Show the drift velocity is ExB/B^2

[shedrick94]

A charged particle drifts in uniform, constant magnetic and electric fields. The electric field, E, is perpendicular to the magnetic field, B.

Show that the drift velocity is given by v_d = (E×B)/B²

Heres where I get to:
F=e(E+vxB)=0 as v is uniform.Therefore E+vxB=0.

Take vector product of B with both sides.

BxE +Bx(vxB)=0.

Using identity Ax(BxC) = B(A.C)-C(A.B)

I get BxE+v(B.B)-B(B.v)=0
Then I don't know where to go from here.

 

[mfb]

You can assume that the drift happens in two dimensions, so you can say something about the relative direction of v and B which simplifies (B.v).
In three dimensions the total velocity does not have to follow the initial equation, so you need that assumption.

 

[shedrick94]

You can assume that the drift happens in two dimensions, so you can say something about the relative direction of v and B which simplifies (B.v).
In three dimensions the total velocity does not have to follow the initial equation, so you need that assumption.

Why are you allowed to assume the drift is in two dimensions though?

 

[mfb]

The component in the third dimension is not a drift.

 

[shedrick94]

The component in the third dimension is not a drift.

Would that be an acceleration then?

 

[mfb]

A motion in the third direction would stop quickly in matter. In vacuum, the charged particle could freely keep moving in that direction, without influencing the two-dimensional motion in the other directions.

 

[shedrick94]

A motion in the third direction would stop quickly in matter. In vacuum, the charged particle could freely keep moving in that direction, without influencing the two-dimensional motion in the other directions.

Can I ask what it means by a drift velocity then?

 

[mfb]

The velocity in the two-dimensional plane I mentioned.

 

[shedrick94]

The velocity in the two-dimensional plane I mentioned.

yes but what does it specifically mean by 'drift'. You said the velocity in the 3rd dimension is not a drift.

 

[mfb]

I guess it is just convention to call that drift.