5.1 Lorentz force

The Lorentz force is a fundamental concept in electromagnetism, describing how charged particles interact with electric and magnetic fields. It combines the effects of both fields, explaining the motion of charges in various scenarios, from particle accelerators to cosmic rays.

Understanding the Lorentz force is crucial for grasping the behavior of charged particles in electromagnetic fields. This knowledge forms the basis for many practical applications, including Hall effect sensors, cyclotrons, and magnetic mirrors used in plasma confinement.

Definition of Lorentz force

• The Lorentz force is a fundamental force in electromagnetism that describes the force experienced by a charged particle moving through an electromagnetic field
• It is named after the Dutch physicist Hendrik Antoon Lorentz, who first formulated the equation in the late 19th century
• The Lorentz force plays a crucial role in understanding the behavior of charged particles in various applications, such as particle accelerators, plasma physics, and astrophysics

Force on moving charge

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• The Lorentz force acting on a moving charged particle is given by the equation $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$, where $\vec{F}$ is the force, $q$ is the charge of the particle, $\vec{E}$ is the electric field, $\vec{v}$ is the velocity of the particle, and $\vec{B}$ is the magnetic field
• The force depends on both the charge and the velocity of the particle, as well as the strength and direction of the electric and magnetic fields
• The Lorentz force can cause charged particles to accelerate, decelerate, or change their trajectory depending on the field configurations

Magnetic vs electric force

• The Lorentz force consists of two components: the electric force $q\vec{E}$ and the magnetic force $q\vec{v} \times \vec{B}$
• The electric force acts on a charged particle regardless of its motion and is always parallel or antiparallel to the electric field
• The magnetic force acts only on moving charged particles and is always perpendicular to both the velocity of the particle and the magnetic field
• In the absence of an electric field, the Lorentz force reduces to the magnetic force alone

Mathematical formulation

• The Lorentz force equation $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ is a vector equation that describes the force acting on a charged particle in an electromagnetic field
• The equation combines the effects of both the electric and magnetic fields on the particle's motion

Vector cross product

• The magnetic force component of the Lorentz force involves the vector cross product between the velocity $\vec{v}$ and the magnetic field $\vec{B}$
• The cross product $\vec{v} \times \vec{B}$ results in a vector that is perpendicular to both $\vec{v}$ and $\vec{B}$, following the right-hand rule
• The magnitude of the cross product is given by $|\vec{v} \times \vec{B}| = vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$

Magnitude and direction

• The magnitude of the Lorentz force is given by $|\vec{F}| = |q||\vec{E} + \vec{v} \times \vec{B}|$
• The direction of the Lorentz force depends on the sign of the charge and the directions of the electric and magnetic fields
• For a positive charge, the force is in the same direction as $\vec{E}$ and $\vec{v} \times \vec{B}$, while for a negative charge, the force is in the opposite direction

Units of Lorentz force

• The SI unit of the Lorentz force is the newton (N), which is the same as the unit of any other force
• The units of the components in the Lorentz force equation are: charge in coulombs (C), electric field in volts per meter (V/m), velocity in meters per second (m/s), and magnetic field in teslas (T)
• Dimensional analysis of the Lorentz force equation yields: $[F] = [q][E] + [q][v][B] = C \cdot V/m + C \cdot m/s \cdot T = N$

Magnetic field interactions

• The magnetic force component of the Lorentz force, $q\vec{v} \times \vec{B}$, describes how a moving charged particle interacts with a magnetic field
• The magnetic force depends on the velocity of the particle relative to the magnetic field and the strength of the field itself

Perpendicular velocity component

• When a charged particle moves perpendicular to a magnetic field, it experiences the maximum magnetic force
• The magnitude of the force is given by $F = qvB\sin 90^\circ = qvB$, where $q$ is the charge, $v$ is the velocity, and $B$ is the magnetic field strength
• This perpendicular force causes the particle to move in a circular path, as the force is always perpendicular to the velocity (magnetic force does no work)

Parallel velocity component

• When a charged particle moves parallel to a magnetic field, it experiences no magnetic force
• The cross product of parallel vectors is zero, so $\vec{v} \times \vec{B} = 0$ when $\vec{v}$ is parallel to $\vec{B}$
• As a result, the particle continues to move along the magnetic field lines without any deviation

Stationary charges in magnetic fields

• Stationary charges do not experience any magnetic force, as the velocity is zero
• The magnetic force term in the Lorentz force equation becomes zero when $\vec{v} = 0$
• This is consistent with the fact that magnetic fields do not do work on charged particles, as work requires displacement

Charged particle motion

• The Lorentz force equation governs the motion of charged particles in electromagnetic fields
• The trajectory of a charged particle depends on its initial velocity, the strength and orientation of the fields, and the particle's charge-to-mass ratio

Circular paths in uniform fields

• When a charged particle moves perpendicular to a uniform magnetic field, it experiences a constant magnetic force that causes it to follow a circular path
• The radius of the circular motion is given by $r = \frac{mv}{qB}$, where $m$ is the mass of the particle, $v$ is the velocity, $q$ is the charge, and $B$ is the magnetic field strength
• The angular frequency of the circular motion, called the cyclotron frequency, is given by $\omega = \frac{qB}{m}$

Helical paths in uniform fields

• When a charged particle has a velocity component parallel to a uniform magnetic field, it follows a helical (spiral) path
• The parallel component of the velocity remains constant, while the perpendicular component results in circular motion
• The pitch of the helix depends on the ratio of the parallel velocity component to the perpendicular velocity component

Velocity selector applications

• The principle of charged particle motion in uniform fields is used in velocity selectors, which are devices that filter charged particles based on their velocity
• A velocity selector consists of perpendicular electric and magnetic fields, where the strengths are adjusted such that $E = vB$ for the desired velocity $v$
• Particles with the selected velocity pass through undeflected, while particles with other velocities are deflected and filtered out

Magnetic force on current-carrying wire

• A current-carrying wire in a magnetic field experiences a force due to the Lorentz force acting on the moving charges (electrons) within the wire
• The force on a current-carrying wire depends on the current, the length of the wire, and the strength and orientation of the magnetic field

Direction of force

• The direction of the force on a current-carrying wire is determined by the right-hand rule
• Point your thumb in the direction of the current (conventional current, opposite to electron flow) and your fingers in the direction of the magnetic field; the force will be perpendicular to both and in the direction of your palm
• The direction of the force can be found using the vector cross product $\vec{F} = I\vec{L} \times \vec{B}$, where $I$ is the current, $\vec{L}$ is the length vector of the wire, and $\vec{B}$ is the magnetic field

Magnitude of force

• The magnitude of the force on a current-carrying wire is given by $F = ILB\sin\theta$, where $I$ is the current, $L$ is the length of the wire, $B$ is the magnetic field strength, and $\theta$ is the angle between the wire and the magnetic field
• When the wire is perpendicular to the magnetic field ($\theta = 90^\circ$), the force is maximum and given by $F = ILB$
• When the wire is parallel to the magnetic field ($\theta = 0^\circ$ or $180^\circ$), the force is zero

Magnetic torque on current loops

• A current loop (a coil of wire) in a magnetic field experiences a magnetic torque due to the forces acting on the individual sides of the loop
• The magnetic torque tends to align the magnetic dipole moment of the loop with the external magnetic field
• The magnetic dipole moment of a current loop is given by $\vec{\mu} = IA\hat{n}$, where $I$ is the current, $A$ is the area of the loop, and $\hat{n}$ is the unit vector normal to the plane of the loop
• The magnetic torque on a current loop is given by $\vec{\tau} = \vec{\mu} \times \vec{B}$, where $\vec{B}$ is the external magnetic field

Hall effect

• The Hall effect is the production of a voltage difference (Hall voltage) across an electrical conductor transverse to an electric current in the conductor and a magnetic field perpendicular to the current
• It is a result of the Lorentz force acting on the charge carriers (electrons or holes) in the conductor

Charge carrier drift

• When a current flows through a conductor in a magnetic field, the charge carriers (electrons for n-type semiconductors or holes for p-type semiconductors) experience a Lorentz force perpendicular to both the current and the magnetic field
• This force causes the charge carriers to drift away from the direction of the current, resulting in a buildup of charge on one side of the conductor and a depletion of charge on the other side
• The charge accumulation creates an electric field (Hall electric field) that opposes the further drift of charge carriers, resulting in a steady-state condition

Hall voltage and Hall coefficient

• The Hall voltage is the potential difference across the conductor in the direction perpendicular to both the current and the magnetic field
• The Hall voltage is given by $V_H = \frac{IB}{ntq}$, where $I$ is the current, $B$ is the magnetic field strength, $n$ is the charge carrier density, $t$ is the thickness of the conductor, and $q$ is the charge of the carriers (positive for holes, negative for electrons)
• The Hall coefficient is defined as $R_H = \frac{E_y}{J_xB_z} = \frac{1}{nq}$, where $E_y$ is the Hall electric field, $J_x$ is the current density, and $B_z$ is the magnetic field
• The Hall coefficient depends on the type and density of charge carriers in the conductor and can be used to determine the properties of the material

Hall effect applications

• The Hall effect has numerous applications in various fields, including:
  • Magnetic field sensors (Hall sensors) used in position and motion detection, current sensing, and proximity switches
  • Characterization of semiconductors, such as determining the type (n or p), density, and mobility of charge carriers
  • Hall thrusters used in spacecraft propulsion systems, where ions are accelerated by the combined effect of electric and magnetic fields

Cyclotron motion

• Cyclotron motion refers to the circular motion of a charged particle in a uniform magnetic field, where the particle is accelerated by an alternating electric field
• It is the principle behind the operation of cyclotron particle accelerators

Cyclotron frequency and radius

• The cyclotron frequency is the angular frequency of the circular motion of a charged particle in a uniform magnetic field, given by $\omega = \frac{qB}{m}$, where $q$ is the charge of the particle, $B$ is the magnetic field strength, and $m$ is the mass of the particle
• The cyclotron radius is the radius of the circular motion, given by $r = \frac{mv}{qB}$, where $v$ is the velocity of the particle perpendicular to the magnetic field
• The cyclotron frequency and radius depend on the charge-to-mass ratio of the particle and the strength of the magnetic field

Resonance condition

• In a cyclotron accelerator, an alternating electric field is applied across the gap between the "dees" (D-shaped electrodes) to accelerate the charged particles
• The frequency of the alternating electric field must match the cyclotron frequency of the particles for efficient acceleration, a condition known as the resonance condition
• When the resonance condition is met, the particles gain energy from the electric field each time they cross the gap, resulting in a spiral path of increasing radius

Principle of cyclotron accelerator

• A cyclotron accelerator consists of two hollow D-shaped electrodes (dees) placed in a uniform magnetic field, with a gap between them
• Charged particles are injected into the center of the cyclotron and are accelerated by the alternating electric field each time they cross the gap
• As the particles gain energy, their velocity and radius of motion increase, but the cyclotron frequency remains constant due to the uniform magnetic field
• The particles spiral outward until they reach the maximum radius, where they are extracted from the cyclotron and directed towards a target or further acceleration stages

Magnetic mirrors

• Magnetic mirrors are devices that use strong inhomogeneous magnetic fields to reflect charged particles, confining them in a region of space
• They are used in various applications, such as plasma confinement in fusion reactors and particle traps

Reflection of charged particles

• When a charged particle moves into a region of increasing magnetic field strength, it experiences a force that opposes its motion, causing it to slow down and eventually reflect back
• The reflection occurs because the magnetic force on the particle increases as it moves into the stronger field region, while the parallel component of the velocity decreases due to the conservation of magnetic moment
• The condition for reflection is that the particle's velocity parallel to the magnetic field becomes zero at some point, which depends on the initial pitch angle (the angle between the velocity and the magnetic field)

Loss cone angle

• The loss cone angle is the critical angle that determines whether a charged particle will be reflected by a magnetic mirror or escape the confinement
• Particles with pitch angles smaller than the loss cone angle will escape the magnetic mirror, while particles with pitch angles larger than the loss cone angle will be reflected
• The loss cone angle is given by $\sin^2\theta_L = \frac{B_0}{B_m}$, where $B_0$ is the magnetic field strength at the center of the mirror and $B_m$ is the maximum field strength at the mirror points

Magnetic mirror applications

• Magnetic mirrors have several applications, including:
  • Plasma confinement in fusion reactors, where they are used to confine high-temperature plasmas for sustained fusion reactions
  • Particle traps, such as the Penning trap and the Paul trap, which use magnetic and electric fields to confine charged particles for precision measurements and quantum computing
  • Astrophysical phenomena, such as the Van Allen radiation belts around Earth, where charged particles are trapped by the planet's magnetic field