The magnetic field existing in a region is given by $$\vec{B} = 0.2(1 + 2x)\hat{k}\ T$$. A square loop of edge 50 cm carrying 0.5 A current is placed in x−y plane with its edges parallel to the x-y axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is ______ mN.

For horizontal segments parallel to x-axis:

$$d\vec{F} = I(dx\,\hat{i} \times \vec{B})$$

$$\vec{F}_{\text{top}} + \vec{F}_{\text{bottom}} = 0$$

For vertical segment at $$x_1 = 2$$:

$$\vec{F}_1 = I\vec{L}_1 \times \vec{B}(x_1)$$

$$\vec{F}_1 = 0.5(-0.5\hat{j}) \times [0.2(1 + 2(2))\hat{k}]$$

$$\vec{F}_1 = -0.25\hat{j} \times 1\hat{k} = -0.25\hat{i}$$

For vertical segment at $$x_2 = 2 + 0.5 = 2.5$$:

$$\vec{F}_2 = I\vec{L}_2 \times \vec{B}(x_2)$$

$$\vec{F}_2 = 0.5(0.5\hat{j}) \times [0.2(1 + 2(2.5))\hat{k}]$$

$$\vec{F}_2 = 0.25\hat{j} \times 1.2\hat{k} = 0.30\hat{i}$$

$$\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2$$

$$\vec{F}_{\text{net}} = -0.25\hat{i} + 0.30\hat{i} = 0.05\hat{i}$$

$$|\vec{F}_{\text{net}}| = 0.05\text{ N} = 50\text{ mN}$$