A connector `AB` can slide without frictiion along a `II`-shaped conductor located in a horizontal plane (Fig). The connector has a length `l`, mass `m` and resistance `R`. The whole system is located in a unifrom magnetic field of induction `B` directed vertically. At the moment `t = 0` a constant horizontal force `F` starts acting on the connector shifting it translationwise to the right. Find how th e velocity o fhte connector varies with time `t`. The inductance of the loop and the resistance of the II-shaped conductor conductor are assumed to be negligible

Explanation

With the help of the circulation, done in the previous problem, Ampere's force on the connector,
`vec(F)_(amp) = (v B^(2) l^(2))/(R)` directed toward left.
Now from Newton's secound law,
`F - F_(amp) = m (dv)/(dt)`
So, `F = (v B^(2) l^(2))/(R) + m (dv)/(dt)`
or, `int_(0)^(t) dt = m int_(0)^(v) (dv)/(F - (v B^(2) l^(2))/(R))`
or, `(l)/(m) = (R)/(B^(2) l^(2)) In ((F - (v B^(2) l^(2))/(R))/(F))`
Thus, `v = (1 - e (-t B^(2) l^(2))/(Rm)) (RF)/(B^(2) l^(2))`