A positively charged particle of mass m and charge q is projected on a rough horizontal x-y plane surface with z-axis in the vertically upward direction. Both electric and magnetic fields are acting in the region and given by `vec E = - E_0 hat k and vec B= -B_0 hat k` , respectively. The particle enters into the field at `(a_0, 0, 0)` with velocity `vec v = v_0 hat j`. The particle starts moving in some curved path on the plane. If the coefficient of friction between the particle and the plane is `mu`. Then calculate the
(a) time when the particle will come to rest
(b) distance travelled by the particle when it comes to rest.

Explanation

(a) Force of electric field will act in downward direction, so
normal force between particle and surface will be as
`N=mg+qE_0….(i)`
Let at any time velocity of the particle is v, then
`-m(dv)/(dt)=muN….(ii)`
From Eqs.(i) and (ii),
`-m(dv)/(dt)=mu(mg+qE_0)`
`-mint_(v_0)^0dv=mu(mg+qE_0) int_0^tdt`
Thus, `t=(mv_0)/(mu(mg+qE_0)`
(b) From Eq.(ii),
`-m(dv)/(dt) (dl)/(dt)=muN`
`implies -mv(dv)/(dl)=mu(mg+qE_0)`
`implies -mint_(v_0)^0 vdv=mu(mg+qE_0)int_0^ldl`
`implies l=(mv_0^2)/(2mu(mg+qE_0)`.