Magnetism: Force and Field

26 Essential University Physics Richard Wolfson Magnetism:Force and Field

In this lecture you’ll learn • To describe magnetism in relation to electric charge • To calculate magnetic forces on charges and currents • And to describe the trajectories of charged particles in magnetic fields • To explain the origin of magnetic fields • And to calculate the magnetic fields of simple current distributions • To describe the effects of magnetism in matter

Magnetic field and magnetic force • The magnetic field, designated , exerts a force on moving electric charges. • The force depends on the charge q, the magnetic field the charge velocity , and on the orientation between • The magnitude of the force is given by F = qvBsin, and its direction follows from the right-hand rule. • The magnetic force may be written in terms of the vector cross product:

Clicker question • The figure shows a proton moving in a magnetic field. What will the direction of the magnetic force on the proton be in both cases? • parallel to • into the page • out of the page

Charged particles in magnetic fields • Since the magnetic force is always at right angles to a charged particle’s velocity… • A particle moving in a plane perpendicular to the field undergoes uniform circular motion. • The cyclotron frequencyf of the motion is independent of the particle’s speed:f = qB/2pm(for speeds much less than that of light). • When the particle has a component of motion along the field, its trajectory is a spiral.

The force is ,where is a vector describing the length and orientation of a straight conductor. The magnetic force on a current • An electric current consists of moving charges, so a current-carrying conductor experiences a magnetic force. • The force actually involves both magnetic forces on the moving charges, and an electric force associated with charge separation.

Clicker question • The figure shows a flexible conducting wire passing through a magnetic field that points out of the page. The wire is deflected upward, as shown. In which direction is current flowing in the wire? • to the left • to the right

Origin of the magnetic field • The magnetic field not only produces forces on moving electric charges… • The magnetic field also arises from moving electric charge. • The Biot-Savart law gives the magnetic field arising from an infinitesimal current element: • The field of a finite current follows by integrating: • Here 0 is the permeability constant, equal to 4p 10–7 N/A2.

Behavior of magnetic field lines • Magnetic fields originate in moving charge. • But unlike static electric fields, whose field lines begin and end on charges, magnetic field lines don’t begin or end on the moving charges and currents that are their source. • Instead, magnetic field lines generally encircle the moving charges or currents. • Their direction follows from the right-hand rule. • In special cases, field lines may extend to infinity in both directions, but they don’t begin or end.

Using the Biot-Savart law: a line current • Integrating the contributions from current elements along an infinite line gives a field that falls off as the inverse of the distance y from the wire:B = 0I/2py • The field encircles the current. • Two parallel wires experience forces from each other’s magnetic field: • Parallel currents attract. • Antiparallel currents repel.

Clicker question • A flexible wire is wound into a flat spiral as shown in the figure. If a current flows in the direction shown, will the coil tighten or become looser? • The coil will tighten. • The coil will become looser.

Using the Biot-Savart law: a current loop • Integrating the contributions from current elements along a circular loop of current gives a field on the loop axis that depends on the distance x along the axis: • For large distances (x >> a), this reduces to

Magnetic dipoles • The 1/x3 dependence of the current-loop’s magnetic field is the same as the inverse-cube dependence of the electric field of an electric dipole. • In fact, a current loop constitutes a magnetic dipole. • Its dipole moment is  = IA, with A the loop area. • For an N-turn loop,  = NIA. • The direction of the dipole moment vector is perpendicular to the loop area. • The fields of electric and magnetic dipoles are similar far from their sources, but differ close to the sources.

Dipoles and monopoles:Gauss’s law for magnetism • There do not appear to be any magnetic analogs of electric charge. • Such magnetic monopoles, if they existed, would be the source of radial magnetic field lines beginning on the monopoles, just as electric field lines begin on point charges. • Instead, the dipole is the simplest magnetic configuration. • The absence of magnetic monopoles is expressed in Gauss’s law for magnetism: • Gauss’s law for magnetism is one of the four fundamental laws of electromagnetism. O • Gauss’s law ensures that magnetic field lines have no beginnings or endings, but generally form closed loops. • If monopoles are ever discovered, the right-hand side of Gauss’s law for magnetism would be nonzero.

Clicker question • The figure shows two sets of field lines. Which set could be a magnetic field? • (a) • (b)

Electric motors • The electric motor is a vital technological application of the torque on a current loop. • A current loop spins between magnet poles. • In a DC motor, the commutator keeps reversing the current direction to keep the loop spinning in the same direction.

Ampère’s law • Gauss’s law for electricity provides a global description of the electric field in relation to charge that is equivalent to Coulomb’s law. • Analogously, Ampère’s law provides a global description of the magnetic field in relation to moving charge that is equivalent to the Biot-Savart law. • But where Gauss’s law involves a surface integral over a closed surface, Ampère’s law involves a line integral around a closed loop. • For steady currents, Ampère’s law sayswhere the integral is taken around any closed loop, and Iencircled is the current encircled by that loop. O

Ampère’s law and current • Ampère’s law says wherever the integral of the magnetic field around a closed loop is nonzero, then there must be current flowing through the area bounded by the loop. • The oppositely directed magnetic fields in these structures in the solar corona necessarily involve currents flowing perpendicular to the image, as application of Ampère’s law to the rectangular amperian loop shows. • Currents in the three wires shown are the same, but one is opposite the other two. If O around loop 2, which current is the opposite one?

Using Ampère’s law • Ampère’s law is always true, but it can be used to calculate magnetic fields only in cases with sufficient symmetry. • Then it’s possible to choose an amperian loop around whichcan be evaluated in terms of the unknown B. • An example: Ampère’s law quickly gives the 1/r field of a line current—or outside any current distribution with line symmetry. O Cross section of a long cylindrical wire. Any field line can serve as an amperian loop, for evaluating the field both outside and inside the wire.

A current sheet • An infinite current sheet is an idealization of a wide, flat distribution of current. • Application of Ampère’s law shows that the magnetic field outside the sheet is uniform and has magnitudewhere Js is the current per unit width: • However, the field direction reverses across the current sheet. • Far from a finite current sheet, the field begins to resemble that of a line current.

Solenoids • A solenoid is a long, tightly woundcoil of wire. • When a solenoid’s length is much greater than its diameter, the magnetic field inside is nearly uniform except near the ends, and the field outside is very small. • In the ideal limit of an infinitely long solenoid, the field inside the solenoid is uniform everywhere, and the field outside is zero. • Application of Ampère’s law shows that the field of an infinite solenoid is B = 0nI, where n is the number of turns per unit length.

Electric and magnetic fields of common charge and current distributions

Summary • Magnetism involves moving electric charge. • Magnetic fields exert forces on moving electric charges: • The magnetic force on a charge q moving with velocityin a magnetic field • The magnetic force on a length L of current-carrying conductor is • Magnetic fields arise from moving electric charge, as described by • The Biot-Savart law: • Ampère’s law: • Magnetic fields encircle the currents and moving charges that are their sources. • Unlike static electric fields, magnetic field lines don’t begin or end. • This fact is expressed in Gauss’s law for magnetism: O O

Magnetism: Force and Field