Problem Consider a particle of charge q and mass m, initially at rest at (0,0,0), in the presence of a static magntic field →B=Bˆz and →E=Eˆy.
(a) Taking E,B>0, sketch the orbit of the particle when q>0.
(b) Derive an exact expression for the orbit [x(t),y(t),z(t)] or the particle. Express your answer in terms of E, B, and ωc
(c) Find the drift velocity after averaging the motion in time. If there were many particles of various charges and masses present, would there be any net current?
(d) Suppose the electric field were replaced by a force F in the y-direction. What would be the drift velocity?
Answer (a)
Answer (b)
→B=Bˆz and →E=Eˆy. From Lorentz equation,
mdvdt=q(→E+→v×→B)
˙vx=qmvyB˙vy=qm→Ey−qmvxB˙vz=0
In z-direction z(t)=constant=0
dvxdt=qBmvy=ωcvydvydt=qEm−qBmvx=qEym−ωcvx
d2vydt2=−ω2cvyd2vxdt2=−ω2c(vx−EB)
since EB is constant, d2dt2[vx−EB]=d2vxdt2=−ω2c(vx−EB)
vx=iv⊥eiωct+EBvy=v⊥eiωct
x(t)=x(0)+rLcosωct+EBty(t)=y(0)+rLsinωct
Answer (c)
<vd>=∫∞−∞vdf(v)dv∫∞−∞f(v)dv=EB
There will be no current because →E×→B drifts the ions and electrons in the same direction.
Answer (d)
→vf=1q→F×→BB2=FEqBˆy
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