Charged Particle Motion in Electric and Magnetic Fields

Theory: Electromagnetism

3. Forces on a Charge in Uniform Fields

a) Is it possible for an electric charge in a uniform magnetic field to move without any force acting on it?

Answer: Yes. If the velocity v is parallel (or antiparallel) to the magnetic field B, then the magnetic force q v × B is zero and no magnetic force acts on the charge.

b) Is it possible for the kinetic energy not to change?

Answer: Yes. In a purely magnetic field the magnetic force is always perpendicular to the velocity, so it does no work and the kinetic energy of the particle does not change.

5. Charged Particle Entering a Uniform Electric Field

a) A charged particle enters a uniform electric field with velocity perpendicular to the field. Describe the path followed and explain how the particle changes its energy.

Answer: The trajectory is similar to a projectile: the particle follows a curved (parabolic) path due to the constant electric force. Whether the particle gains or loses kinetic energy depends on the sign of the charge and the direction of the electric field. If the electric force accelerates the particle in the direction of its velocity, the kinetic energy increases; if it opposes the velocity, the kinetic energy decreases.

b) Repeat the above for a magnetic field.

Answer: In a uniform magnetic field with velocity perpendicular to B, the particle undergoes uniform circular motion in the plane perpendicular to the field. The magnetic force is always perpendicular to the velocity, so it does no work and the kinetic energy remains constant.

11. Electron Perpendicular to a Magnetic Field

a) An electron is incident perpendicular to a magnetic field with a given speed. Determine the magnetic field intensity necessary so that the period of motion is ¹⁰_-6 s.

Answer: For the given data, the required magnetic field intensity is B = 3.573 × 10^-5 T.

b) Reason how this would change if the incident particle were a proton.

Answer: If the incident particle were a proton (same magnitude of charge but much larger mass), the direction of curvature would be opposite to that of the electron (because the sign of the charge is opposite) and the radius and period would be larger, since both radius and period scale with mass.

15. Work and Energy Variation Due to Magnetic Force

a) A particle of charge q and velocity v enters a magnetic field in a direction perpendicular to the field. Analyze the work done by the magnetic force and the variation of the particle's kinetic energy.

Answer: The magnetic force is always perpendicular to the velocity, so the work done by the magnetic force is zero. Therefore, the kinetic energy is conserved and the particle undergoes uniform circular motion.

b) Repeat the above if the particle moves in a direction parallel to the field and explain the differences.

Answer: If the particle's velocity is parallel to the magnetic field, the magnetic force is zero (q v × B = 0). The work is still zero and the kinetic energy is unchanged; the particle continues in a straight line at constant speed rather than performing circular motion.

23. Electron Beam in Electric and Magnetic Regions

b) An electron beam penetrates a zone where both electric and magnetic fields exist. If the electrons do not deflect, can we affirm both the electric and magnetic fields are zero? Argue the response.

Answer: No. If there is no deflection, it means the electric and magnetic forces balance each other: qE = q v B (in magnitude) and their directions are opposite. Thus, both fields can be nonzero but must satisfy E = v B with appropriate directions so the total transverse force is zero.

30. Electron Beam Crossing a Region Without Deviation

a) If an electron beam passes through a region without deviating, can you say that the region has no magnetic field? What must the magnetic field be?

Answer: Not necessarily. A magnetic field may exist but produce no force on the electrons if it is parallel to their velocity (so q v × B = 0). Alternatively, if both electric and magnetic fields are present, they may cancel each other (qE + q v × B = 0).

b) In a uniform magnetic region pointing downward, if two protons are launched horizontally in opposite directions, describe their motions.

Answer: Both protons will describe uniform circular trajectories in the plane perpendicular to the magnetic field. Because their velocities are opposite, the directions of their circular motion will be opposite (they will curve in opposite senses), but both move in circles of the same radius if their speeds are equal.

31. Circular Motion of an Electron in a Magnetic Field

a) Why does an electron continue in a circular path and what determines the sense (direction) of that motion?

Answer: The magnetic force acts perpendicular to the tangential displacement at each instant and thus provides the centripetal force that keeps the particle on a circular path. The sense (clockwise or counterclockwise) of the motion depends on the sign of the charge and the direction of the magnetic field.

b) Derive the relationships for circular motion in a magnetic field.

Derivation and formulas:

• Balance of centripetal and magnetic forces: q v B = m v² / r.
• Radius of circular motion: r = m v / (|q| B).
• Period of revolution: T = 2 π m / (|q| B), independent of speed for a given B and particle.