MAGNETIC FORCE

Magnetic force is the same as gravitational and electrical forces in that no one knows truly what it is. Magnetic force is different from gravitational and electrical forces in that its potential and momentic energy is at the expense of an electrical field in time.

The existence of a magnetic field is a necessary consequence of the laws of electrostatics, and the principles of special relativity. One of the prerequisites of special relativity is there must be motion, and it is in effect when the observer is in motion relative to the system from the observer's point of view.

The classical development of magnetics started when Oersted deflected a compass needle with a electric current carrying wire. Thus, it became believed, all so called magnetic phenomena result from forces between electric charges in motion. The charges in motion relative to an observer set up a magnetic field and an electric field which exerts canceling forces on the second charge relative to the observer.

Because it is very easy to move charges through an wire, most all knowledge about magnetics comes from the study of a current through a wire. However, this is the same as the movement of small electric fields through the wire.

In the early development of magnetic theory, bar magnetics were used to provide a fixed magnetic source. The magnetic field in a bar magnetic had two definite poles. For the lack of any thing better, early scientists referred to these poles as north and south. More modern terminology is positive and negative. One thing that is agreed upon is that there is two distinct opposite in characteristic poles.

It was found fairly simply that a force is developed between two bar magnetics. When the poles are alike; N-N or S-S, the force is repelling. When the poles are opposite; N-S or S-N, the force is attracting.

Till recently, moving electric fields outside wires, which corresponds to a constant net electric field with respect to an observer has not occurred. However, with the advent of nuclear reactions, it is possible to observe an expanding electric field from a point origin. This field is called the ElectroMotive Pulse (EMP). Because of the military value of an electromotive pulse, the phenomena is being studied relatively secretly. In electrostatics the electric intensity is a ratio of the force on a charge at rest to the magnitude of charge. In magnetodynamics the magnetic intensity or induction is the ratio of the force on a moving charge to the product of the charge and the velocity.

Because force and velocity are vectored quantities magnetic induction is a vectored quantity. Thus, the direction of v and polarity of B is taken into account:

F force direction is directly relative to q's polarity with respect to two charges.

General experiments using proton and electron beams in a vacuum perpendicular to a magnetic field, show the direction of deflection of a positively charged particle and a negatively charged particle are 180 degrees from each other

In a circular wire conducting a current, the magnetic field created is about the wire in circular form. The magnetic field about a moving charge is circular about the charge. The magnetic field is created outside the boundary area.

In relating force to magnetic fields and currents in a length of wire L, within a uniform magnetic field B, carrying a current of density J, moving from left to right can be used. The flux density B is perpendicular and inward to the plane of the wire when the observer is looking down on the wire. A positive charge q1 within the wire drifting right with a velocity v1 is acted upon by an upward force F, of magnitude q1*v1*B, the drift velocity of a negative charge is in the opposite direction. Thus the force is in the same direction as negative times negative is plus. This is because the electron is a negative charge and is moving opposite to the charge q1. Therefore F2 = q2*v2*B.

The same forces are present in a wire which is carrying an electric current. In a current carrying wire the forces on charges q1(+) & q2(-) are in the same direction, because the velocity of the charges is in opposite direction. To cause a current J in this conductor requires an electric field. The affect of a current flowing through a conductor in a uniform magnetic field is to force the conductor perpendicular to the magnetic field and perpendicular to the electric current. In 1879, using a flat conductor, it was found that not only was a force detected but also a charge was detected in the direction of the force by majority carriers. This was called the Hall Effect. Later this led to the "hole conduction" theory. The hole conduction theory is the name given to the concept of positive charges being the carriers.

Generally speaking, the unit of magnetic induction B, is 1 newton per ampere-meter. Where the magnetic field F, is uniform and normal to the finite area A, the magnetic flux across the area is given by: Flux = B * A, and the unit of magnetic flux is 1 newton-meter per ampere, or 1 Wb in honer of Mr. Wilhelm Edward Weber, (1804-1891). Thus, B can be termed flux density and is 1 weber per square meter if the element of delta A, is at right angles to the lines of induction. In the cgs system of magnetic values 4 * * = 1, thus, B tended to be called gauss. In the mks system 1 Oersted gave rise to a field of 4 * * 10^-7 Gauss. Named after Mr. Karl Friedrich Gauss, (1777-1855).

A good example of how this is put to use is an electromagnetic pump used in nuclear reactors. It can be shown that metal material can have a force exerted on it if it is subjected to a current and strong magnetic field. This pump is a device which moves metal by means of the interaction between an electric current, liquid metal, and a magnetic field. A pipe containing liquid metal is put in a strong magnetic field and a current is forced perpendicular to the magnetic field and direction of liquid metal flow. The liquid metal has a force on it perpendicular to the magnetic field and electric current, thus, it flows and there is no moving mechanical parts.

If n1 and n2 represent the number of charges per unit volume then:

Since J = sum of n * q * v and the product of J * A = I then force for a length l:

For a conductor of any shape making any angle with the magnetic field, the force is a product of q times the vector product v cross B. Therefore, the force on an element of conductor of length dl is:

This can be shown easily using the Hall effect. For example; given a flat ribbon conductor with a current density J within a magnetic field perpendicular to the direction of current and to the plane of the ribbon, the force acting on the current tends to move electrons to one side of the ribbon and positive charge to the other.

It is interesting to also note that hole flow and positive charge current are different. The concept of hole flow, as the electron moves from one body to another, it leaves a hole where it used to be. This of course, is a positive charged body. However, it was not the body which move. It was the electron that moved. The concept of hole flow has confused many people when evaluating magnetic inductance and forces.

Since magnetics is a vectored quantity and its field and force are perpendicular to charge motion, a rule of hand is in order to visualize what is happening.

Rule 1. In the case of an electron current in a wire, grab the wire with the right hand with the thumb in the direction of current flow; the inducted field will be in the direction of the fingers about the wire.

Rule 2. In the case of an electron moving in a magnetic field; the index finger is in the direction electron motion, the thumb is in the direction of the magnetic field, and the force is in the direction of the middle finger. (Thumb, index finger and middle finger are all perpendicular.)

Remember what is moving. Many books describe current as a flow of positive charges. There is a logical and scientific reason for positive current flow which is commonly referred to as hole flow, however, this can be confused with electron flow. So, it is very important to know what is moving

Now, comes on of my pet things in magnetics. In all the books on the subject of magnetics, the field is referred to as lines of force. In my study of magnetics, I must have read two dozen books on the subject. Each and every one of these books refers to the magnetic field as lines of force. I have read whole dissertations on the subject of magnetic lines. One professor, when attempting to explain how the Sun spots occur, referred to the magnetic lines as snapping like a rubber band. Another very famous person came up with a number of lines for a given area. In all the classes I have had on the subject of physics and magnetics, the teachers referred to the magnetic field as lines of force. Unfortunately, I have been so conditioned that I use the term "lines" too.

The truth is; there are no lines. Magnetic flux is a continuum. The word "lines" comes from primitive magnetic observations. Primarily, those observations which used magnetic materials as graphite to observe the action of the magnetic field. The graphite lines up. This is because the material used creates a special situation where separate units of alignment occur. These units are lines. This is caused by material immediately adjacent to a unit being magnetically polarized with the same polarization. Since magnetic flux of the same polarity creates a force which is directed away from the material, material with the same polarity flux is force apart, thus, separation. The material attracts in one direction and repels in the other causing the illusion of lines.

In the development of magnetic theory there are many terms. These terms can be very confusing. Many really smart people get confused and I can't say I have everything straight either. Magnetic terms fall into the category of equivalent field strength. Because in the early days of magnetic study it was easier to create a magnetic field by a current through a wire, magnetic field strength is in terms of current through a wire rather than some known magnetic field. Because I have seen so many different standards, I will use Van Nostrands's Scientific Encyclopedia Fifth Edition's definition of terms. On occasions I feel they don't quite got a grasp on the situation either.

Maxwell (Mx) mostly written as (M) however it gets mixed up with meters. A maxwell is a unit of magnetic flux. 1 maxwell equals 1 gauss per square centimeter, or, 1 magnetic line of force. Note: the use of lines of force.

Gauss: (G) A unit of magnetic flux density or magnetic induction. The ratio of the flux in any cross section to the area of that cross section, the cross section being taken normal to the direction of flow. 1 Gauss equals 1 Maxwell per square centimeter.

Oersted: (Oe) A unit of magnetic field strength. The magnetic field produced at the center of a plane circular coil of 1 turn and of a radius 1 centimeter, which carries a current of 1/2 abamperes. (1 abampere = 10 amperes)

Tesla: (T) A unit of magnetic flux density (magnetic induction). The magnetic flux density of a uniform field that produces a torque of 1 newton-meter on a plane current loop carrying 1 ampere and having a projected area of 1 square meter on the plane perpendicular to the field. T = (N/A) m.

Weber: (Wb) A unit of magnetic flux. The magnetic flux passing through an area of 1 square meter placed normal to a uniform magnetic field of magnetic flux density equal to 1 tesla. Wb = T m^2. If the flux linked by a circuit changes at a uniform rate of 1 weber per second, a voltage of 1 volt is induced in the circuit. Wb = V s.

Writers, scientists and students can get very confused with magnetics. If you believe positive or negative current flow is confusing, and you might be thinking the definition of terms seem confusing, the next statement should be perfectly clear as mud to you. Believe it or not: 1 Gauss = 10000 gauss & 1 Oersted = 10000 oersted. Now what would you think an Oersted is in terms of Gauss? If you said that to give rise to 1 Gauss requires a field of 10^7 Oersteds you would be might right. A lot of competent people have gone astray with this concept of Oersted & oersted, Gauss & gauss. What does mean regarding Webers and webers, Tesla and tesla, or what? Well; one weber per square meter = 10^4 gauss & 1 maxwell per square centimeter = 1 gauss. Confused? Don't feel lonely.

Can you imagine how easy it would be to write G instead of g? Can you imagine how a reader views this problem? I have read papers from NASA that got the terms Gauss and gauss wrong.

Consider the situation where there are two parallel conductors. When opposite direction current is flowing in both conductors, the resulting magnetic field repels the conductors. When the current is in the same direction in both wires, the resulting magnetic field attracts

Force On Two Parallel Conducting Wires + being current into the page & - being current out of the page.

Commonly associated with magnetic fields is the magnetic moment. The magnetic moment is a term given to the torque applied to a loop carrying a current in a magnetic field. The magnetic moment is analogous to the electric moment of a dipole. It seems the magnetic moment of a loop is independent of the shape of the loop and is dependent upon the area of the loop. Most loops are in multiple loop configuration.

When the magnetic field of external origin is applied to a current carrying loop a torque is developed. This is a force on the loop which tends to twist it. All electric motors work on this principle.

Torque is given by:

The terms of the magnetic moment are the same as gyroscopic moment and electro dipole moment. It is a vectored quantity given by:

Thus, the vectored torque is:

Another characteristic of magnetism is its ability to induce a charge. In effect the induced charge is an electric field.

Suppose there is a conductor of length l, in a magnetic field B, perpendicular to the length of the conductor. Now, move the conductor with a velocity v, such that the ends are perpendicular to the velocity. The force on the free charges would be q * (v X B). Thus, an electric field would be set up that has a force equal to q * Ee. Since the force due to the magnetic field would generate a nonelectrostatic field Fn = q * En. Because the electric field is due to motion, the emf is call motional electromotive force. Customarily stated as the rate of change of flux and given by:

This equates to an electromotive force which is equal to the rate of change of flux across the area bounded by the system in the opposite direction. Either the flux can be moving or the conductor can be moving but if they are both moving at the same speed in the same direction, emf = 0. Further simplification: An electric field which is changing in time sets up a magnetic field. A magnetic field changing with time sets up an electric field. The mobility of the charge affects things immensely. In all the books I have read on the Earth's magnetic field one thing is missing. This little detail is very important. Is the Earth's magnetic field moving with the spin of the Earth or is it stationary?

Emf is a scaler measured in volts. Emf can also be defined as:

Further reduction:

If two conductors were connected at the ends such that loops were formed, one end of a loop in a magnetic field, and the other loop out of the field, then the resultant electric field equals the electric field due to a electrostatic field, plus the electric field due to the nonelectrostatic field. The second loop has an induced current in it. Thus, the general expression for motional emf in an element of conductor dl is:

This could be looked at in terms of two wires.

Over simplified, mutual induction is the source magnetic field affecting an adjacent material. The most common mutual inductance device is the transformer. A transformer is in almost everything electrical. Transformers are in power plants, their on telephone poles, there in telephones, T.V.s, radios, and computers; transformers are all over.

Even in a coil, mutual inductance affects the adjacent portion of the loop. But more importantly, mutual inductance affects any charge particle adjacent to the magnetic source. If the charged particles have a motion of freedom in a linear environment then they will move with that environment linearly. If the charged particles have three degrees of motion freedom than they move accordingly. This mutual induction principle works on any charge from atoms to stars.

From these Michael Faraday (1791-1867), a English chemist and physicist deduced: the line integral of an induced electric field around a closed path, or the electromotive force e in the path, equals the time rate of change of magnetic flux across the area bounded by the path. Hence, The induced electromotive force in a circuit equals the negative of the time rate of change of magnetic flux through the area bounded by the circuit. Also, Lenz's law which is a good rule states: The direction of an induced current is such as to oppose the cause producing it.

Suppose there were a disk of a conductor with a radius R rotating at an uniformed velocity w about an axis which is perpendicular to a magnetic disk. With out regard to energy, what is the maximum emf which this system can induce? Where is the induced emf?

The integral from zero to R of the nonelectrostatic field dotted with the change in radius gives the electric intensity or force by electric field disk segment:

Supposing this is the Earth. Now this is not the way it is, this is only a supposition. Earth has a magnetic field of .3 oersteds at the equator and .6 oersteds at the poles (or, is it .3 Oersteds & .6 Oersteds?).

Given the volts and that the Earth interior is a conductor, current would be induced. There would also be a power transfer. Because the Earth's interior materials are not superconducting, there would be heat induced.

An interesting experiment with the disk shows eddy currents are set up which affect the force and torque on the disk. These eddy currents are very much the same as the Earth's eddy currents and are perpendicular to the rotation.

The terms inductance and magnetic power were developed primarily for current carrying wires, however, they are close to the characteristics of any components associated with magnetics.

The term inductance pertains to a coil of wire. As a current is started in the coil the magnetic field builds up. As the magnetic field builds up, it induces a counter electromotive force in the same wire. Also, as the magnetic field builds up, it will induce a electromotive force in any coil near by. As time becomes close to infinity compared to the time rate, the magnetic field remains constant, and there is no induced electromotive forces. When the current is stopped, the magnetic field collapses, and induces an electromotive force in the opposite direction as starting up, which magnitude depends upon the rate of change current as the current is shut off.

An interesting note here, is that current can not be cut off instantly, because this would create an infinite counter electromagnetic force. This would usually causes an arc. Which is why points on relays and switches ware out.

Ampere's law states that the line integral of the magnetic induction B around any closed path is equal to 4 * * k / c^2 times the net current across the area bounded by the path. Let = 4 * * k / c^2 equals 1.257 x 10^-6 webers per ampere meter.

In a coil of length l the flux = (B * A or * N * I * A) / l

The coil's inductance L is = ( * N^2 * A) / l

The mutual inductance of M = ( * A * N1 * N2) / l

Inductance and mutual inductance is in terms of Volts per ampere per second or henry. The effect on the current is:

The energy stored in the magnetic field is:

The work which is stored in the coil:

In the development of electrical systems, it became important to increase mutual inductance, and decrease losses. The transformer became a very important component in electronics. A transformer is one or more coils of wire used to transform electrical power. The ways transformers are used is unimportant to this paper however, there are a lot of ways.

To improve the transformer requires a grater degree of mutual inductance, which is dependent upon the magnetic field. Therefore, the magnetic properties of various materials was studied. For example, it was known that iron had greater magnetic properties than other materials, so someone stuck iron in the center of the transformer (called core), and as expected the inductance increased. However, because iron has free electrons, iron had eddy currents associated with it because of mutual inductance. Of course, any induced unnecessary currents are losses, the eddy currents were studied. Such was reason for laminating transformer cores, the lamination are perpendicular to the current and insulated from each other, thus, increasing resistance to the eddy current induction, reducing losses.

At this time the reader should be visualizing the Earth's core as the core of a transformer. The difference is that the Earth's core is a hot liquid continuous medium capable of currents and magnetization by induction.

Magnetization M in a material is defined as the magnetic moment per unit volume. Magnetic intensity H in a material is the vector difference between B/ and M

H = B/ - M ampere / meter

To define the nature of material we can introduce a magnetic field in, the term magnetic susceptibility Ms was developed and is the ratio of magnetization to intensity.

To better represent a material, * (1 + Ms) was given the term permeability . Thus,

Another expression is Km = / for relative permeability

In a vacuum the susceptibility is 0 thus Km = 1

To within reason the affects of temperature on susceptibility is proportional to C/T where C is Curies constant and T is the Kelvin temperature. Diamagnetic materials are independent of temperature, thus, only paramagnetic material which increase vibration molecularly with temperature are susceptible to temperature.

There are many material susceptibility types; diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, superparamagnetic, and others which have not been classified yet.

Possibly over simplified, diamagnetic material opposes the development and paramagnetic material helps the development of a magnetic field. Ferromagnetic materials are like paramagnetic materials only the effects are very strong. I, my self would like to define antiferromagnetic materials as ferromagnetic only on diamagnetic materials but that isn't the case. Antiferromagnetic materials have their magnetic moments aligned in an antiparallel fashion in addition to being diamagnetic. Ferrimagnetic materials are like antiferromagnetic materials in that their magnetic moments align in an antiparallel fashion, but their moments have different magnitudes. Superparamagnetic materials are ferromagnetic particles in a nonferromagnetic matrix.

Typical diamagnetic materials are hydrogen, helium, copper, gold, silicon, germanium, graphite, and sulphur. Typical paramagnetic materials are Potassium, oxygen, and tungsten. Typical ferromagnetic materials are iron, nickel, and cobalt. A typical antiferromagnetic material is manganese oxide. Typical ferrimagnetic materials are iron oxide magnetite , nickel-zinc ferrite , and nickel ferrite . Obviously, not only elements exhibit magnetic properties but compounds do to.

Typical susceptibility figures for some common elements and their compound substances found in abundance on or in several planets: Iron & compounds 10,000 x 10^-6 cgs to 20,000 x 10^-6 cgs; Hydrogen & compounds -3.98 x 10^-6 cgs to -63 x 10^-6 cgs, Nickel & compounds 2,000 x 10^-6 cgs to 7,000 x 10^-6 cgs; Silicon & compounds -3 x 10^-6 cgs to -128 x 10^-6 cgs, Aluminum & compounds 16 x 10^-6 cgs to -320 x 10^-6 cgs. The problem with defining susceptibility is it changes with temperature, pressure, ionic state, and what it is mixed with. These values were measured at room temperature and pressure.

Susceptibility of a substance is like an amplification factor. A positive susceptibility would enhance the present magnetic field and a negative susceptibility could reverse it however a negative value would more than likely simply reduce the field.

The truth of the matter is that these magnetic material differences is not well understood. The only common link that all the authorities have in common is the magnetic moments were caused by both the obit of the electron and its spin. Other than the, is all subjective theory.

There is also an effect which allows a material to keep the magnetic field when the inducing magnetic field has been removed. The permeability of these materials is their ability to retain some of the magnetic field which was introduced to them. Since this is in effect a history of the material the term hysteresis was given to the characteristic. Iron is a material which has this characteristic. However, this characteristic decreases with temperature and is nonexistent in a molten material.

Magnetic susceptibility is at the atomic level and hysteresis is at the molecular level.

For the Earth's core there is no hysteresis because of the liquid and high temperature but, the material is still highly susceptible to magnetization from a moving electric field. The Iron at the Earth's surface however, is capable of hysteresis and susceptible to magnetization from a moving electric field.

The Earth's mantle is a Silicon, Oxygen, Aluminum, and Iron semi-liquid mixture. Oxygen's susceptibility varies dramatically with temperature and the elements around it. The Earth's mantle is more or less diamagnetic by volume with a weak susceptibility. But even a small percentage of Iron will out affect Silicon, Oxygen and Aluminum because Iron's susceptibility is more or less 10,000 times that of Si, O, or Al. The Earth's mantle is going to be subject to all sorts of magnetic anomalies due to its variation of materials.

A magnetic field is the vectored sum of all the fields present at any point. Generally, surface magnetic effects are the effect of various types of magnetic fields acting on the Earth's surface at any point. Magnetic fields outside the Earth will affect both the surface and interior magnetic action and magnetic moments. Magnetic field inside the Earth will affect both the surface and near field magnetic action and magnetic moments.

Faraday's work was quoted by James Clerk Maxwell edited by W.D. Niven and I'm using it again. I quote this work not because it is the foremost authority on electromagnetic systems but because both Faraday and Maxwell were trying to explain the Earth's magnetic system but couldn't put a name to the source. A lot of explanations were left out along with some detail. The book is two inches thick and I just don't have room.

The second problem I had with this is "should I embellish it with some of my own explanations?" Laws of electro-tonic nature:

Law I. The entire electro-tonic intensity round the boundary of an element of surfaces incases the quantity of magnetic induction which passes through that surface.

Law II. The magnetic intensity at any point is connected with the quantity of magnetic induction by a set of equations called equations of conduction.

Law III. The entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes thought that surface.

Law IV. The quantity and intensity of electric currents are connected by a system of equations of conduction.

Law V. The total electro magnetic potential of a closed current is measured by the product of the quantity of the current times the entire electro-tonic intensity in the same direction.

Law VI. The electro-motive force on any element of a conductor is measured by the instantaneous rate of change of the electro-tonic intensity on that element whether in magnitude or direction.

Equations of Magnetic force:

F, G, & H is electromagnetic momentum in the x, y, & z directions. The value of force will be given by , , & ; where , , & are magnetic intensity.

Equations of current:

The above and being the angle impact to the field. Ix, Iy, & Iz being where current exists and represent total current including true conduction and displacement currents.

Equations of electromotive force:

Above & = electric potential.

Where s is the displacement:

Equations of Emf intensity:

Where = electric displacement & k is a proportionality constant.

Equations of resistance:

Where p, q & r current due to true conduction & is a proportionality constant.

Equations of conductivity:

Where e being quantity of free electricity & above.

In a conductor of length l parallel to x axis where dx/dt, dy/dt, & dz/dt exist and ds/dt is velocity; where Quantity is:

the increments due to motion of conductor are:

the increments due to the lengthening are:

The total Emf from the current:

One complete chapter in the Scientific Papers of James Clerk Maxwell by Faraday was dedicated to the magnetics of a sphere within a magnetic field. I include one specific section on the affects of a conducting spherical shell revolving in a magnetic field because it is to point. Also, it was this chapter which led me to believe that Faraday was thinking "EARTH" rather than any old sphere. It was the use of the words Terrestrial Magnetism. The word terrestrial was used a lot. Now, Faraday's works were interpreted twice before I got them and I could be reading something into this that isn't there. But, I don't think so. I think Faraday was trying to get a point across.

The core is not only highly susceptible but a good conductor. Faraday's Force on a revolving spherical conduction shell in a magnetic field will cause a magnetic field because it generates a current.

This rendition of the affects of a conducting sphere revolving in a magnetic field is rather short. R is the radius of the shell and T is the thickness of the conducting area. The coefficient of resistance k used in this section is resistance to a magnetic field and not to the conduction of electrons. It turns out that k = 4 R P, where P is the internal magnetic pressure of the material which is a tensor. Such a pressure is responsible for a material being diamagnetic, paramagnetic, or ferromagnetic. The relationship of k is if k is larger than k' then the material is paramagnetic, and if k is smaller than k' then the material is diamagnetic. It seems that Faraday use I as his variable for total terrestrial magnetism, I'll use B because it fits with the first part of this chapter. Omega is a function of rotation and not necessarily angular velocity. As these formulas move to the end, it is seen that changes, thus a constant angular velocity it is not. X, Y, & Z are positions in space, , , & are electro-tonic functions at points x, y, & z. Bx, By, & Bz are symbolic magnetic quantity and intensity at xB, yB, & zB. Ex, Ey, & Ez are electric intensity at positions xe, ye, & ze. is the angle the spin vector has with B.

where;

at;

the Emf of:

where;

and;

Current equals:

The magnetic moment:

The axis along y is such that the magnetic field tends to turn back to the axis of x by electric tension. According to Faraday and some common sense a spinning in a loop tends to create a dipole which wants to align with the magnetic field. This is assuming the spin and field are not aligned to begin with. Another observation is that the current is opposite the direction of spin. This also stands to reason as the electrons are more mobile than the proton so they tend to slide backwards easier. This isn't to say the current is going backwards, it means that the current is a displacement current. Such a current would have a breaking affect on the sphere and that is given by:

As such, because it is a breaking force work must be done and work is:

The force - work relationship for a conducting sphere spinning in magnetic field is in part what the Earth is doing. As the magnetic field intensity goes up the breaking force goes up by the square factor. As the Earth gets closer to the Sun the spin would slow down. This is just the opposite of the affect from gravity. It turns out, this affect is more like what is happening with the Earth's spin.

Another part of this equation is that as the body gets larger the affect is larger. In the case of size, the affect is not just squared but cubed. Also, the Earth is not a conducting sphere with a defined thickness but a partially conducting spherical solid.

The last things which should be noted are the affects of angular velocity and the material's magnetic resistance. As both these change the affect is dependent upon the relative relationship between the two. To state this as a function of speed, assume both the angular velocity and k are 1, the result would be 1 times whatever the rest of the equation yielded. Now, increase both the speed and magnetic resistance by two times and the affect of the force halves. Whereas, if the angular velocity were doubled and the k factor remained the same then the change would be 2/5.

I believe this particular affect is seen with artificial satellites. Both the Russians and NASA had to compensate the satellites spin for reasons which didn't make sense. That is the force on the satellite changed from time to time, day to day, and moment to moment.

What would demonstrate this characteristic best is a drastic change in the electromagnetic medium of space. Early observations show that such an affect is present when a satellite undergoes a field altered by a magnetic storm on the Sun. Even satellites very close to the Earth are affected drastically by the Sun's magnetic changes. The following graph shows a measured spin anomaly of an orbiting satellite immediately after large solar flares.

Interesting enough, the early satellites were conducting spheres housing the electronics and they were set spinning in order to improve trajectory. Later model satellites tended to be totally different than metal spheres and they generally are stationary pointing some antenna toward Earth but the problem of stabilization still exists.