5pir^(2)i_(0)B_(0)`
C. `-3pir^(2)i_(0)B_(0)`
D. `-2pir^(2)i_(0)B_(0)`

Explanation

To solve the problem, we need to calculate the initial magnetic energy stored in the ring that is carrying a current and is placed in a uniform magnetic field. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Identify the Magnetic Moment (μ)**:
The magnetic moment (μ) of a current-carrying loop is given by the formula:
\[ \mu = I \cdot A \]
where \(I\) is the current flowing through the loop and \(A\) is the area of the loop. For a ring of radius \(r\):
\[ A = \pi r^2 \]
Therefore, the magnetic moment of the ring is:
\[ \mu = I_0 \cdot \pi r^2 \]
The direction of the magnetic moment will be along the z-axis (k direction) since the ring lies in the XY plane.

2. **Given Magnetic Field (B)**:
The magnetic field is given as:
\[ \mathbf{B} = B_0 (2\hat{i} - 3\hat{j} + 5\hat{k}) \, \text{T} \]

3. **Calculate the Magnetic Energy (U)**:
The magnetic energy stored in a magnetic moment in a magnetic field is given by the formula:
\[ U = -\mathbf{\mu} \cdot \mathbf{B} \]
Substituting the values we have:
\[ U = -\left(I_0 \cdot \pi r^2 \hat{k}\right) \cdot \left(B_0 (2\hat{i} - 3\hat{j} + 5\hat{k})\right) \]

4. **Evaluate the Dot Product**:
The dot product can be calculated as follows:
\[ \mathbf{\mu} \cdot \mathbf{B} = (I_0 \cdot \pi r^2) \cdot B_0 \cdot 5 \]
Since the only component that contributes to the dot product is along the k direction:
\[ \mathbf{\mu} \cdot \mathbf{B} = I_0 \cdot \pi r^2 \cdot 5B_0 \]

5. **Final Expression for Magnetic Energy**:
Thus, the magnetic energy stored in the ring is:
\[ U = -5 I_0 \pi r^2 B_0 \]

### Final Answer:
The initial magnetic energy stored in the ring is:
\[ U = -5 \pi r^2 I_0 B_0 \, \text{Joules} \]