No, velocity is generally not constant for a charged particle moving within a uniform magnetic field, even though its speed remains unchanged. Velocity is a vector quantity, encompassing both magnitude (speed) and direction. While the magnetic field does not alter the speed of a moving charged particle, it can significantly change its direction of motion.
The fundamental reason for this lies in the nature of the magnetic force. The magnetic force experienced by a charged particle is always perpendicular to its velocity vector and the magnetic field direction. This crucial characteristic ensures there is no work done by the magnetic field on the particle, preventing any change in the magnitude of its velocity. Therefore, while speed remains constant in any case, the continuous reorientation of the force causes the direction of motion to change, leading to a varying velocity.
Understanding Magnetic Force and its Effects
The force experienced by a charged particle moving in a magnetic field is known as the Lorentz force, given by the equation:
F = q(v × B)
Where:
- F is the magnetic force vector
- q is the charge of the particle
- v is the velocity vector of the particle
- B is the magnetic field vector
- × denotes the cross product
Since the force F is always perpendicular to the velocity v, it acts as a centripetal force, continuously deflecting the particle's path without accelerating or decelerating it. This results in the particle moving in a curved trajectory, such as a circle or a helix, while maintaining its original speed.
Comparing Velocity vs. Speed in a Magnetic Field
| Feature | Velocity (Vector) | Speed (Scalar) |
|---|---|---|
| Magnitude | Can remain constant (speed) | Always remains constant |
| Direction | Changes due to the perpendicular magnetic force | Not applicable |
| Constancy | Generally NOT constant (due to changing direction) | ALWAYS constant (no work done by magnetic field) |
| Impact | Determines the particle's trajectory (circular, helical) | Determines the kinetic energy of the particle (remains same) |
Scenarios in a Uniform Magnetic Field
The specific motion of a charged particle in a uniform magnetic field depends on the angle between its initial velocity and the magnetic field lines:
-
Motion Perpendicular to the Field:
When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a constant centripetal force. This causes the particle to follow a perfect circular path. In this scenario, the speed is constant, but the direction of velocity continuously changes as the particle moves around the circle. -
Motion Parallel or Anti-Parallel to the Field:
If a charged particle moves exactly parallel (θ = 0°) or anti-parallel (θ = 180°) to the magnetic field lines, the magnetic force on it is zero (since sin 0° = sin 180° = 0). In this special case, with no other forces acting, the particle will continue to move in a straight line with constant velocity (both speed and direction remain unchanged). -
Motion at an Angle to the Field (Other than 0°, 90°, 180°):
When a charged particle enters the field at an angle, its velocity vector can be resolved into two components: one parallel to the field and one perpendicular.- The parallel component of velocity remains unaffected, as it experiences no magnetic force.
- The perpendicular component undergoes circular motion, as described above.
The combination of these two motions results in a helical (spiral) path along the magnetic field lines. Again, while the speed is constant, the velocity is not due to the changing direction in the circular component of motion.
Key Concepts and Practical Insights
- Work-Energy Theorem: The work done by a force is given by W = F ⋅ d. Since the magnetic force is always perpendicular to the displacement (velocity), the work done by the magnetic field on the charged particle is zero. According to the Work-Energy Theorem, W = ΔKE (change in kinetic energy). Since W=0, ΔKE=0, meaning the kinetic energy (1/2 mv²) remains constant, which directly implies that the speed remains constant.
- Lorentz Force Application: The principle of magnetic force guiding charged particles is fundamental to many technologies.
- Mass Spectrometers: Used to separate ions based on their mass-to-charge ratio by observing their deflection in magnetic fields.
- Cyclotrons and Particle Accelerators: Employ magnetic fields to bend the paths of charged particles, allowing them to gain energy from electric fields without leaving the confines of the apparatus.
- Fusion Reactors (e.g., Tokamaks): Use strong magnetic fields to confine extremely hot plasma, preventing it from touching the reactor walls.
In summary, while the speed of a charged particle in a uniform magnetic field remains constant due to the perpendicular nature of the magnetic force, its velocity typically changes because its direction of motion is altered, leading to curved or helical trajectories. The only exception is when the particle moves exactly parallel or anti-parallel to the magnetic field lines.
