The Magnetic Field

 Elevator to problems

Natural Magnets

Iron bearing rocks known as lodestones were known to early civilizations to have the property of attracting, and being attracted to, iron implements. Many of these lodestones were discovered in a territory in Asia minor known as Magnesia, from which of course we get the word magnet. Besides the property of attracting iron, an interesting feature of lodestones shaped into thin needles is that they always orient themselves in a north-south direction. References to the magnetic compass can be found in very early eastern writings; similar references appeared in the west sometime during the twelfth century.
The first formal, thorough study of magnets was done by Sir William Gilbert at the turn of the 17th century. When not engaging in his duties as court physician to Queen Elizabeth I, Gilbert carried out basic research on the nature of the Earth's magnetic field. He suggested in his book, de Magnete, that the Earth was a giant lodestone with a field strong enough to cause a compass needle to be deflected.
See this applet

While electric fields arise from stationary charges, magnetic fields arise from charges in motion, either by spinning on an axis, or by moving along a line. Electrons in atoms spin as they orbit their nuclei. For most elements, the magnetic fields caused by the spin of some of the electron population are canceled by the fields of electrons spinning in the opposite direction. For three elements, iron, cobalt and nickel, some of the spinning electrons are unpaired, giving each atom a residual magnetic effect. These atoms try to line up into ferromagnetic domains so that their magnetic fields add. This process is hampered by the vibration of atoms due to thermal effects. However, these tiny domains can be made to align themselves when placed in a strong magnetic field. When this happens, an ordinary piece of iron becomes a very strong magnet. Ironically (pun intended), because we cannot  turn them on or off or otherwise control them, natural magnets get relatively little attention in physics

For a discussion of the earth as a magnet, go to
The Exploration of the Earth's Magnetosphere


The first connection between electricity and magnetism was made by the Danish physicist, Hans Christian Oersted. In 1820, he announced that surrounding a current carrying wire was a magnetic field strong enough to deflect a compass needle. Further, he showed that the field direction depended on current direction in the wire and that a compass placed near a wire carrying a current would line up tangent to a circle that had the wire at its center. This discovery caused other scientists to shift their attention from what was happening inside the wire to what was happening outside the wire.
Magnetic field strength (B) about a current carrying wire is a vector quantity given by the equation



see the applet at


where I is the magnitude of the current and R is the distance from the wire to the point in question. There is also a constant involved, uo ( also known as the permeability of free space) uo = 4 pi x 10^-7 Tm/A The are several units that are used for B. The SI unit is the Tesla (T) which is equivalent to a N/A-m. Also, 1 T = 10^4 Gauss = 1 Weber/m^2 = 10^4 Maxwell/cm^2. (These last units are very out-of-vogue and are seen only in old text books.)
The direction of the B field is very curious. Whereas the field lines around a stationary charge radiate outward (positive charge) or inward (negative charge) as the spokes of a wheel, the field lines around a long straight conductor carrying a current are set up tangent to a circle that has the wire at the center. There are two possibilities for a directed line segment to be shown this way. A way to remember which way nature works is to apply a version of the right hand rule for determining the directions of B-fields. Please understand that the direction of a particular magnetic has nothing to do with your hand; it is just a "handy" way to remember.
The right hand rule says "Grab the wire with your right hand, thumb pointing in the direction of the current. Your fingers will wrap around the wire in the same sense as the magnetic field. Note that we said "sense" (clockwise or counter-clockwise) as opposed to same direction , say due north. See the box below.


 Right Hand Rule
Grasp the wire with your right hand so that your thumb points in the direction of the current. Your fingers will wrap around the wire in the same sense as B.
current out of page current into page






See the diagram at left. Consider two long straight parallel wires 10 cm apart and carrying currents of 6 A (top) and 4 A (bottom) in the same direction..Find Bnet A) 5 cm above the top wire; B) 5 cm below the bottom wire; C) midway between the wires; D) Find the point(s) where B net = 0. what happens here?
See the answers in the boxes below.

What happens to these answers if the currents are oppositely directed?

This is the solution to part A  This is the solution to part B This is the solution to part C  This is the solution to part D

Helmholtz Coils


The magnitude of  the B-field falls off quickly as we move away from the wire. We find that if we turn the wire into a loop, the center of the loop is equidistant from the wire and the B-field is constant near the center.  Add more (N) turns of wire and B is multiplied by N.  A special arrangement of two identical coils of some radius R and separated by that same distance R are known as Helmholtz coils.  See the figure.
 A second coil called a solenoid consists of many (N) turns of wire wrapped around a long, narrow cylinder. The B-field inside is very nearly uniform everywhere inside the coil except near the ends.

The applications of the magnetic field to generate electric current, to transport it efficiently over great distances, and then turn it to mechanical energy represent one of the great stories in the history of the development of civilization. We shall look at these applications in the next chapter on Electrodynamics.

See these Applets

Induced current in a coil

The Lorentz Equation
we have already seen that an electric field can propel a charged particle regardless if it is at rest or moving. This is not the case with a magnetic field. A magnet will neither attract nor repel a static charged particle; a magnet will exert a force on a moving charge, but only if the particle is moving across field lines rather than parallel to them. We determine empirically that the force acting on moving charged particles is given by

the magnitude of the force ia given by the bottom line in the box.



 The direction of the force is a vector perpendicular to the plane formed by v and B as guided by a different right hand rule which states, for positive charge carriers;1) point your right thumb in the direction of the first vector mentioned (v) on the right-hand side; 2) point your right index finger in the direction of the second vector mentioned (B); the direction of the resultant vector (F) is perperdicular to the plane formed by V abd B and points in the direction of the middle finger of the right hand.

1) Before continuing, convince yourself that in the diagram at left, if the
blue vector is first and the black vector is second, the resultant points in the direction of the red vector.

2) Notice that the vector cross product is not commutative; the magnitude of the vectors is the same, but the direction of the resultant is different. Show that v x B is not equal to B x v. (Magnitude is the same; direction is different.)

3) Note that if the charge carrier is negative, a left hand rule must be applied. That makes a magnetic field a very useful mechanism for separating a beam of mixed (read: positive and negative) charges



 A variation on the Lorentz equation is

Consider the two wires 10 cm apart seen earlier.


 The lower wire creates a B-field which points out of the page at the site of the upper wire. Because the upper wire carries a current, the field from the lower wire exerts a force on the upper wire. The magnitude of this force is F = IlBsin theta (theta is the angle between the current direction and the B-field.). The direction of the force is given by a right-hand rule. Thumb poits in the direction of the current. Index finger points in the direction of B. Convince yourself that the force vector points down.

But wait!!!

The upper wire carries a current which creates a field at the site of the lower wire. This field points into the page and the force it creates points up. But you knew this would happen because of Newton's third law.

See the excellent applets at these sites. Note that Internet Explorer may not allow you to view these applets. We suggest viewing them with a Netscape browser.

See the Lorentz force act on a wire at

Go to this site and proceed through Prof. Hershfeld's slides

Go to magnetic field problems


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 Thhis page was last reviewed 03/02/09