$(30$ points$)$ The electric motor.

A copper wire square with sides of length b is placed in a uniform magnetic field B along the x$-$axis as shown in the figure. The square is free to rotate around the y axis. The wire has cross sectional area A and contains free charges q at density n$($m$^(-3))$ moving with constant speed v$.$

a$)$ What is the current density j in the wire?

b$)$ What is the current I in the wire?

c$)$ What is the magnitude f of the Lorentz force on each charge carrier?

d$)$ What is the magnitude and direction of the total Lorentz force F on each side of the square? Show force directions on the figure.

e$)$ What is the torgue N around the y axis on each side of the square? What is the total torque N$\_("$tot $")?($This torque keeps the wire square rotating around the y axis.$)$

Please do d$)$ and show it$.$ Explain in details