$6.$ A conducting rod with mass, $\backslash ($ m $\backslash ),$ and length, $\backslash ($ L $\backslash ),$ moves on the top of two horizontal rails that are connected to a battery. The rails and the rod are placed in an upward uniform magnetic field, B$.$ The battery maintains a constant current, I, in the circuit. Assuming friction, air resistance, and electrical resistance are negligible answer the following:

a$.$ Find the direction and magnitude of the net force on the rod.

b$.$ Determine the speed of the rod as a function of time.

c$.$ Find the distance, $\backslash ($ d $\backslash ),$ that the rod must move from rest to reach speed, $\backslash ($ v $\backslash ),$ in terms of $\backslash (\backslash$mathrm$\{$I$\},\backslash$mathrm$\{$L$\}\backslash ),\backslash ($ B$,$ m $\backslash ),$ and fundamental constants.

d$.$ If the coefficient of friction is $\backslash (\backslash$mu $\backslash ),$ find the new distance that the rod must move to attain the same speed, $\backslash ($ v $\backslash ),$ found in part $($c$).$