Static Magnetic Field In Vacuum

Magnetism: History And Applications

Our knowledge of magnetism and magnetic phenomena is as old as science itself. According to the writings of the great Greek philosopher Aristotle, the attractive power of magnets was known by Thales of Miletus at about 600 B.C. From these origins with the Greek philosophers, the sciences of electricity and magnetism developed separately for centuries.

Longer later, in the 16th century A.D., English physician William Gilbert (1544 – 1603) studied the properties of magnets. He realized that a magnetic field existed around the earth’s sphere. He could magnetize an iron sphere and showed that the magnetic field around the sphere was similar to that around the earth.

In the 18^th century, the theory of magnetism developed in parallel to electrostatics. The basic law that evolved was the inverse-square law of attraction and repulsion between unlike and like magnetic poles. With the development of the voltaic cell (battery) by Alessandro Volta (1745 –1827), an Italian chemist and physicist, the magnetic effects of currents were discovered in 1820 by Hans Christian Oersted (1777 – 1851), a Danish chemist and physicist. At that time, he found that an electric current in a wire can deflect a magnetic compass needle.

This was followed by the formulation of the Biot–Savart law for the magnetic field from a long straight current-carrying wire in 1820. Further studies by André-Marie Ampère (1775 –1836), a French physicist and mathematician, led to the law of force between conductors carrying currents.

Later, James Clerk Maxwell (1831 – 1879), a Scottish physicist and mathematician, was the first to formulate the four equations that describe the propagation of the electromagnetic field in the 19th century.

The applications of magnetic fields and magnetic materials are countless and changing rapidly every year. Magnets are used in many devices like Electric machines and analog meters. They also drive the speaker cones of loudspeakers and headphones in TVs, computers, and telephones. A modern car comes equipped with dozens of magnets because they are required in the motors for engine ignition, automatic window control, sunroof control, and windshield wiper control. Most security alarm systems, doorbells, and automatic door latches employ magnets. Large electromagnets are used to pick up heavy loads. Magnetic tapes are also used routinely in sound- and video-recording equipment, and also to store computer data. Finally, intense magnetic fields are used in magnetic resonance imaging (MRI) devices to explore the human body with better resolution and greater safety than x-rays can provide.

Figure 1 shows a magnetic model of the Earth sphere. Earth behaves like a very large permanent magnet, creating a magnetic field that extends from within the planet out into space. Also, the Earth’s magnetic poles are different from its geographic poles.

Overview On Magnetism

Magnetism is closely linked with electricity. Oersted was the first scientist who discovered that electric currents create magnetic fields and also magnetic fields affect moving charges.

There are two main ways that magnetic fields are produced. One way is using elementary particles such as electrons because these particles have an intrinsic magnetic field around them. It means, the magnetic field, just as mass and electric charge, is a basic characteristic of each particle. In certain materials, the magnetic fields of the electrons add together to give a net magnetic field around the material. Such addition is the reason why a magnet has a permanent magnetic field. In other materials, the magnetic fields of the electrons cancel out, giving no net magnetic field surrounding the material.

Let us assume a magnet which has the shape of a bar. Iron objects are most strongly attracted to either end of such a bar magnet, called its poles. One end is called the north pole (N) and the other the south pole (S). The names come from the behavior of a magnet in the presence of Earth’s magnetic field. If a bar magnet is suspended from its midpoint by a piece of string so that it can swing freely in a horizontal plane, it will rotate until its magnetic north pole points to the geographical north of Earth and its south pole points to the south of Earth, like a compass.

Simple experiments with two bar magnets show that like poles repel each other and unlike poles attract each other. Figure 2 shows such phenomena in the presence of 2 bar magnets.

In Figure 2 (a) the N-end of one magnet attracts the S-end of another magnet and the force F is attractive. But, in Figure 2 (b) the N-end of one magnet repels the N-end of the other magnet and the force F is then repulsive.

Although the force between opposite magnetic poles is similar to the force between positive and negative electric charges, there is an important difference: positive and negative electric charges can exist in isolation of each other; north and south poles don’t. No matter how many times a permanent magnet is cut, each piece always has a north pole and a south pole.

The other way to produce a magnetic field is to use moving electrically charged particles, such as a current in a wire, or in a solenoid or a loop, to make an electromagnet.

It is possible to repeat the experiment in Figure 2 in the presence of current. For example, one of the permanent magnets is replaced by a coil of wire bearing a current from a current source ‘i’ to influence the other permanent magnet in the same way. This is shown in Figure 3.

The coil is a circuit element of current-carrying wire, with one or more turns, usually roughly circular or cylindrical, designed to produce a magnetic field. From this experiment, we conclude that the underlying mechanism is the same – i.e., the force generated by a current-bearing coil is the same phenomenon as the force associated with a permanent magnet.

The extent to which a piece of material retains its magnetism depends on whether it is classified as magnetically hard or soft. Soft magnetic materials are easily magnetized but tend to lose their magnetization easily. These materials, such as iron, are used in the cores of transformers, generators, and motors. Other magnetically soft materials include ferrites. Ferrites are used in high-frequency applications, such as radios.

Hard magnetic materials are used in permanent magnets. Such magnets provide magnetic fields without the use of electricity. In the case of a permanent magnet, the magnetic field arises from mechanisms occurring at the scale of the atoms comprising the material. Permanent magnets are used in many devices, including loudspeakers, and the read/write heads of computer hard drives. There are a large number of different materials used in permanent magnets. For example, ‘Alnico’ is a family of permanent magnet alloys made primarily of aluminum (Al), nickel (Ni), and cobalt (Co).

Magnetic Field

On a microscopic scale, an electric field surrounds any stationary electric charge. When the charge is in motion, the surrounding region of space also includes a magnetic field. On a macroscopic scale, a magnetic field surrounds a properly magnetized material. A static magnetic field remains constant over time, meaning it does not vary in magnitude or direction.

Experiments show that a stationary charged particle doesn’t interact with a static magnetic field. When a charged particle is moving through a magnetic field, however, a magnetic force acts on it. This is quite different from the electric force, which exerts a force on a charged particle whether it’s moving or at rest. Further, the electric force is directed parallel to the electric field whereas the magnetic force on a moving charge is directed perpendicular to the magnetic field.

Let us consider the effect of a magnetic field on an electrically-charged particle. First, imagine a region of free space with no electric or magnetic fields. Next, imagine that a positively-charged particle ‘q’ appears. This motionless particle will experience no force (v = 0 then F = 0). Next, a magnetic field (B) appears; perhaps this is due to a permanent magnet or a current in the vicinity. This situation is shown in Figure 4a.

Still, no force is applied to the particle. Nothing happens until the particle is set in motion with velocity v > 0. In our assumption, the direction of v is perpendicular to this page and toward the reader. Figure 4b shows this situation. Suddenly, the particle perceives a force F. Then, the magnetic field applies a force to a charged particle in motion.

It is worth noting that a single charged particle in motion is the simplest form of current. Therefore, not only is the current the source of the magnetic field, the magnetic field also exerts a force on current.

In vacuum, the magnetic Field represented by the vector field B, which is also called magnetic flux density. To fully describe a vector field, we must define both its magnitude (or strength) and direction at every point in space. When describing magnetic fields, we occasionally refer to the concept of magnetic field lines, defined as the curves in space traced out by following the direction in which the magnetic field vector points.

The direction of a magnetic field B at any location is the direction in which the north pole of a compass needle points at that location. Figure 5 shows how the magnetic field of a permanent bar magnet can be traced with the aid of a compass.

Magnetic field lines have some properties:

• They are directed from the north pole to the south pole of the magnet;
• Lines bend at the surface of magnetic materials and make closed loops;
• Lines can penetrate magnetic materials;
• Their density is proportional to the magnetic field strength.

The same rules were followed in drawing Figure 1. In fact, Earth’s geographic north pole is actually near its magnetic south pole and vice versa. Therefore, magnetic field lines emerge from Earth’s surface near the geographic south pole (in vicinity of the magnetic north pole) and end near the geographic north pole. Once outside Earth, the field lines curve around the planet, extending far into space.

Magnetic field lines are remarkable for the reason that a magnetic field line always forms a closed loop, unlike the electric field, it means there are no magnetic monopoles.

Let us return to Figure 4b. We can describe the properties of the magnetic field B at some point in terms of the magnetic force F exerted on a test charge at that point. Our test object is a charge q moving with velocity v. It is found experimentally that the magnetic force F on the particle is proportional to the magnitude of the charge q, the velocity of the particle v, and the external magnetic field B.

Then, the magnetic force applied to the particle can be calculated by Equation 1:

where ‘×’ denotes the cross product between the two vectors v and B. The magnetic force F in Equation 1 is called the Lorentz force.

In Mathematics, the cross product (or vector product) of two vectors V₁ and V₂ is a third vector (with a hypothetical name C) that is perpendicular to both V₁ and V₂. The magnitude of this resultant vector (|C|) is a product of magnitudes of the 2 vectors, times the sine function of the angle θ between directions of V₁ and V₂. The resultant vector is defined as Equation 2:

where:

• ∣V₁∣ and ∣V₂∣ are the magnitudes (lengths) of the vectors.
• θ is the angle between V₁ and V₂.h
• sinθ determines how much of the vectors are perpendicular to each other.
• n^ is a unit vector perpendicular to both V₁ and V₂ (following the right-hand rule).

Thus, for the positive charge q moving with the velocity v in the presence of the magnetic field B, we can realize that the magnetic force F is always perpendicular to both v and B. The magnitude of the vector F is a product of magnitudes of the 2 vectors, times the sine function of the angle θ between the directions of v and B. Finally, Equation 3 explains the magnitude and the direction of the force F:

Figure 6 shows how the direction of the magnetic force F is the same as the unit vector n^.

To determine the direction of the force, we employ right-hand rule number 1:

1. Point your right-hand fingers in the direction of the velocity v.
2. Rotate your fingers toward the magnetic field B, moving through the smallest angle
3. Your thumb is now pointing in the direction of the magnetic force F (= v × B) exerted on a positive charge.

Figure 7 shows how to use the right-hand rule for a positive moving charge.

So, if q is negative, simply use the right-hand rule to find the direction for positive q and then reverse that direction for the negative charge. Figure 8 illustrates the effect of a magnetic field on charged particles with opposite signs moving at the same velocity v in the presence of the magnetic field B.

Let us go back to Equation 3, we recall that the magnitude of the magnetic force F can be calculated by Equation 4:

This expression can be used to find the magnitude of the magnetic field B as explained in Equation 5:

If F is in newtons, q in coulombs, and v in meters per second, the SI unit of the magnetic field is the tesla (T), named after Nikola Tesla (1856–1943), a Serbian-American engineer, and inventor known for his contributions to electromagnetism and electrical engineering.

Then, if a 1-C charge moves in a direction perpendicular to a magnetic field of magnitude 1 T with a speed of 1 m/s, the magnetic force exerted on the charge is 1 N.

For example, conventional laboratory magnets can produce magnetic fields as large as about 2.5 T. This value can be compared to the small value of Earth’s magnetic field near its surface, which is only about 0.5 x 10^-4 T.

From Equation 4 we see that the force on a charged particle moving in a magnetic field has its maximum value when the particle’s motion is perpendicular to the magnetic field, corresponding to θ = 90°, so that sin θ = 1. The magnitude of this maximum force has a value as explained in Equation 6:

Also from Equation 1, F is zero when v is parallel to B (corresponding to θ = 0° or 180°), so no magnetic force is exerted on a charged particle when it moves in the direction of the magnetic field or opposite the field.

Summary

• Moving charges produce magnetic fields.
• Magnetic fields can be produced by a constant current or by permanent magnets.
• Magnetic forces are produced by magnetic fields and act on other permanent magnets and charges in motion (or currents).
• Magnetic fields are an intrinsic property of some materials, most notably permanent magnets.
• A bar magnet has poles identified as “N” (north) and “S” (south).
• Magnetic poles also exert attractive or repulsive forces on each other similar to the electrical forces between charged objects.
• The magnetic field or magnetic flux density is a vector field which we identify using the symbol B and which has SI units of tesla (T).
• The answer to the question of how we can quantify a magnetic field, involves an experimentally-derived equation that predicts force F as a function of charge, velocity vector, magnetic field vector, and the sine function of the angle between them
• The force F exerted by the magnetic field is perpendicular to both the direction of charge motion and the direction in which the magnetic field points.
• If the charge is negative rather than positive, the force F is directed opposite.
• The magnetic force F has its maximum value when the charge moves in a direction perpendicular to the magnetic field lines, decreases in value at other angles, and becomes zero when the particle moves along the B field lines.
• Magnetic field lines always form closed loops.

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