Two long parallel conductors, X and Y $,$ having distance d between them are inclined at the angle $\backslash (\backslash$delta $\backslash )$ to the horizontal. The conductors are connected by a conducting wire $\backslash ($ W $\backslash )$ at the top. This whole setup is inside a vertically upward magnetic field. Flux density of the field is B $($Figure $1).$ An iron rod $($Z$)$ with resistance $\backslash (\backslash$mathbf$\{$R$\}\backslash )$ and mass $\backslash ($ m $\backslash )$ is initially held on the parallel conductors at the top. When the rod is just released it starts to move downward with an acceleration and after a while it continues to move with a constant velocity $\backslash ($ u $\backslash ).$ The resistance of the conductors $\backslash (\backslash$mathrm$\{$X$\},\backslash$mathrm$\{$Y$\}\backslash )$ and wire W are negligible.

a$)$ What is the force influencing to move the rod with acceleration just after its release?

b$)$ Briefly explain why the rod attains a constant speed.

c$)$ Derive expression and determine the direction of

i$.$ induced emf in the rod

ii$.$ current through the rod

iii. magnetic force acting on the rod

d$)$ If $\backslash (\backslash$delta$=10^\{\backslash$circ$\}$ ; $\backslash$mathrm$\{$m$\}=50\backslash$mathrm$\{$~g$\}$ ; $\backslash$mathrm$\{$d$\}=25\backslash$mathrm$\{$~cm$\}$ ; $\backslash$mathrm$\{$R$\}=5\backslash$Omega $\backslash ),$ calculate the required magnetic flux density B to move the rod at the constant speed u of $\backslash (10\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).[$consider $\backslash ($ g$=9\mathrm{.}8\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )]$

e$)$ Now $($with the rod moving at the constant velocity u$),$ what would be the impact to the velocity, if

i$.$ magnetic flux density is doubled

ii$.$ direction of the magnetic field is reversed