# Magnetism: Nature of Magnetism, Magnetic Field, Magnetic Field of a Straight Current, Magnetic Field of a Current Loop, Earth’s Magnetic Field, Magnetic Force on a Moving Charge Magnetic Force on a Current, Force Between Two Currents, Ferromagnetism and Magnetic Intensity

Magnetism

In This Chapter:

Nature of Magnetism

Magnetic Field

Magnetic Field of a Straight Current

Magnetic Field of a Current Loop

Earth’s Magnetic Field

Magnetic Force on a Moving Charge

Magnetic Force on a Current

Force Between Two Currents

Ferromagnetism

Magnetic Intensity

Nature of Magnetism

Two electric charges at rest exert forces on each other according to Coulomb’s law. When the charges are in motion, the forces are different, and it is customary to attribute the differences to magnetic forces that occur between moving charges in addition to the electric forces between them. In this interpretation, the total

force on a charge Q at a certain time and place can be divided into two parts: an electric force that depends only on the value of Q and a magnetic force that depends on the velocity v of the charge as well as on Q. In reality, there is only a single interaction between charges, the electromagnetic interaction. The theory of relativity provides the link be- tween electric and magnetic forces: Just as the mass of an object moving with respect to an observer is greater than when it is at rest, so the electric force between two charges appears altered to an observer when the charges are moving with respect to the observer. Magnetism is not distinct from electricity in the way that, for example, gravitation is.

##### You Need to Know
###### Despite the unity of the electromagnetic interaction, it is convenient for many purposes to treat electric and magnetic effects separately.

Magnetic Field

A magnetic ﬁeld B is present wherever a magnetic force acts on a moving charge. The direction of B at a certain place is that along which a charge can move without experiencing a magnetic force; along any other direction that the charge would be acted on by such a force. The magnitude of B is equal numerically to the force on a charge of 1 C moving at 1 m/s perpendicular to B.

The unit of magnetic ﬁeld is the tesla (T), where

The gauss (G), equal to 10−4 T, is another unit of magnetic ﬁeld some- times used.

Magnetic Field of a Straight Current

The magnetic ﬁeld a distance s from a long, straight current I has the magnitude

where m is the permeability of the medium in which the magnetic ﬁeld exists. The permeability of free space m0 has the value

The ﬁeld lines of the magnetic ﬁeld around a straight current are in the form of concentric circles around the current. To ﬁnd the direction of B, place the thumb of the right hand in the direction of the current; the curled ﬁngers of that hand then point in the direction of B (Figure 15-1).

Figure 15-1

Magnetic Field of a Current Loop

The magnetic ﬁeld at the center of a current loop of radius r has the magnitude

The ﬁeld lines of B are perpendicular to the plane of the loop, as shown in Figure 15-2(a). To ﬁnd the direction of B, grasp the loop so the curled ﬁngers of the right hand point in the direction of the current; the thumb of that hand then points in the direction of B [Figure 15-2(b)].

Figure 15-2

A solenoid is a coil consisting of many loops of wire. If the turns are close together and the solenoid is long compared with its diameter, the magnetic ﬁeld inside it is uniform and parallel to the axis with magnitude

In this formula, N is the number of turns, L is the length of the solenoid, and I is the current. The direction of B is as shown in Figure 15-3.

Figure 15-3

Figure 15-4

Earth’s Magnetic Field

The earth has a magnetic ﬁeld that arises from electric currents in its liq- uid iron core. The ﬁeld is like that which would be produced by a current loop centered a few hundred miles from the earth’s center whose plane is tilted by 11° from the plane of the equator (Figure 15-4). The geomagnetic poles are the points where the magnetic axis passes through the earth’s surface. The magnitude of the earth’s magnetic ﬁeld varies from place to place; a typical sea-level value is

3 × 10−5 T.

Solved Problem 15.1 In what ways are electric and magnetic ﬁelds similar? In what ways are they different?

Solution.

Similarities: Both ﬁelds originate in electric charges, and both ﬁelds can exert forces on electric charges.

Differences: All electric charges give rise to electric ﬁelds, but only a charge in motion relative to an observer gives rise to a magnetic ﬁeld.

Figure 15-5

Electric ﬁelds exert forces on all charges, but magnetic ﬁelds exert forces only on moving charges.

###### Magnetic Force on a Moving Charge

The magnetic force on a moving charge Q in a magnetic ﬁeld varies with the relative directions of v and B. When v is parallel to B, F = 0; when v is perpendicular to B, F has its maximum value of

The direction of F in the case of a positive charge is given by the right- hand rule, shown in Figure 15-5; F is in the opposite direction when the charge is negative.

###### Magnetic Force on a Current

Since a current consists of moving charges, a current-carrying wire will experience no force when parallel to a magnetic ﬁeld B and maximum force when perpendicular to B. In the latter case, F has the value

where I is the current and L is the length of wire in the magnetic ﬁeld. The direction of the force is as shown in Figure 15-6.

Figure 15-6

Note!

Owing to the different forces exerted on each of its sides, a current loop in a magnetic ﬁeld always tends to rotate so that its plane is perpendicular to

B. This effect underlies the operation of all electric motors.

Force Between Two Currents

Two parallel electric currents exert magnetic forces on each other (Fig- ure 15-7). If the currents are in the same direction, the forces are attractive; if the currents are in opposite directions, the forces are repulsive. The force per unit length F/L on each current depends on currents I1 and I2 and their separation s:

Solved Problem 15.2 A positive charge is moving virtually upward when it enters a magnetic ﬁeld directed to the north. In what direction is the force on the charge?

Solution. To apply the right-hand rule here, the ﬁngers of the right hand are pointed north and the thumb of that hand is pointed upward. The palm of the hand faces west, which is therefore the direction of the force on the charge.

###### Ferromagnetism

The magnetic ﬁeld produced by a current is altered by the presence of a substance of any kind. Usually the change, which may be an increase or a decrease in B, is very small, but in certain cases, there is an increase in B by hundreds or thousands of times. Substances that have the latter effect are called ferromagnetic; iron and iron alloys are familiar examples.

##### Remember
###### An electromagnet is a solenoid with a ferromagnetic core to increase its magnetic ﬁeld.

Ferromagnetism is a consequence of the magnetic properties of the electrons that all atoms contain. An electron behaves in some respects as though it is a spinning charged sphere, and it is therefore magnetically equivalent to a tiny current loop. In most substances, the magnetic ﬁelds of the atomic electrons cancel, but in ferromagnetic substances, the cancellation is not complete and each atom has a certain magnetic ﬁeld of its own. The atomic magnetic ﬁelds align themselves in groups called do- mains with an external magnetic ﬁeld to produce a much stronger total B. When the external ﬁeld is removed, the atomic magnetic ﬁelds may re- main aligned to produce a permanent magnet. The ﬁeld of a bar magnet has the same form as that of a solenoid because both ﬁelds are due to parallel current loops (Figure 15-7).

Magnetic Intensity

A substance that decreases the magnetic ﬁeld of a current is called dia- magnetic; it has a permeability m that is less than m0. Copper and water

Figure 15-7

are examples. A substance that increases the magnetic ﬁeld of a current by a small amount is called paramagnetic; it has a permeability m that is greater than m0. Aluminum is an example. Ferromagnetic substances have permeabilities hundreds or thousands of times greater than m0.

##### Note!
###### Diamagnetic substances are repelled by magnets; paramagnetic and ferromagnetic ones are attracted by magnets.

Because different substances have different magnetic properties, it is useful to deﬁne a quantity called magnetic intensity H, which is independent of the medium in which a magnetic ﬁeld is located. The magnetic intensity in a place where the magnetic ﬁeld is B and the permeability is m is given by

The unit of H is the ampere per meter. Magnetic intensity is sometimes called magnetizing force or magnetizing ﬁeld.

The permeability of a ferromagnetic material at a given value of H varies both with H and with the previous degree of magnetization of the material. The latter effect is known as hysteresis.