Magnetic Force on Moving Charges

• Moving charges and magnetism are intricately linked, forming the basis of electromagnetism.
• Electric currents generate magnetic fields.
• The force experienced by a moving charge in a magnetic field is fundamental.

Magnetic Force on a Single Moving Charge

• The magnetic force (F) on a charge (q) moving with velocity (v) in a magnetic field (B) is given by F = q(v x B).
• This force is perpendicular to both the velocity and the magnetic field.
• The magnitude of the force is F = qvBsin(θ), where θ is the angle between v and B.
• If a charge is moving parallel (θ = 0°) or anti-parallel (θ = 180°) to the magnetic field, the magnetic force is zero.
• The magnetic force is maximum when the charge moves perpendicular to the magnetic field (θ = 90°).
• The direction of the force is given by the right-hand rule: point your fingers in the direction of v, curl them towards B, and your thumb points in the direction of F (for positive charges; for negative charges, the force is in the opposite direction).
• The unit of magnetic field strength (B) is the Tesla (T), where 1 T = 1 N/(A·m).
• An older, non-SI unit for B is the Gauss (G), where 1 T = 10,000 G.

Motion of a Charged Particle in a Uniform Magnetic Field

• If a charge moves perpendicular to a uniform magnetic field, it experiences a constant magnetic force perpendicular to its velocity.
• This results in uniform circular motion.
• The magnetic force provides the centripetal force required for circular motion: qvB = mv²/r.
• The radius of the circular path is given by r = mv/(qB).
• The period of the circular motion is T = 2πr/v = 2πm/(qB).
• The frequency of the circular motion (cyclotron frequency) is f = 1/T = qB/(2πm).
• The period and frequency are independent of the particle's speed.
• If the velocity has a component parallel to the magnetic field, the charge will move in a helical path.
• The radius of the helix is determined by the perpendicular component of the velocity, while the pitch (distance between helix turns) is determined by the parallel component.

Magnetic Force on a Current-Carrying Wire

• A current-carrying wire consists of moving charges, so it experiences a force in a magnetic field.
• The force on a straight wire of length L carrying current I in a uniform magnetic field B is given by F = I(L x B).
• The magnitude of the force is F = ILBsin(θ), where θ is the angle between the wire and the magnetic field.
• The direction of the force is given by the right-hand rule: point your fingers in the direction of the current (conventional current, positive to negative), curl them towards B, and your thumb points in the direction of F.
• If the wire is not straight or the magnetic field is not uniform, the force must be calculated by integrating the force on small segments of the wire.

Torque on a Current Loop

• A current loop in a magnetic field experiences a torque that tends to align the loop's magnetic dipole moment with the magnetic field.
• The magnetic dipole moment (μ) of a current loop is defined as μ = IA, where I is the current and A is the area vector of the loop (direction perpendicular to the loop, determined by the right-hand rule).
• The torque (τ) on the current loop is given by τ = μ x B.
• The magnitude of the torque is τ = μBsin(θ), where θ is the angle between μ and B.
• The potential energy (U) of a magnetic dipole in a magnetic field is U = -μ · B = -μBcos(θ).

Magnetic Fields Produced by Currents

• Electric currents create magnetic fields around them.
• The Biot-Savart Law describes the magnetic field dB created by a small current element Idl: dB = (μ₀/4π) (Idl x r)/r², where μ₀ is the permeability of free space (4π x 10⁻⁷ T·m/A) and r is the distance from the current element to the point where the field is being calculated.
• The magnetic field due to a long, straight wire carrying current I at a distance r from the wire is B = (μ₀I)/(2πr). The field lines form circles around the wire. The direction is given by the right-hand rule: point your thumb in the direction of the current, and your fingers curl in the direction of the magnetic field.
• The magnetic field at the center of a circular loop of radius R carrying current I is B = (μ₀I)/(2R).
• Ampere's Law provides a convenient way to calculate the magnetic field in situations with high symmetry: ∮ B · dl = μ₀Ienc, where the integral is taken around a closed loop (Amperian loop) and Ienc is the net current enclosed by the loop.
• For a solenoid (a long coil of wire), the magnetic field inside the solenoid is approximately uniform and parallel to the axis, with a magnitude B = μ₀nI, where n is the number of turns per unit length.
• For a toroid (a donut-shaped coil), the magnetic field is confined to the interior of the toroid and is approximately B = (μ₀NI)/(2πr), where N is the total number of turns and r is the distance from the center of the toroid.

Magnetic Materials

• Materials respond differently to magnetic fields.
• Diamagnetic materials are weakly repelled by magnetic fields. They have a negative magnetic susceptibility. The induced magnetic dipole moments oppose the external field.
• Paramagnetic materials are weakly attracted to magnetic fields. They have a positive magnetic susceptibility. The magnetic dipole moments tend to align with the external field.
• Ferromagnetic materials are strongly attracted to magnetic fields and can become permanently magnetized. They exhibit hysteresis, meaning their magnetization depends on their past history. Examples include iron, nickel, and cobalt.
• The magnetic field inside a material is modified by the material's magnetization.

Applications of Magnetic Forces and Fields

• Electric motors use the torque on current loops in magnetic fields to produce rotational motion.
• Mass spectrometers use magnetic fields to separate ions based on their mass-to-charge ratio.
• Magnetic resonance imaging (MRI) uses strong magnetic fields and radio waves to create images of the inside of the body.
• Particle accelerators use magnetic fields to steer and focus beams of charged particles to high energies.
• Magnetic confinement fusion uses strong magnetic fields to confine plasma at high temperatures for nuclear fusion research.

Examine the connection between moving charges and magnetism. Learn about the magnetic force exerted on a single moving charge within a magnetic field. Understand the relationship between charge, velocity, magnetic field, and force direction.