11.3 Motion of a Charged Particle in a Magnetic Field
3 min read•june 24, 2024
Charged particles in magnetic fields move in fascinating ways, creating circular or helical paths depending on their initial velocity. The magnetic force acts perpendicular to both the particle's motion and the field, causing without changing the particle's speed.
Understanding these motions is crucial for many applications, from particle accelerators to plasma confinement in fusion reactors. The radius and period of the circular path depend on the particle's properties and the , while helical trajectories occur when particles enter at an angle.
Motion of a Charged Particle in a Uniform Magnetic Field
Motion of charged particles in magnetic fields
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- Charged particle experiences magnetic force when moving through magnetic field
- Magnetic force is perpendicular to both particle's velocity and magnetic field ()
- Force acts as centripetal force, causing particle to move in circular path (cyclotron motion)
- Direction of depends on sign of particle's charge
- Positive charges () circle counterclockwise
- Negative charges () circle clockwise
- Particle's speed remains constant as magnetic force does no work on particle
- Magnetic force only changes direction of particle's velocity, not magnitude ( conserved)
Circular path calculations for charged particles
- Radius of circular path depends on particle's mass (), charge (), velocity (), and magnetic field strength ()
- Radius () given by: (also known as )
- Larger mass (heavy ions) or velocity results in larger radius
- Larger charge () or magnetic field strength results in smaller radius
- Radius () given by: (also known as )
- Period () of circular motion is time for particle to complete one revolution
- Period given by:
- Period depends on particle's mass, charge, and magnetic field strength
- Period independent of particle's velocity (frequency of revolution constant)
- The inverse of the period is known as the
- Period given by:
Helical trajectories in uniform magnetic fields
- When charged particle enters magnetic field with velocity not perpendicular to field, path becomes helix
- Particle's velocity decomposed into components parallel and perpendicular to magnetic field
- Perpendicular component causes particle to move in circular path (as described above)
- Parallel component unaffected by magnetic field, causing particle to move along field lines (uniform motion)
- Resulting motion combines circular motion in plane perpendicular to magnetic field and constant velocity along field lines
- Combination creates (spiral motion)
- Pitch of helix depends on ratio of parallel velocity component to perpendicular velocity component
- Larger parallel component results in longer pitch (stretched helix)
- Larger perpendicular component results in shorter pitch (compressed helix)
- The angle between the particle's velocity vector and the magnetic field lines is called the
Particle behavior in non-uniform magnetic fields
- In non-uniform magnetic fields, particles can experience magnetic mirroring
- of a particle is conserved in slowly varying magnetic fields
- As a particle moves into a stronger magnetic field region, its pitch angle increases
- If the magnetic field strength increases sufficiently, the particle can be reflected back towards weaker field regions
Key Terms to Review (36)
Alpha Particles: Alpha particles are a type of ionizing radiation consisting of two protons and two neutrons, making them the heaviest of the common types of particle radiation. They are emitted during the radioactive decay of certain heavy elements, such as uranium and radium.
Aurorae: Aurorae are natural light displays in the Earth's sky, predominantly seen in high-latitude regions around the Arctic and Antarctic. They occur due to interactions between charged particles from the solar wind and the Earth's magnetic field.
Charge-to-Mass Ratio: The charge-to-mass ratio, often denoted as e/m, is a fundamental physical quantity that describes the ratio of the electric charge of a particle to its mass. This ratio is an important characteristic that determines the behavior of charged particles in electric and magnetic fields, and has numerous applications in various fields of physics.
Circular Motion: Circular motion is the movement of an object in a circular path, where the object continuously changes direction but maintains a constant distance from the center of the circular path. This type of motion is characterized by a centripetal force that acts on the object, causing it to continuously deviate from a straight-line trajectory.
Cosmic rays: Cosmic rays are high-energy protons and atomic nuclei that travel through space at nearly the speed of light. They originate from various sources, including the Sun, distant stars, and supernovae.
Cyclotron Frequency: Cyclotron frequency is the frequency at which a charged particle orbits in a magnetic field, determined by the charge of the particle and the strength of the magnetic field. This frequency is crucial in understanding how charged particles move and interact within magnetic environments, particularly in the context of their circular motion and applications such as magnetic confinement in fusion reactors and astrophysical phenomena.
Cyclotron Motion: Cyclotron motion refers to the circular trajectory of a charged particle moving in a uniform magnetic field. This motion is characterized by the particle's acceleration in a perpendicular direction to both the magnetic field and the particle's velocity, resulting in a circular path.
Electron Mass: The electron mass is the rest mass of an electron, which is the fundamental unit of negative electric charge in an atom. It is a crucial parameter in the study of the motion of charged particles in a magnetic field, as it determines the acceleration and trajectory of the electron under the influence of magnetic forces.
Electrons: Electrons are subatomic particles that carry a negative electric charge and are found in all atoms. They play a crucial role in various physical phenomena, including Coulomb's law, electrical current, the motion of charged particles in magnetic fields, and the Hall effect.
Elementary Charge: The elementary charge is the smallest known unit of electric charge, carried by a single electron or proton. It serves as the fundamental unit of electric charge in many physics concepts and equations.
F = qvB sin θ: The formula F = qvB sin θ describes the force experienced by a charged particle moving through a magnetic field. It represents the magnitude of the force, where F is the force, q is the charge of the particle, v is the velocity of the particle, B is the strength of the magnetic field, and θ is the angle between the velocity vector and the magnetic field vector.
Free electrons: Free electrons are electrons that are not bound to atoms and can move freely within a material. In conductors, these free electrons enable the flow of electric current.
Gyroradius: Gyroradius, also known as the Larmor radius, is the radius of the circular motion of a charged particle moving perpendicular to a magnetic field. This motion occurs due to the Lorentz force acting on the particle, resulting in a spiral path as it also moves forward. The gyroradius is a crucial concept in understanding how charged particles behave in magnetic fields, especially in fields like plasma physics and astrophysics.
Hall effect: The Hall effect describes the production of a voltage difference (the Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current. This phenomenon is used to measure magnetic fields and carrier density in materials.
Hall Effect: The Hall effect is a phenomenon in which a voltage difference is produced across an electrical conductor transverse to an electric current flowing through the conductor and to an applied magnetic field perpendicular to the current. This effect has important applications in various areas of physics and technology.
Helical motion: Helical motion describes the spiral trajectory of a charged particle moving through a magnetic field, combining uniform circular motion with linear motion along the magnetic field lines. The resulting path resembles a helix.
Helical path: A helical path refers to the three-dimensional spiral trajectory that a charged particle follows when it moves in a magnetic field, influenced by the Lorentz force. This motion is characterized by circular motion in a plane perpendicular to the magnetic field while simultaneously translating along the direction of the field, resulting in a corkscrew-like movement. The combination of these motions leads to unique properties in particle dynamics, particularly in fields like electromagnetism and plasma physics.
Kinetic Energy: Kinetic energy is the energy of motion possessed by an object due to its movement. It is the energy that an object has by virtue of being in motion and is directly proportional to the mass of the object and the square of its velocity.
Lorentz Force: The Lorentz force is the force exerted on a charged particle when it moves through a magnetic field. It is a fundamental concept in electromagnetism that describes the interaction between electric and magnetic fields and the motion of charged particles.
Lorentz force equation: The Lorentz force equation describes the force experienced by a charged particle moving through an electric and magnetic field. It is given by $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$, where $q$ is the charge, $\mathbf{E}$ is the electric field, $\mathbf{v}$ is the velocity of the particle, and $\mathbf{B}$ is the magnetic field.
Magnetic Bottle: A magnetic bottle is a device that uses magnetic fields to confine charged particles, effectively trapping them within a specified region of space. This confinement occurs due to the Lorentz force, which acts on the charged particles when they move through a magnetic field, resulting in a stable orbiting motion that prevents them from escaping. Magnetic bottles are essential in various applications, including plasma confinement in fusion research and space physics.
Magnetic Field Strength: Magnetic field strength, also known as magnetic flux density or magnetic induction, is a measure of the force exerted by a magnetic field on a moving charged particle or a current-carrying conductor. It is a fundamental concept in the study of electromagnetism and is crucial in understanding the behavior of charged particles and the interactions between magnetic fields and matter.
Magnetic mirror: A magnetic mirror is a phenomenon in which a charged particle experiences a change in magnetic field strength that causes it to be reflected back toward a region of higher magnetic field intensity. This effect occurs when charged particles move through varying magnetic fields, typically found in magnetized plasma environments like those in space. Magnetic mirrors are essential in understanding the confinement of charged particles and their motion in magnetic fields.
Magnetic moment: The magnetic moment is a vector quantity that represents the strength and direction of a magnetic source, such as a current-carrying loop or a magnetic dipole. It quantifies how much torque a magnetic field exerts on the magnetic source, indicating its tendency to align with an external magnetic field. This concept is crucial for understanding the behavior of charged particles in magnetic fields and the interaction between currents in parallel conductors.
Mass Spectrometer: A mass spectrometer is an analytical instrument used to identify the chemical composition of a sample by measuring the mass-to-charge ratio of ionized particles. It is a powerful tool for studying the properties and structure of molecules in various fields, including physics, chemistry, and biology.
Pitch Angle: The pitch angle is the angle between the velocity vector of a charged particle and the direction of the applied magnetic field. This angle is a critical parameter in determining the motion of the charged particle within the magnetic field.
Protons: Protons are positively charged subatomic particles found in the nucleus of an atom. They play a crucial role in determining the atomic number of an element, which defines its identity and chemical properties. Protons, along with neutrons, make up the mass of an atom and interact with electrons to form the basis of chemical bonding.
Right-hand rule: The right-hand rule is a mnemonic used to determine the direction of the magnetic field surrounding a current-carrying conductor. Point your thumb in the direction of the current and curl your fingers; your fingers indicate the direction of the magnetic field lines.
Right-Hand Rule: The right-hand rule is a mnemonic device used to determine the direction of various quantities related to electromagnetism, such as the direction of magnetic fields, the motion of charged particles in magnetic fields, and the direction of the magnetic force on a current-carrying conductor. It provides a simple and intuitive way to visualize and remember these directional relationships.
Tesla: The tesla (T) is the SI unit of magnetic flux density, representing the strength of a magnetic field. One tesla is defined as one weber per square meter.
Tesla: The tesla (T) is the unit of magnetic flux density, or magnetic field strength, in the International System of Units (SI). It is named after the Serbian-American inventor Nikola Tesla, who made significant contributions to the field of electromagnetism. The tesla is a fundamental unit that is essential in understanding and describing various electromagnetic phenomena and their applications.
Van Allen radiation belts: The Van Allen radiation belts are two layers of charged particles trapped by Earth's magnetic field. These belts contain protons and electrons that spiral along the magnetic field lines, creating regions of intense radiation.
Velocity selector: A velocity selector is a device that uses perpendicular electric and magnetic fields to filter charged particles based on their velocities. It ensures that only particles with a specific velocity can pass through undeflected.
Velocity Selector: A velocity selector is a device used to select charged particles with a specific velocity from a beam of charged particles. It utilizes the combined effects of electric and magnetic fields to filter out particles with undesired velocities, allowing only those with the desired velocity to pass through.
Weber: The weber (Wb) is the SI unit of magnetic flux, representing the quantity of magnetism. One weber is equal to one tesla meter squared ($1 \, \text{Wb} = 1 \, \text{T} \cdot m^2$).
Weber: The weber (symbol: Wb) is the unit of magnetic flux in the International System of Units (SI). It is named after the German physicist Wilhelm Eduard Weber. The weber is a fundamental unit that is closely related to the concepts of magnetic field, electromagnetic induction, and the functioning of various electrical and electronic devices.