Magnetism

In this lecture we closely follow the textbook of D.Halliday, R.Resnick and J.Walker which is one of the best in the field.

In addition to an electric field, the region of space surrounding any moving electric charge also contains a magnetic field. One can define a magnetic field B at some point in space in terms of the magnetic force FB that the field exerts on a test object, for which we use a charged particle moving with a velocity v.

The magnetic force that acts on a charge q moving with a velocity v in a magnetic field B is

The magnitude FB of the magnetic force exerted on the particle is proportional to the charge q and to the speed v of the particle.

The magnetic force exerted on a positive charge is in the direction opposite the direction of the magnetic force exerted on a negative charge moving in the same direction

The direction of this magnetic force is perpendicular both to the velocity of the particle and to the magnetic field. The magnitude of this force is

where _ is the smaller angle between v and B. The SI unit of B is the tesla (T), where 1 T = 1 N/(A·m).

When a charged particle moves in a magnetic field, the work done by the magnetic force on the particle is zero because the displacement is always perpendicular to the direction of the force. The magnetic field can alter the direction of the particle’s velocity vector, but it cannot change its speed.

There are several important differences between electric and magnetic forces:

• The electric force acts in the direction of the electric field, whereas the magnetic force acts perpendicular to the magnetic field.

• The electric force acts on a charged particle regardless of whether the particle is moving, whereas the magnetic force acts on a charged particle only when the particle is in motion.

• The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a particle is displaced.

The current is a collection of many charged particles in motion; hence, the resultant force exerted by the field on the wire is the vector sum of the individual forces exerted on all the charged particles making up the current. The force exerted on the particles is transmitted to the wire when the particles collide with the atoms making up the wire.

If a straight conductor of length L carries a current I, the force exerted on that conductor when it is placed in a uniform magnetic field B is

where the direction of L is in the direction of the current

If a charged particle moves in a uniform magnetic field so that its initial velocity is perpendicular to the field, the particle moves in a circle, the plane of which is perpendicular to the magnetic field. The radius of the circular path is

where m is the mass of the particle and q is its charge.

The magnetic field at a distance a from a long, straight wire carrying an electric current I is

where

is magnetic constant.

The field lines are circles concentric with the wire.

The magnetic force per unit length between two parallel wires separated by a distance a and carrying currents I1 and I2 has a magnitude

The force is attractive if the currents are in the same direction and repulsive if they are in opposite directions.

The magnetic flux ÔB through a surface is defined by

In general, any current loop has a magnetic field and thus has a magnetic dipole moment, including the atomic-level current loops described in some models of the atom. Thus, the magnetic moments in a magnetized substance may be described as arising from these atomic-level current loops. For the Bohr model of the atom, these current loops are associated with the movement of electrons around the nucleus in circular orbits. In addition, a magnetic moment is intrinsic to electrons, protons, neutrons, and other particles; it arises from a property called spin. The magnetic moment of the electron is proportional to its orbital angular momentum.

All substances contain electrons, but not all substances are magnetic. The main reason is that in most substances, the magnetic moment of one electron in an atom is canceled by that of another electron orbiting

in the opposite direction. The net result is that, for most materials, the magnetic effect produced by the orbital motion of the electrons is either zero or very small.

In addition to its orbital magnetic moment, an electron has an intrinsic property called spin that also contributes to its magnetic moment. In this regard, the electron can be viewed as spinning about its axis while it orbits the nucleus (this is a simplified description,in real spin arises from relativistic dynamics that must be incorporated into a quantum-mechanical analysis).

The magnetic state of a substance is described by a quantity called the magnetization vector M. The magnitude of this vector is defined as the magnetic moment per unit volume of the substance. As you might expect, the total magnetic field B at a point within a substance depends on both the applied (external) field B0 and the magnetization of the substance. The total magnetic field in the region becomes

When analyzing magnetic fields that arise from magnetization, it is convenient to introduce a field quantity, called the magnetic field strength H within the substance. The magnetic field strength represents the effect of the conduction currents in wires on a substance.

Substances can be classified as belonging to one of three categories, depending on their magnetic properties. Paramagnetic and ferromagnetic materials are those made of atoms that have permanent magnetic moments. Diamagnetic materials are those made of atoms that do not have permanent magnetic moments.

For paramagnetic and diamagnetic substances, the magnetization vector M is proportional to the magnetic field strength H. For these substances placed in an external magnetic field, one can write

where χ is a dimensionless factor called the magnetic susceptibility. For paramagnetic substances, χ is positive and M is in the same direction as H. For diamagnetic substances, χ is negative and M is opposite H.

A small number of crystalline substances in which the atoms have permanent magnetic moments exhibit strong magnetic effects called ferromagnetism. Some examples of ferromagnetic substances are iron, cobalt, nickel, gadolinium, and dysprosium.

These substances contain atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field. Once the moments are aligned, the substance remains magnetized after the external field is removed.

This permanent alignment is due to a strong coupling between neighboring moments, a coupling that can be understood only in quantum-mechanical terms.

Let us consider a loop of wire connected to a galvanometer. When a magnet is moved toward the loop, the galvanometer needle deflects in one direction. When the magnet is moved away from the loop, the needle deflects in the opposite direction. When the magnet is held stationary relative to the loop, no deflection is observed. Finally, if the magnet is held stationary and the loop is moved either toward or away from it, the needle deflects. From these observations, one can  conclude that the loop “knows” that the magnet is moving relative to it because it experiences a change in magnetic field. Thus, it seems that a relationship exists between current and changing magnetic field.

As a result of these observations, Faraday concluded that an electric current can be induced in a circuit (the secondary circuit in our setup) by a changing magnetic field.

Faraday’s law of induction states that the emf induced in a circuit is directly proportional to the time rate of change of magnetic flux through the circuit:

Lenz’s law states that the induced current and induced emf in a conductor are in such a direction as to oppose the change that produced them.

An emf and a current are induced in a circuit by a changing magnetic flux. In the same manner, circulating currents called eddy currents are induced in bulk pieces of metal moving through a magnetic field.

Electric generators are used to produce electrical energy. Alternating current (ac) generator is a device that converts mechanical energy to electrical energy. In its simplest form, it consists of a loop of wire rotated by some external means in a magnetic field

In commercial power plants, the energy required to rotate the loop can be derived from a variety of sources. For example, in a hydroelectric plant, falling water directed against the blades of a turbine produces the rotary motion; in a coal-fired plant, the energy released by burning coal is used to convert water to steam, and this steam is directed against the turbine blades. As a loop rotates in a magnetic field, the magnetic flux through the area enclosed by the loop changes with time; this induces an emf and a current in the loop according to Faraday’s law. The ends of the loop are connected to slip rings that rotate with the loop. Connections from these slip rings, which act as output terminals of the generator, to the external circuit are made by stationary brushes in contact with the slip rings.

Motors are devices that convert electrical energy to mechanical energy. Essentially, a motor is a generator operating in reverse. Instead of generating a current by rotating a loop, a current is supplied to the loop by a battery, and the torque acting on the current-carrying loop causes it to rotate.