The magnetic field existing in a region is gicen by `vecB =B_0(1+(x)/(l))vec k.` A square loop of edge l and carrying a current I, is placed with its edges parallel to the x-y axes. Find the magnitude of the net magnetic force experienced by the loop.

Options

A. `2B_(0)Il`
B. zero
C. `B_(0)Il`
D. `4B_(0)Il`

Answer

To solve the problem, we need to find the net magnetic force experienced by a square loop carrying a current \( I \) in a magnetic field defined by \( \vec{B} = B_0 \left(1 + \frac{x}{l}\right) \hat{k} \). The loop has an edge length \( l \) and is oriented with its edges parallel to the x-y axes.

### Step-by-Step Solution:

1. **Identify the Magnetic Field at Different Edges of the Loop:**
- The magnetic field is given by \( \vec{B} = B_0 \left(1 + \frac{x}{l}\right) \hat{k} \).
- For the edge \( AB \) (where \( x = 0 \)):
\[ B_1 = B_0 \left(1 + \frac{0}{l}\right) = B_0 \]
- For the edge \( CD \) (where \( x = l \)):
\[ B_2 = B_0 \left(1 + \frac{l}{l}\right) = B_0 \left(1 + 1\right) = 2B_0 \]

2. **Determine the Magnetic Force on Each Edge:**
- The magnetic force on a straight current-carrying conductor in a magnetic field is given by \( \vec{F} = I \vec{L} \times \vec{B} \).
- For edge \( AB \) (current flows in the positive y-direction):
\[ \vec{F}_{AB} = I l \hat{j} \times B_1 \hat{k} = I l B_0 \hat{i} \]
- For edge \( CD \) (current flows in the negative y-direction):
\[ \vec{F}_{CD} = -I l \hat{j} \times B_2 \hat{k} = -I l (2B_0) \hat{i} = -2 I l B_0 \hat{i} \]

3. **Calculate the Net Force on the Loop:**
- The forces on edges \( AD \) and \( BC \) will cancel each other out because the magnetic field does not vary along these edges (the magnetic field is the same at both edges).
- Therefore, the net force on the loop is:
\[ \vec{F}_{net} = \vec{F}_{CD} + \vec{F}_{AB} = (-2 I l B_0 + I l B_0) \hat{i} = -I l B_0 \hat{i} \]
- The magnitude of the net magnetic force is:
\[ F_{net} = | -I l B_0 | = I l B_0 \]

4. **Conclusion:**
- The magnitude of the net magnetic force experienced by the loop is:
\[ F_{net} = I l B_0 \]

### Answer:
The magnitude of the net magnetic force experienced by the loop is \( I l B_0 \).