20.1 Magnetic Fields, Field Lines, and Force

Magnets fascinate us with their invisible forces. They have north and south poles that attract or repel, creating magnetic fields around them. These fields are generated by moving charges or magnetic dipoles, and can be visualized using field lines.

Magnetic forces affect charged particles and current-carrying wires. The force on a moving charge is given by the equation F = qv × B. This force causes charged particles to move in circular paths in uniform magnetic fields, a principle used in particle accelerators.

Magnetic Fields and Properties

Properties and creation of magnets

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• Magnets have two poles: north and south poles attract each other while like poles repel
• Magnetic monopoles (isolated north or south poles) have not been observed in nature
• Magnetic fields are generated by moving electric charges or magnetic dipoles, which are created by current loops or electron spin
• Ferromagnetic materials (iron, nickel, cobalt) can be permanently magnetized by aligning their magnetic domains (regions with aligned magnetic dipoles) through exposure to strong external magnetic fields
• Paramagnetism and diamagnetism are weak forms of magnetism where paramagnetic materials (aluminum, platinum) are weakly attracted to magnetic fields and diamagnetic materials (copper, water) are weakly repelled by magnetic fields
• The degree to which a material can be magnetized is described by its magnetization, which is the magnetic moment per unit volume

Magnetic Field Lines and Forces

Magnetic field lines around objects

• Magnetic field lines represent the direction of the magnetic field at any point, pointing from north to south outside the magnet and forming continuous closed loops that never cross each other
• The density of field lines indicates the strength of the magnetic field, with more closely spaced lines representing a stronger field
• Magnetic field lines around a bar magnet exit the north pole and enter the south pole, with the field being strongest near the poles
• Magnetic field lines around a current-carrying wire form circular patterns surrounding the wire, with the direction determined by the right-hand rule (point thumb in current direction, fingers curl in field direction)
• The total number of magnetic field lines passing through a surface is called the magnetic flux

Forces in magnetic fields

• Magnetic force on a moving charge is given by $\vec{F} = q\vec{v} \times \vec{B}$, where $q$ is the charge, $\vec{v}$ is the velocity, and $\vec{B}$ is the magnetic field (this is part of the more general Lorentz force)
  1. The force is perpendicular to both velocity and magnetic field
  2. The direction is determined by the right-hand rule (point fingers in $\vec{v}$ direction, curl in $\vec{B}$ direction, thumb points in $\vec{F}$ direction)
• Magnetic force on a current-carrying wire is given by $\vec{F} = I\vec{L} \times \vec{B}$, where $I$ is the current, $\vec{L}$ is the wire length vector, and $\vec{B}$ is the magnetic field
  1. The force is perpendicular to both wire and magnetic field
  2. The direction is determined by the right-hand rule (point fingers in $I$ direction, curl in $\vec{B}$ direction, thumb points in $\vec{F}$ direction)
• Charged particles in uniform magnetic fields experience circular motion with radius $r = \frac{mv}{qB}$ (where $m$ is mass, $v$ is speed, $q$ is charge, $B$ is magnetic field strength) and period $T = \frac{2\pi m}{qB}$
  • This circular motion is known as cyclotron motion and is the basis for particle accelerators

Magnetic Moments and Torque

• A current loop in a magnetic field experiences a torque, which is related to its magnetic moment
• The magnetic 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 perpendicular to the loop
• The torque on a magnetic dipole in a magnetic field is given by $\vec{\tau} = \vec{\mu} \times \vec{B}$