- #1

Adesh

- 735

- 188

*magnetic forces do no work*is given in

**Introduction to Electrodynamics**by

**David J. Griffiths**like this

Consider a charged particle [itex]q[/itex] moving with velocity [itex]\mathbf{v}[/itex] in a magnetic field [itex]\mathbf{B}[/itex] then the force on it is [itex]\mathbf{F}= q(\mathbf{v}\times {B})[/itex] and if [itex]q[/itex] moves an amount [itex]d\mathbf{l}=\mathbf{v}dt[/itex], the work done is $$ dW= \mathbf{F} \cdot d\mathbf{l} = q(\mathbf{v}\times \mathbf{B}) \cdot \mathbf{v}dt =0$$ hence, magnetic forces do no work.

My problem is why he has replaced [itex]d\mathbf{l}[/itex] with [itex]\mathbf{v}dt[/itex]? This substitution implies that the charged particle was moving with [itex]\mathbf{v}[/itex] only and no force acted on it because when that magnetic force will act on it will cause displacement in a direction perpendicular to [itex]\mathbf{v}[/itex] and force is also in the same direction there the expression for work should not become zero. When he made that substitution he didn't take into account that the direction of velocity will get changed as soon as force comes into the action.

I think that particle should move in the direction of the force and we should take [itex]dl[/itex] in the direction of the force and not in the direction of the previous velocity. My main problem is that substitution, [itex]d\mathbf{l} = \mathbf{v}dt[/itex], beaucse what I think is that as force is applied the direction of velocity will get changed and the new velocity [itex]\mathbf{v'}[/itex] will have some direction in the direction of [itex]\mathbf{F}[/itex] and [itex]d\mathbf{l}= \mathbf{v'}dt[/itex]. If we assume that [itex]v' = v[/itex] isn't the same thing that we want to prove? However,

**v'**and

**v**and can't have the same direction, the magnetic force would at least change the direction of the velocity.

The definition of work says

*Force times the displacement in the direction of force*and my thinking urges me to believe that the magnetic force would cause a displacement in it’s direction (because particle changes it’s path when magnetic field does work on it).

Thank you.