To analyze the motion of a charged particle in the presence of both electric and magnetic fields, we need to consider the forces acting on the particle and how they influence its velocity over time. Given that the electric field \( \mathbf{E} \) and magnetic field \( \mathbf{B} \) are parallel, and the initial velocity \( \mathbf{V_0} \) is perpendicular to \( \mathbf{E} \), we can derive the velocity \( \mathbf{V} \) of the particle at time \( t \).
Understanding the Forces
When a charged particle moves in an electric field, it experiences an electric force given by:
Electric Force: \( \mathbf{F_E} = q \mathbf{E} \)
In a magnetic field, the particle experiences a magnetic force that depends on its velocity:
Magnetic Force: \( \mathbf{F_B} = q (\mathbf{V} \times \mathbf{B}) \)
Setting Up the Problem
Since \( \mathbf{E} \) and \( \mathbf{B} \) are parallel, the magnetic force will always be perpendicular to the velocity of the particle. Initially, the velocity \( \mathbf{V_0} \) is perpendicular to \( \mathbf{E} \), which means the particle will experience a constant electric force in the direction of \( \mathbf{E} \) and a magnetic force that will change direction as the particle moves.
Equations of Motion
The net force acting on the particle is the sum of the electric and magnetic forces:
Net Force: \( \mathbf{F} = \mathbf{F_E} + \mathbf{F_B} = q \mathbf{E} + q (\mathbf{V} \times \mathbf{B}) \)
Using Newton's second law, we have:
Newton's Second Law: \( m \frac{d\mathbf{V}}{dt} = q \mathbf{E} + q (\mathbf{V} \times \mathbf{B}) \)
Solving the Motion
To solve for the velocity \( \mathbf{V} \) at time \( t \), we can break this down into components. Since \( \mathbf{E} \) and \( \mathbf{B} \) are parallel, we can assume without loss of generality that:
- Electric field \( \mathbf{E} = E \hat{i} \)
- Magnetic field \( \mathbf{B} = B \hat{i} \)
- Initial velocity \( \mathbf{V_0} = V_{0y} \hat{j} \) (perpendicular to \( \mathbf{E} \))
As the particle moves, the electric force will cause it to accelerate in the direction of \( \mathbf{E} \), while the magnetic force will cause it to move in a circular path due to the Lorentz force. The equations of motion can be expressed as:
Acceleration: \( \frac{d\mathbf{V}}{dt} = \frac{q}{m} \mathbf{E} + \frac{q}{m} (\mathbf{V} \times \mathbf{B}) \)
Velocity at Time t
To find the velocity at time \( t \), we can integrate the acceleration. The electric field contributes a constant acceleration in the direction of \( \mathbf{E} \), while the magnetic field causes a circular motion. The resulting velocity can be expressed as:
Velocity: \( \mathbf{V}(t) = \left( V_{0y} \cos(\omega t) + \frac{E}{B} \right) \hat{i} + \left( V_{0y} \sin(\omega t) \right) \hat{j} \)
where \( \omega = \frac{qB}{m} \) is the angular frequency of the circular motion due to the magnetic field.
Final Expression
Thus, the velocity of the particle at time \( t \) can be summarized as:
Final Velocity: \( \mathbf{V}(t) = \left( \frac{qE}{m} t + V_{0y} \cos(\omega t) \right) \hat{i} + \left( V_{0y} \sin(\omega t) \right) \hat{j} \)
This expression shows how the particle's velocity evolves over time, influenced by both the electric and magnetic fields. The electric field contributes a linear increase in the velocity component along its direction, while the magnetic field induces oscillatory motion perpendicular to that direction.
