- #1
Fernando Mourao
- 3
- 0
Homework Statement
Describe semiquantitatively the motion of an electron under the presence
of a constant electric field in the x direction,
E =E0x^
and a space varying magnetic field given by
B = B0 a(x + z)x^ + B0 [1 + a(x - z)]z^
where Eo, Bo, and a are positive constants, lαxl « 1 and lαzl « 1.
Assume that initially the electron moves with constant velocity in the
z direction, v(t = 0) = v0z^. Verify if t his magnetic field satisfies the
Maxwell equation ∇ x B = 0
Homework Equations
[/B]
Equation of motion:
m dv/dt = q[E + v × B]
The Attempt at a Solution
I've proven that the B field satisfies the Maxwell equation.
and considering B(0,0,0) = B0 z^
I got the B field as a first order approximation about the origin as
B(r) = B0z^ + (B0αx + B0αz)x^ + (B0αx - B0αz )z^
So from the equation of motion I get:
m dv/dt = q[E + v × B] = q[E + v × B0 + v×[r⋅∇B]]
The first two terms on the right hand side show that the particle would have a uniform acceleration along the x direction and a circular motion (varying with the instantaneous velocity) with x and y components; the last term is a force term and results in a combined gradient-curvature drift of the particle.
I feel like I'm missing something regarding the divergent term of ∇B. How far should i go with the resolution of the equation of motion?