This is a nice problem because it sounds like you'd have to solve some differential equations to get the answer, but in fact there is some simple reasoning that avoids all that.

Let's say the electric field points down, the magnetic field points out of your monitor, and the particle is positively charged. Then when we release the particle from rest, it feels a downward force from the electric field and starts accelerating down.

Moving downward makes the particle feel a leftward force from the magnetic field, so it starts curving left. Will it asymptote out to move purely left, or start moving upwards after a while, or what?

One thing to notice is that there's a solution to the equations of motion where the particle moves only left. When moving left, the magnetic field pushes up and the electric field pushes down. If the particle moves at speed E/B E/BE/B, these forces cancel and there is no acceleration, so that particle just keeps moving left in a straight line.

Another way to say this is that from its own reference from (let's call that the primed frame, and the original the unprimed frame), the particle does not experience an electric field. That's because in the primed frame, the particle can't feel a magnetic force because its velocity is zero, so if there's no acceleration, there's no electric field. Evidently, when we transform into this frame, the magnetic field transforms so as to make an upward-pointing electric field that cancels the downward electric field.

That's not the motion of our particle because the initial conditions are wrong, but it's a nice frame to use for the analysis. In the primed frame there is only a magnetic field, and the particle starts off moving to the right at v=E/B v=E/Bv = E/B. A charged particle in a magnetic field will undergo uniform circular motion.

Transforming back to the unprimed frame, we take the circular motion in the primed frame and add a constant horizontal motion of equal magnitude to the particle's speed in the unprimed frame. This is kinematically the same as the motion of a point on the edge of a circle that's rolling without slipping. The path traced out is a cycloid.

Answer by Mark Eichenlaub:

This is a nice problem because it sounds like you'd have to solve some differential equations to get the answer, but in fact there is some simple reasoning that avoids all that.Let's say the electric field points down, the magnetic field points out of your monitor, and the particle is positively charged. Then when we release the particle from rest, it feels a downward force from the electric field and starts accelerating down.Moving downward makes the particle feel a leftward force from the magnetic field, so it starts curving left. Will it asymptote out to move purely left, or start moving upwards after a while, or what?One thing to notice is that there's a solution to the equations of motion where the particle moves only left. When moving left, the magnetic field pushes up and the electric field pushes down. If the particle moves at speed [math]E/B[/math], these forces cancel and there is no acceleration, so that particle just keeps moving left in a straight line.Another way to say this is that from its own reference from (let's call that the primed frame, and the original the unprimed frame), the particle does not experience an electric field. That's because in the primed frame, the particle can't feel a magnetic force because its velocity is zero, so if there's no acceleration, there's no electric field. Evidently, when we transform into this frame, the magnetic field transforms so as to make an upward-pointing electric field that cancels the downward electric field.That's not the motion of our particle because the initial conditions are wrong, but it's a nice frame to use for the analysis. In the primed frame there is only a magnetic field, and the particle starts off moving to the right at [math]v = E/B[/math]. A charged particle in a magnetic field will undergo uniform circular motion.Transforming back to the unprimed frame, we take the circular motion in the primed frame and add a constant horizontal motion of equal magnitude to the particle's speed in the unprimed frame. This is kinematically the same as the motion of a point on the edge of a circle that's rolling without slipping. The path traced out is a cycloid.