A particle of mass m and charge q is moving in a region where uniform, constant electric and mangetic fields `vec E and vec B` are present. `vec E and vec B` are parallel to each other. At time `t=0,` the velocity `vec v_0` of the particle is perpendicular to `vec E` (Assume that its speed is always `lt lt c`, the speed of light in vacuum). Find the velocity `vec v` of the particle at time `t`. You must express your answer in terms of `t, q, m,` the vector `vec v_0, vec E` and `vec B` and their magnitudes `vec v_0, vec E` and `vec B`.

Explanation

Because the forces to parallel electric and magnetic fields on a charged particle moving perpendicular to the fields will be at right angles to each other `(` electric force being along the direction of `vec(E)` while magnetic force perpendicular to the plane containing `bar(v)` and `bar(B))` so magnetic force will not affect the motion of charged particle in the direction of electric field and vice- versa. So the problem is equivalent to superposition of two independent motion as shown in the figure.
So for motion of the particle under electric field alone.
`a_(y)=(qE)/(m)i.e.,(dv_(y))/(dt)=(qE)/(m)`
or `int_(0)^(y)dv_(y)=int_(0)^(t)dv_(y)=int_(0)^(t)(qE)/(m)dt " "i.e.v_(y)=(qE)/(m)t.........(1)`
While at the same instant , the charged particle under the action of magnetic field will describe a circle in the `x-z` plane
With `r=(mv_(0))/(qB)i.e., omega=(v_(0))/(r)=(qB)/(m)`
So angular position of the particle at time t in the `x-y` plane
will be given by ` theta= omega t =(qB)/(m) t ` and therefore in accordance with figure.
and
`v_(z)=v_(0)sin theta =v_(0)sin omegat =v_(0)sin ((qB)/(m))t..................(3)`
So in the light of equations `(1),(2)` and `(3)` , we get
`vec(v)=hat(i)v_(x)+hat(j)v_(y)+hat(k)v_(z)`
`vec(v)=[v_(0)cos ((qB)/(m))t]hat(i)+((qE)/(m)+)hat(j)+(v_(0)sin .(qB)/(m)+)hat(k)`
But because here, `hat(i)=(vec(v)_(0))/(v_(0)),hat(j)(vec(E))/(E)=(vec(B))/(B)` And
`hat(k)=(bar(v)xxbar(B))/(v_(0)B)`
So
`bar(v)_(0)=((bar(v)_(0))/(v_(0)))v_(0)cos ((qBt)/(m))+((vec(E))/(E))(qE)/(m)t+((vec(v)_(0)xxvec(B))/(v_(0)B))v_(0)sin ((qB)/(m)t)`