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    Vectors and Velocity Problem

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    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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
    https://brainmass.com/physics/velocity/vectors-velocity-problem-504724

    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

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:45 pm ad1c9bdddf>
    https://brainmass.com/physics/velocity/vectors-velocity-problem-504724

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