Problem
A rectangular loop of wire carries $5 \mathrm{~A}$ current and lies in a plane that also contains a very long straight wire carrying 15 A current as shown in the figure. Determine the net force on the loop due to the current flowing through the straight wire. The arrows in the figure show the direction of each current. The permeability of the free space is $4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}$. Give your answer in units of $\mu \mathrm{N}$ (micro newton). Take the direction to the right and to the up as positive for $x$ and $y$ coordinates.
Solution
Step 1 ：Calculate the magnetic field and force on each segment of the loop using Biot-Savart law and force formula: $$B = \frac{\mu_0 I}{2 \pi r}$$ and $$F = I' l B \sin{\theta}$$
Step 2 ：Sum up the forces on each segment to find the net force on the loop: $$F_{net} = F_{left} + F_{right} + F_{top} + F_{bottom}$$
Step 3 ：Simplify the final answer using Python code: $$F_{net} = 40 \, \mu\text{N}$$
Step 4 ：\(\boxed{40 \, \mu\text{N}}\)
