With your configuration of the electric and magnetic fields, neither of them changes the y-component of the velocity. so as the particle moves along the cycloid, it also keeps moving along the y-direction with constant velocity.Samson4 said:
Cycloid motion is a type of motion in which a charged particle moves along a curved path, known as a cycloid, due to the influence of an external magnetic field. This type of motion is commonly observed in particle accelerators and in certain types of mass spectrometers.
In cycloid motion, a charged particle moves along a cycloid path because of the interaction between its electric charge and an external magnetic field. The magnetic field exerted on the particle causes it to experience a force, known as the Lorentz force, which causes the particle to move in a curved path.
The equation for cycloid motion is given by the Lorentz force equation: F = qvBsin(θ), where F is the force exerted on the particle, q is the charge of the particle, v is its velocity, B is the strength of the magnetic field, and θ is the angle between the velocity and the magnetic field.
The cycloid motion of a charged particle can be affected by several factors, including the strength and direction of the external magnetic field, the velocity of the particle, and the charge of the particle. Additionally, any external forces acting on the particle can also affect its cycloid motion.
Cycloid motion of charged particles is used in a variety of scientific research applications, such as in particle accelerators to study subatomic particles and in mass spectrometry to analyze the composition of chemical samples. It is also used in medical imaging techniques, such as magnetic resonance imaging (MRI).