In the figure of the charge q is released at the origin. Find its x and y components of velocity as a function of time t and give the type of trajectory of the charged particle.

Given:
Electric field E acts along the +ve y axis and in xy plane there exists a uniform magnetic field B acting along +ve z axis. Mass of charged particle is m.


Ans:
Trajectory: Cycloid
v_x=(E/B){1-cos(qBt/m)}
v_y=(E/B)sin(qBt/m)

To analyze the motion of a charged particle in the presence of both an electric field and a magnetic field, we can break down the problem into manageable parts. The charged particle, with charge q and mass m, is subjected to an electric field E directed along the positive y-axis and a magnetic field B directed along the positive z-axis. This setup leads to interesting dynamics that can be described mathematically.

Understanding the Forces at Play

When the charged particle is released from the origin, it experiences two main forces:

• Electric Force (F_E): This force acts on the particle due to the electric field and is given by F_E = qE. Since E is in the y-direction, this force will accelerate the particle in that direction.
• Magnetic Force (F_B): The magnetic force acts perpendicular to both the velocity of the particle and the magnetic field. This force is given by F_B = q(v × B), where v is the velocity vector of the particle.

Equations of Motion

To find the components of velocity as functions of time, we need to set up the equations of motion. The electric force will cause the particle to accelerate in the y-direction, while the magnetic force will influence its motion in both x and y directions.

Acceleration Due to Electric Field

The acceleration in the y-direction due to the electric field can be expressed as:

a_y = F_E/m = (qE)/m

Magnetic Force and Motion

The magnetic force will cause the particle to move in a circular path in the xy-plane. The Lorentz force equation gives us the necessary information to derive the velocity components:

F_B = q(v_x B - v_y 0) = qv_xB

Since the magnetic force acts perpendicular to the velocity, it will not change the speed of the particle but will change its direction.

Velocity Components as Functions of Time

From the dynamics involved, we can derive the x and y components of velocity:

• Velocity in the x-direction: v_x = (E/B)(1 - cos(qBt/m))
• Velocity in the y-direction: v_y = (E/B)sin(qBt/m)

Characterizing the Trajectory

The trajectory of the charged particle can be described as a cycloid. This is because the combination of the linear acceleration due to the electric field and the circular motion induced by the magnetic field results in a path that resembles a cycloid. The particle will oscillate back and forth while also moving in a circular path, creating this unique trajectory.

Visualizing the Motion

Imagine a point on the edge of a rotating wheel. As it moves forward, it also goes up and down in a wave-like motion. Similarly, the charged particle moves in a circular path while being influenced by the electric field, leading to the cycloidal trajectory.

In summary, the motion of the charged particle in the presence of both electric and magnetic fields results in a cycloidal trajectory, with the x and y components of velocity expressed as functions of time. This interplay of forces creates a fascinating dynamic that illustrates the principles of electromagnetism in action.