Lorentz Force Question - Where did the y dot come from ?

[Physics Enemy]

Basically the question is about a penning ion trap. You need to use the equation for the Lorentz Force, which I have. It says the trap electrodes have a potential:

V(x,y,z) = A(2z^2 - x^2 - y^2); There's a superimposed uniform B-Field B = B(z hat)

It then asks you to write down an expression for the z-component of the total electromagnetic force on a particle of charge q, explaining why it doesn't depend on B. I did that. It then says write down the equation of motion. I did that.

It then asks you to do the same thing with the x and y components i.e.) Find the x and y components of the total force, write down the equations of motion, etc.

But the solution shows a y dot in the equation of motion in the x-direction:

x double dot = q/m[2Ax + B(y dot)]

Where did this y dot come into it? Any ideas? I thought a V term is next to B?
Thanks guys, muchly appreciated.

 

[fzero]

$$\dot{y}$$ is the y-component of the velocity.

 

[Physics Enemy]

$$\dot{y}$$ is the y-component of the velocity.

Thanks, I realized a few mins ago lol. The x and z cross to give the y-direction. For some reason I also thought V was potential, not velocity. It's a little v, of course.

 

[Physics Enemy]

I actually disagree with their answer, shouldn't it be -y dot? x cross z gives -y, I think ...

 

[fzero]

I actually disagree with their answer, shouldn't it be -y dot? x cross z gives -y, I think ...

The equation you wrote above was for this $$x$$ component of the force, which contains $$(\vec{v} \times \vec{B})_x = v_y B_z$$.

 

[Physics Enemy]

The equation you wrote above was for this $$x$$ component of the force, which contains $$(\vec{v} \times \vec{B})_x = v_y B_z$$.

Yes you're right. V x B = (V_y * B)xhat - (V_x * B)yhat + (0)zhat
So the x component is B*ydot, and y component is -B*xdot