# Vectors and Velocity Problem

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.

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## 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 âˆ«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) âˆ«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)âˆ«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

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