I need a perfect step by step solution of the following problem:
A particle of charge q is started at the origin with an initial velocity (vector)v_0 = v_0 x(^), in a region where a uniform electric field exists (vector)E = E_0 z(^), and a magnetic field intensity is given by (vector)B = B_0 x(^) + B_0 y(^). HINT: ROTATE THE HORIZONTAL PLANE SO ONE OF YOUR NEW AXES LINES UP WITH THE DIRECTION OF THE RESULTANT B FIELD. Write out and solve the equations for x, y, z as functions of time. Evaluate any/all constants in your solution.© BrainMass Inc. brainmass.com December 24, 2021, 10:45 pm ad1c9bdddf
SOLUTION This solution is FREE courtesy of BrainMass!
** Please see the attached file for the complete solution **
Vectors v, E and B are plotted on the x,y and z rectangular coordinate system above.
Lorentz force experienced by the charged particle is given by:
F = q(E + v X B) [where X represents the cross product]
Substituting for v, E and B we get:
=> F = q[E0 + (v0 X (B0 + B0 )] = q[E0 + (v0 X B0(+ )]
=> F = q[E0 + v0 B0 ( X + X )] ...(1)
As = 0 and X = , equation (1) reduces to:
F = q[E0 + v0 B0 ) = q(E0 + B0) ...(2)
From equation (2) it is evident that the net force acting on the particle is along the z axis; there are no components of the force along x and y axis.
Fx = 0, Fy = 0, Fz = q(E0 + B0)
Force = mass x acceleration. Hence,
Fx = m dvx/dt = 0 (where m = mass of the particle)
=> vx = constant = v0 (As there is no force along x axis, acceleration along x axis is zero. Hence, initial velocity v0 along x axis remains constant)
=> vx = dx/dt = v0
=> x = v0 ∫dt
=> x = v0t + A where A is a constant of integration
As the particle starts at the origin, at t = 0, x = 0. Hence, A = 0
Equation for x as a function of time: x = v0t
Fy = m dvy/dt = 0 or vy = constant = 0 (As there is no force along y axis, acceleration along y axis is zero. Hence, initial velocity along y axis being 0, there will be no component of velocity along y axis)
vy = dy/dt = 0 or y = constant = 0 (as the particle starts at the origin i.e. y = 0)
Equation for y as a function of time: y = 0
Fz = mdvz/dt = q(E0 + B0)
=> dvz/dt = (q/m)(E0 + B0)
=> vz = (q/m)(E0 + B0) ∫dt
=> vz = (q/m)(E0 + B0)t + C where C is the constant of integration
Initial condition: At t = 0, vz = 0 (there is no component of initial velocity along z axis). Hence, C = 0.
=> vz = dz/dt = (q/m)(E0 + B0)t
=> z = (q/m)(E0 + B0)∫t dt
=> z = (q/m)(E0 + B0)t2/2 + D where D is a constant of integration
Initial condition: At t = 0, z = 0. Hence, D = 0.
Equation for z as a function of time: z = (q/m)(E0 + B0)t2/2
Equations for x,y and z as function of time are as follows:
x = v0t
y = 0
z = (q/m)(E0 + B0)t2/2