Question

A plasma of uniform charge density p_o= is contained within a cylindrical shape, of infinite length and radius r_o. With what speed must a particle of mass m and charge q perpendicularly strike the cylinder in order to bring it to rest at the center of the cylinder?

Answer 1

Given that the cylindrical system having charge density o .

Now, from gauss theorem we know that

E(2*pi*x*t) = o*(2*pi*x²*t)/o

E = o*x/o

Here , if a charge 'q' having mass 'm' strikes perpendicular to the cylindrical plasma ,then electric field exerts a force on the charge ,

F =q*E = q* o*x/o

acceleration , a =  q* o*x/o*m

(d/dx) *(dx/dt) =  q* o*x/o*m

v*dv/dt =  q* o*x/o*m

(v_o to 0) v.dv =  q* o/o*m(r_o to 0) x.dx

Here v_o is initial velocity of charge

[v²/2] _{vo to 0} = q* o*/o*m [x²/2]_{ro to 0}

v_o²/2 =  q* o*r_o²/2*o*m

v_o = r_o * sqrt[ q* o/o*m ]