A charged particle of mass m and charge q is projected on a rough horizontally `xy-` 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)` with velocity `vec(v)=upsilon_(0)hat(j)`. The particle starts moving into a circular path on the plane. If the coefficient of friction between the particle and the plane is `mu`. Then calculate tha `:`

`(a)` time when the particle will come to rest
`(b)` time when the particle will hit the centre.
`( c)` distance travelled by the particle when it comes to rest.

Explanation

`(a) N=mg+qE_(0)……………….(1)`
`qB_(0)upsilon=(m upsilon^(2))/( R )………………(2)`
and `-m(d upsilon)/(dt) = mu N …………………(2)`
From equation `(2), " "R=( m upsilon)/(B_(0)q)……………..(4)`
From equations `(1)` and `(3)`

`-m.(d upsilon)/(dt)=mu ( mg +qE_(0))( o r)`
`-m int_(v_(0))^(0)d upsilon=mu(mg+qE_(0))int_(0)^(t)dt`
Thus `t=(m upsilon_(0))/(mu(mg+qE_(0)))`
`(b)` From equation `(4)`
`dR=(m)/(B_(0)q)d upsilon=-(mu(mg+qE_(0))dt)/(qB_(0))`
`int_(R_(1))^(0)dR=(-mu(mg+qE_(0)))/(qB_(0))int _(0)^(t)dt`
or or`t=(qB_(0)R_(i))/(mu(mg+qE_(0)))`
Here `R_(i)=(m upsilon_(0))/(B_(0)q)` Thus`t=(m upsilon_(0))/(mu(mg+qE_(0)))`
`(c )` `-m upsilon.(d upsilon)/(dl)=mu (mg+qE_(0))` or,
`-m int_(upsilon_(0))^(0)upsilond upsilon=mu(mg+qE_(0))int_(0)^(t)dt ` or
`l=(m upsilon_(0)^(2))/(2mu(mg+qE_(0)))`