Premed · Premed · Physics 2
Lecture 9: Magnetic Fields and Forces
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the properties of magnetic fields and their sources
- Calculate the magnetic force on a moving charged particle using the Lorentz force law
- Analyze the motion of charged particles in uniform magnetic fields (circular and helical orbits)
- Calculate the magnetic force on a current-carrying wire
- Determine the torque on a current loop in a magnetic field and explain the magnetic dipole moment
Lecture Content
I. Introduction to Magnetic Fields
Magnetism has been known since antiquity through the behavior of lodestones and compass needles. Magnetic fields are produced by moving electric charges (currents) and by permanent magnets, whose magnetism arises from atomic-level current loops due to electron spin and orbital motion. The magnetic field is denoted by the vector B. Its SI unit is the tesla (T) = 1 kg/(A s^2), and the older unit gauss (G) is related by 1 T = 10^4 G. Earth's magnetic field is roughly 25-65 uT (about 0.5 G), while clinical MRI machines produce fields of 1.5-3 T and research machines can exceed 7 T.
Magnetic fields differ from electric fields in several important ways. No magnetic monopoles exist; isolated north or south poles have never been found. Consequently, magnetic field lines always form closed loops, unlike electric field lines, which begin and end on charges. Furthermore, the magnetic force does no work on charged particles, as will be explained below.
II. Magnetic Force on a Moving Charge
The magnetic force on a charge q moving with velocity v in a magnetic field B is given by the cross product F = qv x B. The magnitude of this force is F = |q|vB sin(theta), where theta is the angle between v and B, and its direction is determined by the right-hand rule.
Several key properties follow from this formula. The force is zero when v is parallel or antiparallel to B (theta = 0 or 180 degrees), and it reaches its maximum when v is perpendicular to B (theta = 90 degrees). Crucially, the force is always perpendicular to both v and B. Because the force is perpendicular to the velocity, the magnetic force does no work on the charge. It changes the direction of motion but not the speed or kinetic energy.
The right-hand rule provides a reliable way to determine the force direction: point your fingers in the direction of v, curl them toward B, and your thumb points in the direction of F for a positive charge. For negative charges, the force is in the opposite direction.
<image>A three-dimensional diagram illustrating the magnetic force on a positive charge. The velocity vector v points to the right, the magnetic field B points into the page (shown with x symbols), and the resulting force F = qv x B points upward (perpendicular to both). The right-hand rule is illustrated with a hand diagram: fingers point along v, curl toward B, and the thumb indicates F. A second panel shows the same scenario for a negative charge, where the force is reversed (pointing downward).</image>
III. Motion of Charged Particles in Magnetic Fields
When a charged particle enters a uniform magnetic field with its velocity perpendicular to B, the magnetic force provides the centripetal acceleration for circular motion. The particle moves in a circle of radius r = mv/(|q|B). The period of revolution is T = 2 pi m/(|q|B), which is independent of the particle's speed. The cyclotron frequency f = |q|B/(2 pi m) is likewise speed-independent. Positive and negative charges orbit in opposite directions.
When the velocity has a component both parallel and perpendicular to B, the parallel component is unaffected (there is no force along B), producing uniform motion along the field lines. The perpendicular component drives circular motion. The combined result is a helical (spiral) path along the field lines, with a pitch of p = v_parallel x T.
These principles underpin several important technologies. A cyclotron accelerates charged particles in a spiral using alternating electric fields while a magnetic field keeps them curving. A mass spectrometer separates ions by their mass-to-charge ratio using the relationship r = mv/(qB). A velocity selector uses crossed electric and magnetic fields to transmit only particles with a specific velocity v = E/B. The aurora borealis is produced when charged particles from the solar wind spiral along Earth's magnetic field lines and collide with atmospheric molecules near the poles.
<image>Panel A: A positive charge moving in a circle in a uniform magnetic field directed out of the page (shown with dots). The radius r = mv/(qB) is labeled, and the force F pointing toward the center is shown at several points along the circular path. Panel B: A helical path of a charged particle when the velocity has components both parallel and perpendicular to B. The magnetic field points along the z-axis, and the particle spirals along the field direction. The pitch p and radius r of the helix are labeled.</image>
IV. Magnetic Force on a Current-Carrying Wire
A current-carrying wire in a magnetic field experiences a force given by F = IL x B, where L is a vector along the wire in the direction of current with magnitude equal to the wire length. The magnitude of this force is F = BIL sin(theta). This result can be derived by summing the forces on all the individual charge carriers within the wire.
For a curved wire, the force on each infinitesimal element is dF = I dL x B, and the total force is obtained by integrating along the wire. The net force on a closed current loop in a uniform field is zero because forces on opposite segments cancel. However, as discussed below, the loop does experience a net torque. The force on current-carrying wires is the operating principle behind electric motors, loudspeakers, and galvanometers.
V. Torque on a Current Loop — Magnetic Dipole Moment
A rectangular current loop with area A, carrying current I, placed in a uniform magnetic field B experiences a torque of tau = nIAB sin(theta), where n is the number of turns and theta is the angle between the normal to the loop and B. The magnetic dipole moment is defined as mu = nIA, a vector whose direction is given by the right-hand rule: curl the fingers in the direction of the current, and the thumb points along mu. The SI units of the magnetic dipole moment are A m^2.
In vector form, the torque is tau = mu x B, and the potential energy of the loop in the field is U = -mu dot B = -mu B cos(theta). The energy is minimized when mu is aligned with B (theta = 0) and maximized when mu is anti-aligned (theta = 180 degrees). The current loop behaves exactly like a magnetic dipole, perfectly analogous to an electric dipole in an electric field.
This torque is the basis of the DC motor: a current loop experiences torque and rotates, and a commutator reverses the current direction each half-turn to maintain continuous rotation.
VI. The Hall Effect
When a current-carrying conductor is placed in a magnetic field perpendicular to the current, the charge carriers are deflected to one side by the magnetic force. This creates a Hall voltage (V_H) across the conductor, perpendicular to both the current and the field. Equilibrium is reached when the electric force from V_H balances the magnetic force.
The Hall voltage is V_H = IB/(nqt), where t is the thickness of the conductor in the direction of B. This measurement can determine the sign, density, and drift velocity of the charge carriers. The Hall effect has many applications: it identifies whether current is carried by electrons or holes in semiconductors, it is the basis of Hall effect sensors used in proximity detectors, speedometers, and current clamps, and historically it proved that current in metals is carried by negative charges (electrons).
<image>A rectangular conductor carrying current I to the right, placed in a magnetic field B pointing into the page. Electrons drift to the left (opposite to conventional current). The magnetic force F = qv x B deflects electrons toward the top of the conductor, creating a negative charge accumulation on top and positive on the bottom. The resulting Hall voltage V_H is shown across the top and bottom surfaces, with the electric field E_H opposing further charge separation. The equation V_H = IB/(nqt) is displayed.</image>
