28.5
Motion Of A Charged Particle In A Magnetic Field
A charged particle experiences a force when moving through a magnetic field. Consider the field to be uniform and the charged particle to move perpend…
Consider a point mass m of charge Q moving perpendicular to a uniform magnetic field B.
The Lorentz force's magnitude is constant and is always perpendicular to its velocity. So, it cannot change the velocity's magnitude, only its direction. Hence, the particle's path is a circle with a radius r.
Using Newton's second law of motion, the Lorentz force equals the centripetal force. Thus, the circle's radius is calculated.
The time period T of the particle's circular path is the circumference by speed. Substituting the value of r, T is obtained.
If the velocity is not perpendicular to the magnetic field, the component parallel to the magnetic field remains unaffected, thus producing a constant motion along the magnetic field. Hence, the particle moves in a helical path.
In the formula for the radius, the perpendicular component of the velocity is substituted. Since T is independent of the speed, it remains the same.
Its pitch is defined as the distance between adjacent turns, given by the product of the parallel component of the velocity and period.
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Q1: Why does a charged particle move in a circle in a magnetic field?
A charged particle moving perpendicular to a uniform magnetic field experiences a magnetic force that is always perpendicular to its velocity. Since the force does no work on the particle, its speed remains constant while only its direction changes. This perpendicular force acts as a centripetal force, causing the particle to follow a circular path.
Q2: How is the radius of a charged particle's circular path calculated?
The radius of the circular path is determined by equating the magnetic force to the centripetal force using Newton's second law. The magnetic force on a moving charge is F = QvB, and the centripetal force is F = mv²/r. Solving for r gives the radius in terms of the particle's mass, charge, velocity, and magnetic field strength.
Q3: What is the period of a charged particle's circular motion in a magnetic field?
The period is the time for one complete circular orbit, calculated as the circumference divided by the particle's speed. Interestingly, the period is independent of the particle's speed and depends only on its mass, charge, and the magnetic field strength. This means particles with different speeds take the same time to complete one cycle.
Q4: What happens when a charged particle's velocity is not perpendicular to the magnetic field?
When velocity has both perpendicular and parallel components, each behaves differently. The perpendicular component produces circular motion, while the parallel component creates constant motion along the magnetic field direction. These motions combine to produce helical motion, where the particle spirals along the field lines.
Q5: How is the pitch of a helical path defined?
The pitch is the distance between adjacent turns of the helix, calculated as the product of the parallel component of velocity and the period of circular motion. Since the period remains constant regardless of speed, the pitch depends only on the parallel velocity component and the magnetic field strength.
Q6: What is a magnetic bottle and how does it trap particles?
A magnetic bottle forms when a charged particle travels between regions of varying magnetic field strength. As the particle moves from a weaker field to a stronger field region, it reflects back before entering the stronger field, similar to a wave reflecting off a wall. If reflection occurs at both ends, the particle becomes trapped in the magnetic bottle region.
Q7: Why does kinetic energy remain constant for a charged particle in a magnetic field?
The magnetic force is always perpendicular to the particle's velocity, so it performs no work on the particle. Since work equals the change in kinetic energy, and no work is done, the particle's kinetic energy and speed remain constant. Only the direction of motion changes, not the magnitude.