How far does a magnet attract?

Magnetism¶

People have known and used magnetic phenomena since ancient times. In particular, the use of thin, rotatable magnetic disks in compasses was in many ways "trend-setting" for the development of mankind. In the meantime, electromagnets, electric motors, generators and transformers in particular have become an indispensable part of our modern society.

Permanent magnets¶

Probably the best known is the attraction of magnets on iron. Substances that, like iron, are attracted by magnets and become magnetized themselves, are called "ferromagnetic" or colloquially as "magnetic" after the Latin name of iron ("ferrum"). These include cobalt, nickel, neodymium and a few alloys.

Magnets have (almost) no effect on non-ferromagnetic materials.

The model of the elementary magnets

Magnets always have two opposing areas where their magnetic effect is particularly strong. These areas are called magnetic poles.

If a magnet breaks into two parts, you can see that both parts have two magnetic poles. Even with further division, the magnetic poles always appear in pairs. Conversely, each magnet can be understood from a large number of the smallest "elementary magnets":

  • If all elementary magnets of a magnet point in the same direction in an orderly manner, their effect is increased - the result is a strong magnetic field.
  • If the elementary magnets point in different directions, their effects cancel each other out.

The elementary magnets of a magnetizable material have different orientations. In a magnetized material, however, the elementary magnets have a uniform magnetic order.

In contrast to electrical charges, there are no “mono-poles” in magnets, only “di-poles”: In a magnet, the north and south poles never appear individually, but always in combination.

Magnetic Influence

In non-magnetized iron, the elementary magnets are arranged in areas. These so-called “Weiss districts”, however, have different orientations, so that a piece of iron does not exert any magnetic effect on other bodies.

Iron can become a magnet itself by approaching a magnet. During this process, known as “magnetic influence”, the individual elementary magnets follow the external magnetic field like compass needles. Since the effects of the individual elementary magnets add up, the iron also acts as a magnet to the outside world.

The extent to which the magnetic order is retained when the external magnet is removed depends on the composition of the magnetized material:

  • In the case of magnetically "soft" materials, for example soft iron, the magnetism caused by influence (largely) disappears again.
  • In magnetically "hard" materials, such as steel, the magnetic effect is permanently retained.

Vibrations or strong heating can destroy the order of the elementary magnets again. A certain temperature (“Curie temperature”) can be specified for all substances, above which the thermal movement of the substance particles prevents a magnetic order. For example, iron is off no longer magnetizable.

The magnetic field¶

The force that a magnet exerts on another magnetic body depends on the distance between the magnets, their shape and their orientation to one another.

A magnet shows the strongest force at its two opposite poles:

  • Magnetizable bodies (e.g. iron paper clips) are equally strongly attracted by both magnetic poles (magnetic influence).
  • When their magnetic poles approach, permanent magnets and electromagnets experience a strong attractive or repulsive effect, depending on their orientation. If the magnetic poles - based on the earth's magnetic field - are referred to as the north and south poles, it can be seen that north and south poles attract each other, while north and north poles or south and south poles repel each other.

The magnetic force is strongest in the immediate vicinity of the magnetic poles and decreases with increasing distance from the magnetic poles.

The easiest way to observe the dependence of the magnetic forces on the shape of the magnet is with the help of iron filings. These are deflected or attracted to different degrees at different points around the magnet, so that in each case a pattern that is characteristic of the shape of the magnet results.

A magnetic needle also deflects at every point in the same direction as the iron filings when it is moved around a magnet, whereby the north pole of the magnetic needle is always oriented towards the south pole of the magnet.

The closed lines that result in the patterns of the iron filings around the magnet thus indicate the direction of the magnetic force. This spatial distribution of the magnetic forces around the magnet is referred to as a "magnetic field" and is represented in technical drawings (somewhat simplified) by a series of closed lines. Often small arrows are drawn along these "magnetic field lines", each at the north pole from the Magnet out and re-enter at the South Pole.

In this way, both the force effect of a magnet can be visualized and its influence on other magnetic or magnetizable bodies can be estimated.

Summary:

  • Magnetic poles always appear in pairs. The magnetic field lines run along these on always closed paths.
  • The arrows of the magnetic field lines point inside the magnet in the direction of the north pole, outside the magnet in the direction of the south pole.
  • The same magnetic poles repel each other, different magnetic poles attract each other.
  • The magnetic effect decreases with increasing distance from the magnetic poles.
  • If several magnets are aligned in the same direction, their effects increase to form an overall magnetic field. If the orientation changes, (almost) no magnetic effect can be detected towards the outside.
  • Magnetic fields can already be shielded by thin magnetizable plates (made of iron, for example), but non-magnetizable materials penetrate almost without attenuation.

Electromagnet¶

In 1820, Hans-Christian Oersted discovered that a current-carrying electrical conductor also exerts a (weak) force on a nearby compass needle. This statement contradicted the prevailing view that electricity and magnetism were two separate phenomena.

By positioning the electrical conductor and the magnetic needle (s) in different positions, Örsted was able to determine the weak magnetic field of the conductor in the same way as with permanent magnets. He could observe the following law:

Right-hand rule (for conductor tracks):

If you hold a straight conductor through which current flows with the right hand so that the outstretched thumb points along the technical direction of the current (from to), then the remaining fingers indicate the direction of the magnetic field lines.

The right-hand rule also applies when the electrical conductor is bent into a circular shape. The thumb must be carried along with the curvature of the conductor in order to always point in the technical direction of the current. It turns out that the resulting magnetic field always points in the same direction at the various points on the conductor loop.

Example:

  • If the conductor loop is on a table and the technical current direction is counterclockwise, the magnetic field inside the conductor loop points vertically upwards at every point. (On the outside of the conductor loop, it points vertically downwards.)

The force effects at the individual points of the electrical conductor add up in a conductor loop to form an overall magnetic field that is reminiscent of the shape of a ring magnet. This effect can be significantly increased by laying several conductor loops on top of one another or by winding a long conductor into a coil.

The magnetic field of a current-carrying coil

A long current-carrying coil with many turns generates a magnetic field on the outside that is very similar to that of a bar magnet. The right-hand rule again provides information about the direction in which the north or south pole of the "electromagnet" is located.

Right-hand rule (for coils):

If you hold the turns of a coil with your right hand in such a way that the fingers point in the technical direction of the current, then the magnetic north pole of the coil is in the direction of the outstretched thumb.

The magnetic effect of a current-carrying coil can be increased by placing a piece of non-magnetized iron through the coil. Due to the magnetic influence, the iron itself becomes magnetic and considerably increases the magnetic effect of the coil.

Electromagnets have the following advantages over permanent magnets:

  • Electromagnets can be switched on and off.
  • The strength of an electromagnet can be regulated by the strength of the current in the coil.
  • The north and south poles of an electromagnet can be swapped by swapping the plus and minus poles of the applied voltage.
  • Electromagnets are almost unbreakable, do not lose strength over time, and are inexpensive to manufacture.

Magnetic field strength and magnetic flux density¶

The force effect of a magnetic field can be quantified by the so-called magnetic field strength to be discribed.

With a straight conductor track, the magnetic field strength is proportional to the flowing current and inversely proportional to the distance from the conductor track. So the following applies:

The unit of the magnetic field strength is according to the above formula .

Inside a conductor loop, the value of the magnetic field strength is a factor of that of a straight conductor path increased, so it applies . Combine a number on conductor loops to form a long coil, the magnetic field strength inside the coil is almost "homogeneous", so the field lines run parallel to each other. For the magnetic field strength of a coil with one number of turns and a length applies:

A second important parameter for describing magnetic fields is the so-called magnetic flux density . It clearly shows how many field lines go through a certain area. There is the following relationship between the magnetic flux density, which characterizes the spatial distribution of the field lines, and the magnetic field strength, which describes the force effect on other magnets:

Here referred to the magnetic field constant. The unit for the magnetic flux density is:

For example, the magnetic flux density of the earth is around .

Strictly speaking, the above relationship (2) only applies in a vacuum, but to a good approximation also in air. However, if there is another material in the magnetic field, its so-called permeability must be taken into account, which indicates whether the field lines running through the material are more likely to be contracted by the material or pushed out of the material. In general, the following applies:

For the size of the permeability number there are essentially three different cases:

  • In diamagnetic materials , The magnetic flux density is therefore slightly reduced compared to the external field. Examples: copper, zinc, water, nitrogen.
  • In paramagnetic materials is , The magnetic flux density is thus slightly increased compared to the external field. Examples: aluminum, platinum, oxygen.
  • In ferromagnetic materials it is , is the magnetic flux density is therefore greatly increased compared to the external field. Examples: iron, cobalt, nickel.

The permeability numbers of dia- and paramagnetic materials are usually only a little different, so that the magnetic properties of these materials rarely play a role in technical applications.

The Lorentz Force¶

While every flow of electrical current results in a magnetic field, every magnetic field has an influence on an electrical current.

If a piece of a current-carrying conductor is in a magnetic field, a force acts on the piece of conductor. It is called the "Lorentz force" after its discoverer Hendrik Antoon Lorentz. The amount of force depends on the strength of the current flow or the magnetic field as well as the angle between the conductor and the magnetic field:

  • The magnitude of the force is greatest when the conductor is oriented perpendicular to the magnetic field.
  • If the conductor and the magnetic field point in the same direction, there is no force.

The Lorentz force always acts perpendicular to the direction of the current and perpendicular to the magnetic field. The following rule of thumb applies to the relationship between current flow, magnetic field and direction of the acting force:

Three-finger rule:

If the thumb of the right hand points in the direction of the technical current direction (from to) and the index finger in the direction of the magnetic field, the middle finger indicates the direction of the force acting.

One denotes the length of the stream traversed conductor piece, which is in the magnetic field is located with so can the Lorentz force can be calculated quantitatively using the following formula:

If the current-carrying conductor runs perpendicular to the magnetic field, the amount of the Lorentz force is the same , otherwise results in the vector product, in which indicates the angle between the conductor piece and the magnetic field. Not just a single piece of conductor, but also a coil, for example If windings are introduced into the magnetic field, this also results in a -fold path length and thus a -fold force effect; this is used, for example, in the construction of electric motors.

Movement of charged particles in magnetic fields

An electric current corresponds to a temporal transport of electric charge. If you put the context between the amperage , the amount of cargo transported and the time it takes into which the above formula (3) follows:

This equation can also be written like this:

The ratio of the distance and time can as speed to be written. Thus it results for one with a charge charged particle moving with speed by a magnetic field with a flux density emotional:

In this case, too, the vector product gives a maximum value when the particle moves perpendicular to the magnetic field; otherwise it can be used for the vector product also be written, where indicates the angle between the direction of movement of the charged particle and the direction of the magnetic field. The above equation can also be written like this:

Is , then applies , and the Lorentz force takes its maximum value at.

Application: mass spectrometer

The Lorentz force can be used to determine the so-called “specific charge” of a particle, that is, the ratio of its charge and its mass to determine. To do this, the particle (or a beam of similar particles) is allowed to travel at one speed enter a magnetic field perpendicular to the magnetic field lines.

If a charged particle, for example an ion or an electron, moves in the horizontal plane and the magnetic field is vertical, the speed remains of the particle even if it is deflected by the Lorentz force perpendicular to the magnetic field. If the arrangement is in a vacuum, there are no frictional forces, so the value of the speed is of the particle constant. If you also keep the magnetic flux density of the magnetic field is constant, the amount of the Lorentz force also remains constant.

The Lorentz force forces the charged particle to follow a circular path through continuous deflection. The necessary radial force must be used as a condition of equilibrium and the Lorentz force be the same size:

For the specific charge of the particle we get:

Is the flux density of the magnetic field as well as the entry velocity known so can by measuring the radius the circular path can be closed to the specific charge of the particle; such an arrangement is called a mass spectrometer.

If the particle beam involves ions each with the same charge, a mass spectrometer can be used in particular to determine the ratio in which individual isotopes of the ion occur; these have largely the same chemical and physical properties, but vary in their mass . In a mass spectrometer, isotopes with a larger mass are directed onto circular paths with a larger radius, so that the frequency distribution of the individual isotopes can be measured using appropriate detectors.

Application: Hall sensors

The Lorentz force is also used in "Hall sensors" named after Edwin Hall, which can be used to measure magnetic flux densities.

A Hall sensor basically has the following structure:

The electrons in a Hall sensor are deflected by the Lorentz force until an equally large, oppositely directed force is established by the electrical field that is building up. If this is the case, the following equilibrium prevails:

The electrical voltage that is established in the case of equilibrium becomes "Hall voltage" called; it can be tapped and measured at corresponding connections along the broad side of the plate. According to the above equation, the following applies to the Hall voltage:

If one knows the speed of the electrons, which occurs depending on the semiconductor material when the current flows through the plate, then the magnetic flux density can be determined directly by measuring the Hall voltage an external magnetic field can be calculated.

Hall sensors have many applications in practice, especially because they are very resistant to water, dirt and vibrations. They are used, for example, in so-called “current clamps” for non-contact measurement of currents in electrical conductors and in cable locators; Both cases take advantage of the fact that electrical currents in conductors cause a magnetic field that can be measured with Hall sensors. Hall sensors are also used for speed measurement in vehicles and turbines; Only small magnets have to be built into the objects rotating past the Hall sensor, which cause a short voltage signal with each rotation of the sensor, which can be registered with a suitable measuring circuit.

Electromagnetic induction¶

If you move a piece of metal perpendicular to the field lines of a homogeneous magnetic field, the Lorentz force inside the conductor causes the electrons to be shifted perpendicular to the direction of movement and also perpendicular to the direction of the magnetic field. Since only the free electrons move, but not the positively charged atomic nuclei, an electric field or voltage builds up inside the metal between the two ends.

This process is called electromagnetic induction. If a measuring device or a consumer (for example a small light bulb) is connected to the ends of the conductor, the induced voltage can in principle be measured or used as a power source. In the case of a single short section of conductor, however, only very low voltage values ​​in the milivolt range result; higher voltage or current values ​​are obtained if the individual conductor section is replaced by a coil with as many turns as possible.

The above figure also shows the technical direction of the current that results from the specified mechanical force when the ladder swing moves (Lorentz force or three-finger rule). If no consumer or measuring device is connected, an electric field builds up in the same direction due to the shifting of charges inside the ladder swing. In the case of equilibrium, the Lorentz force, which the electrons experience inside the piece of conductor moved by the magnetic field, is balanced by the resulting electric field:

The resulting voltage becomes induction voltage called, refers to the length of the conductor that is in the magnetic field. Will follow the above equation dissolved, it results:

Induction voltage only occurs when the conductor is moving at a speed is moved perpendicular to the magnetic field. If the speed has the value zero or if the movement runs parallel to the magnetic field lines, the induced voltage is zero.

Faraday’s law of induction¶

A more general formula for describing the voltage occurring in electromagnetic induction To be able to set up, it makes sense to have a new size to define what is called "magnetic flux":

Between the magnetic flux and the magnetic flux density there is a similar relationship to that between the current intensity and the current density : The latter quantities each indicate the areal density of the reference quantity. The magnetic flux density is not about charge carriers that migrate through a certain area, but about the number of field lines that pass through it. The magnetic flux describes the amount of field lines that pass through an area run away.

Is the magnetic flux The induced voltage is constant over time through an area equals zero. If, on the other hand, the magnetic flux changes by optionally choosing the strength the magnetic flux density and / or the size of the area through which the magnetic field flows changes, it becomes an induction voltage generated, which is greater, the faster this change occurs. The relationship for the induction voltage in a conductor loop is:

This law, according to which the voltage induced in a conductor loop is equal to the change in the magnetic flux over time, was first discovered in 1831 by Michael Faraday.

Examples:

  • In a homogeneous and temporally constant magnetic field, there can be a change in the area through which the magnetic field flows be effected on the one hand by moving the conductor loop into or out of the magnetic field; on the other hand, the same effect, as can be seen in the following figure, can also be brought about by rotating the conductor loop in the magnetic field.

  • If you move a bar magnet through a stationary conductor loop, the magnetic flux density of which is not homogeneous (but increases towards the poles), the conductor loop feels a magnetic field that changes over time. Here, too, a voltage (albeit a very low one) is induced.

The two possibilities described above for generating an induction voltage can be optimized by using a coil consisting of as many turns as possible instead of a single conductor loop. For the induction voltage of a coil with Turns:

Lenz's rule

In the above equations (7) and (8) for describing the induction voltage, there is always the change over time a minus sign. This is a consequence of a law discovered by Emil Lenz, according to which the induction voltage always counteracts its cause.

The cause of the induction voltage is ultimately the Lorentz force, which causes a shift of the electrons and thus an electric field inside the moving conductor loop (s). However, this field does not accelerate the electrons any further (which would result in a perpetual motion machine), but rather slow them down.

... to be continued ...