Question.
A particle having mass m and charge q is released from the origin in a region in which magnetic field and electric field are given by B vector = - Bo*j vector and E vector = Eo* k vector. Find the speed of the particle as a function of its Z - coordinate.
A particle having mass m and charge q is released from the origin in a region in which magnetic field and electric field are given by B vector = - Bo*j vector and E vector = Eo* k vector. Find the speed of the particle as a function of its Z - coordinate.
Answer.
Lorentz force: F = (Fx, Fy, Fz) = q(E + V x B)
Here E = (0, 0, Eo), V = (Vx, Vy, Vz) and B = (0, -Bo, 0) ==>
V x B = (Vz*Bo, 0, -Vx*Bo)
m*(dV/dt) = q(Vz*Bo, 0, Eo -Vx*Bo)
(m/qBo)*(dV/dt) = (Vz, 0, Eo/Bo -Vx) ==>
(m/qBo)*(dVx/dt) = Vz ..... (1)
(m/qBo)*(dVz/dt) = Eo/Bo -Vx ..... (2)
Differentiate (1) wrt t ==> (m/qBo)*(d²Vx/dt²) = dVz/dt ..... (3)
Substitute dVz/dt from (3) in (2) ==>
(m/qBo)² * (d²Vx/dt²) = Eo/Bo -Vx
(m/qBo)² * (d²Vx/dt²) + Vx = Eo/Bo ..... (4)
The solution of the differential equation (4) is
Vx = (Eo/Bo)*[1-cos(qBo*t/m)] ..... (5)
From (1) ==>
Vz = (Eo/Bo)*sin(qBo*t/m) ..... (6)
From (5) and (6) ==>
X = (Eo/Bo)*[t - (m/qBo)sin(qBo*t/m)] ..... (7)
Z = (mEo/qBo²)*[1 - cos(qBo*t/m)] ..... (8)
From (7) and (8) ==>
The particle is moving along the cycloid in the xz coordinate plane.
V² = Vx² + Vz² = (Eo/Bo)² * [2 - 2cos(qBo*t/m)]
V² = (Eo/Bo)² * 2[1 - cos(qBo*t/m)] ..... (9)
From (8) and (9) ==> V² = 2qEo*Z/m
V = √(2qEo*Z/m)
Here E = (0, 0, Eo), V = (Vx, Vy, Vz) and B = (0, -Bo, 0) ==>
V x B = (Vz*Bo, 0, -Vx*Bo)
m*(dV/dt) = q(Vz*Bo, 0, Eo -Vx*Bo)
(m/qBo)*(dV/dt) = (Vz, 0, Eo/Bo -Vx) ==>
(m/qBo)*(dVx/dt) = Vz ..... (1)
(m/qBo)*(dVz/dt) = Eo/Bo -Vx ..... (2)
Differentiate (1) wrt t ==> (m/qBo)*(d²Vx/dt²) = dVz/dt ..... (3)
Substitute dVz/dt from (3) in (2) ==>
(m/qBo)² * (d²Vx/dt²) = Eo/Bo -Vx
(m/qBo)² * (d²Vx/dt²) + Vx = Eo/Bo ..... (4)
The solution of the differential equation (4) is
Vx = (Eo/Bo)*[1-cos(qBo*t/m)] ..... (5)
From (1) ==>
Vz = (Eo/Bo)*sin(qBo*t/m) ..... (6)
From (5) and (6) ==>
X = (Eo/Bo)*[t - (m/qBo)sin(qBo*t/m)] ..... (7)
Z = (mEo/qBo²)*[1 - cos(qBo*t/m)] ..... (8)
From (7) and (8) ==>
The particle is moving along the cycloid in the xz coordinate plane.
V² = Vx² + Vz² = (Eo/Bo)² * [2 - 2cos(qBo*t/m)]
V² = (Eo/Bo)² * 2[1 - cos(qBo*t/m)] ..... (9)
From (8) and (9) ==> V² = 2qEo*Z/m
V = √(2qEo*Z/m)
Maxwell’s theory for electromagnetism:
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