How Do Ions Move Through Perpendicular Electric and Magnetic Fields?

[HPRF]

Homework Statement

A beam of ions enters a region in which the electric field E and magnetic flux density B are normal to each other and both normal to the velocity of the ions. Show that if the velocity of the ions is related to the fields by

v=ExB/B²

then the ions pass through the region undeflected.

Homework Equations

Thinking of using

F=qvxB

but not sure.

The Attempt at a Solution

Substituting v from info into Lornetz force equation but it doesn't seem to answer the question.

 

[lanedance]

so for the ions to pass through unaffected, the force due to the elctric field & the motion through the magnetic must be equal & opposite (ie exactly cancel)

 

[HPRF]

So would it be more accurate to use

qE=qvxB

and then substitute in the v equation?

 

[lanedance]

sounds like a reasonable idea to me, give it a try

couple of pointers though, just be careful with your signs as the forces neecd to cancel out

now you could expand a vector triple product or make use of the orthoganalilty of the vectors in the question & look at the magnitudes...

 

[gabbagabbahey]

So would it be more accurate to use

qE=qvxB

and then substitute in the v equation?

If I were you, I would use the general form of the Lorentz force law $\textbf{F}=q(\textbf{E}+\textbf{v}\times\textbf{B})$, and plug in the velocity you are given...after using some vector triple product identities, you should find that the force on each ion $\textbf{F}$ is zero.