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`Magnetic forces
`Lorentz force
`
`A magnetic field B imparts a force on moving charged particles. The entire
`electromagnetic force on a charged particle with charge q and velocity v is
`called the Lorentz force (after the Dutch physicist Hendrik A. Lorentz) and is
`given by
`
`The first term is contributed by the electric field. The second term is the
`magnetic force and has a direction perpendicular to both the velocity v and
`the magnetic field B. The magnetic force is proportional to q and to the
`magnitude of v × B. In terms of the angle ϕ between v and B, the magnitude
`of the force equals qvB sin ϕ. An interesting result of the Lorentz force is the
`motion of a charged particle in a uniform magnetic field. If v is perpendicular
`to B (i.e., with the angle ϕ between v and B of 90°), the particle will follow a
`circular trajectory with a radius of r = mv/qB. If the angle ϕ is less than 90°,
`the particle orbit will be a helix with an axis parallel to the field lines. If ϕ is
`zero, there will be no magnetic force on the particle, which will continue to
`move undeflected along the field lines. Charged particle accelerators like
`cyclotrons use the fact that particles move in a circular orbit when v and B are
`at right angles. For each revolution, a carefully timed electric field gives the
`particles additional kinetic energy, which makes them travel in increasingly
`larger orbits. When the particles have acquired the desired energy, they are
`extracted and used in a number of different ways, from fundamental studies
`of the properties of matter to the medical treatment of cancer.
`
`The magnetic force on a moving charge reveals the sign of the charge carriers
`in a conductor. A current flowing from right to left in a conductor can be the
`result of positive charge carriers moving from right to left or negative charges
`moving from left to right, or some combination of each. When a conductor is
`placed in a B field perpendicular to the current, the magnetic force on both
`types of charge carriers is in the same direction. This force, which can be seen
`in Figure 5, gives rise to a small potential difference between the sides of the
`conductor. Known as the Hall effect, this phenomenon (discovered by the
`American physicist Edwin H. Hall) results when an electric field is aligned
`with the direction of the magnetic force. As is evident in Figure 5, the sign of
`the potential differs according to the sign of the charge carrier because, in one
`case, positive charges are pushed toward the reader and, in the other, negative
`charges are pushed in that direction. The Hall effect shows that electrons
`dominate the conduction of electricity in copper. In zinc, however, conduction
`is dominated by the motion of positive charge carriers. Electrons in zinc that
`are excited from the valence band leave holes, which are vacancies (i.e.,
`unfilled levels) that behave like positive charge carriers. The motion of these
`holes accounts for most of the conduction of electricity in zinc.
`
`#
`
`magnetic force on moving charges
`Figure 5: Magnetic force on moving charges. The magnetic force F is proportional to the charge and to
`the magnitude of velocity v times the magnetic field B.
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`If a wire with a current i is placed in an external magnetic field B, how will the
`force on the wire depend on the orientation of the wire? Since a current
`represents a movement of charges in the wire, the Lorentz force given in
`equation (5) acts on the moving charges. Because these charges are bound to
`the conductor, the magnetic forces on the moving charges are transferred to
`the wire. The force on a small length dl of the wire depends on the orientation
`of the wire with respect to the field. The magnitude of the force is given by
`idlB sin ϕ, where ϕ is the angle between B and dl. There is no force when ϕ =
`0 or 180°, both of which correspond to a current along a direction parallel to
`the field. The force is at a maximum when the current and field are
`perpendicular to each other. The force is obtained from equation (5) and is
`given by
`
`Again, the cross product denotes a direction perpendicular to both dl and B.
`The direction of dF is given by the right-hand rule illustrated in Figure 6. As
`shown, the fingers are in the direction of B; the current (or in the case of a
`positive moving point charge, the velocity) is in the direction of the thumb,
`and the force is perpendicular to the palm.
`
`#
`
`right-hand rule
`Figure 6: Right-hand rule for the magnetic force on an electric current (see text).
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`Repulsion or attraction between two magnetic dipoles
`
`The force between two wires, each of which carries a current, can be
`understood from the interaction of one of the currents with the magnetic field
`produced by the other current. For example, the force between two parallel
`wires carrying currents in the same direction is attractive. It is repulsive if the
`currents are in opposite directions. Two circular current loops, located one
`above the other and with their planes parallel, will attract if the currents are in
`the same directions and will repel if the currents are in opposite directions.
`The situation is shown on the left side of Figure 7. When the loops are side by
`side as on the right side of Figure 7, the situation is reversed. For two currents
`flowing in the same direction, whether clockwise or counterclockwise, the
`force is repulsive, while for opposite directions, it is attractive. The nature of
`the force for the loops depicted in Figure 7 can be obtained by considering the
`direction of the currents in the parts of the loops that are closest to each other:
`same current direction, attraction; opposite current direction, repulsion. This
`seemingly complicated force between current loops can be understood more
`simply by treating the fields as though they originated from magnetic dipoles.
`As discussed above, the B field of a small current loop is well represented by
`the field of a magnetic dipole at distances that are large compared to the size
`of the loop. In another way of looking at the interaction of current loops, the
`loops of Figure 7 (top) and 7 (bottom) are replaced in Figure 8A and 8B by
`small permanent magnets, with the direction of the magnets from south to
`north corresponding to the direction of the magnetic moment of the loop m.
`Outside the magnets, the magnetic field lines point away from the north pole
`and toward the south pole.
`
`#
`
`$
`
`magnetic force between current loops
`Figure 7: Magnetic force between current loops. In each case shown, the arrow indicates the
`direction of the current i and the magnetic dipole moment m of a loop (see text).
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`READ MORE ON THIS TOPIC
`electromagnetic radiation: Relation
`between electricity and magnetism
`As early as 1760 the Swiss-born mathematician
`Leonhard Euler suggested that the same ether that
`propagates light is responsible for electrical...
`
`It is easy to understand the nature of the forces in Figures 7 and 8 with the
`rule that two north poles repulse each other and two south poles repulse each
`other, while unlike poles attract. As was noted earlier, Coulomb established an
`inverse square law of force for magnetic poles and electric charges; according
`to his law, unlike poles attract and like poles repel, just as unlike charges
`attract and like charges repel. Today, Coulomb’s law refers only to charges,
`but historically it provided the foundation for a magnetic potential analogous
`to the electric potential.
`
`The alignment of a magnetic compass needle with the direction of an external
`magnetic field is a good example of the torque to which a magnetic dipole is
`subjected. The torque has a magnitude τ = mB sin ϑ. Here, ϑ is the angle
`between m and B. The torque τ tends to align m with B. It has its maximum
`value when ϑ is 90°, and it is zero when the dipole is in line with the external
`field. Rotating a magnetic dipole from a position where ϑ = 0 to a position
`where ϑ = 180° requires work. Thus, the potential energy of the dipole
`depends on its orientation with respect to the field and is given in units of
`joules by
`
`Equation (7) represents the basis for an important medical application—
`namely, magnetic resonance imaging (MRI), also known as nuclear magnetic
`resonance imaging. MRI involves measuring the concentration of certain
`atoms, most commonly those of hydrogen, in body tissue and processing this
`measurement data to produce high-resolution images of organs and other
`anatomical structures. When hydrogen atoms are placed in a magnetic field,
`their nuclei (protons) tend to have their magnetic moments preferentially
`aligned in the direction of the field. The magnetic potential energy of the
`nuclei is calculated according to equation (7) as −mB. Inverting the direction
`of the dipole moment requires an energy of 2mB, since the potential energy in
`the new orientation is +mB. A high-frequency oscillator provides energy in
`the form of electromagnetic radiation of frequency ν, with each quantum of
`radiation having an energy hν, where h is Planck’s constant. The
`electromagnetic radiation from the oscillator consists of high-frequency radio
`waves, which are beamed into the patient’s body while it is subjected to a
`strong magnetic field. When the resonance condition hν = 2mB is satisfied,
`the hydrogen nuclei in the body tissue absorb the energy and reverse their
`orientation. The resonance condition is met in only a small region of the body
`at any given time, and measurement of the energy absorption reveals the
`concentration of hydrogen atoms in that region alone. The magnetic field in
`an MRI scanner is usually provided by a large solenoid with B of one to three
`teslas. A number of “gradient coils” insures that the resonance condition is
`satisfied solely in the limited region inside the solenoid at any particular time;
`the coils are used to move this small target region, thereby making it possible
`to scan the patient’s body throughout. The frequency of the radiation ν is
`determined by the value of B and is typically 40 to 130 megahertz. The MRI
`technique does not harm the patient because the energy of the quanta of the
`electromagnetic radiation is much smaller than the thermal energy of a
`molecule in the human body.
`
`The direction of the magnetic moment m of a compass needle is from the end
`marked S for south to the one marked N for north. The lowest energy occurs
`for ϑ = 0, when m and B are aligned. In a typical situation, the compass
`needle comes to rest after a few oscillations and points along the B field in the
`direction called north. It must be concluded from this that Earth’s North Pole
`is really a magnetic south pole, with the field lines pointing toward that pole,
`while its South Pole is a magnetic north pole. Put another way, the dipole
`moment of Earth currently points north to south. Short-term changes in the
`Earth’s magnetic field are ascribed to electric currents in the ionosphere.
`There are also longer-term fluctuations in the locations of the poles. The angle
`between the compass needle and geographic north is called the magnetic
`declination (see Earth: The magnetic field of the Earth).
`
`The repulsion or attraction between two magnetic dipoles can be viewed as
`the interaction of one dipole with the magnetic field produced by the other
`dipole. The magnetic field is not constant, but varies with the distance from
`the dipole. When a magnetic dipole with moment m is in a B field that varies
`with position, it is subjected to a force proportional to that variation—i.e., to
`the gradient of B. The direction of the force is understood best by considering
`the potential energy of a dipole in an external B field, as given by equation (7).
`The force on the dipole is in the direction in which that energy decreases most
`rapidly. For example, if the magnetic dipole m is aligned with B, then the
`energy is −mB, and the force is in the direction of increasing B. If m is
`directed opposite to B, then the potential energy given by equation (7) is
`+mB, and in this case the force is in the direction of decreasing B. Both types
`of forces are observed when various samples of matter are placed in a
`nonuniform magnetic field. Such a field from an electromagnet is sketched in
`Figure 9.
`
`#
`
`electromagnet
`Figure 9: A small sample of copper in an inhomogeneous magnetic field (see text).
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`Magnetization effects in matter
`
`Regardless of the direction of the magnetic field in Figure 9, a sample of
`copper is magnetically attracted toward the low field region to the right in the
`drawing. This behaviour is termed diamagnetism. A sample of aluminum,
`however, is attracted toward the high field region in an effect called
`paramagnetism. A magnetic dipole moment is induced when matter is
`subjected to an external field. For copper, the induced dipole moment is
`opposite to the direction of the external field; for aluminum, it is aligned with
`that field. The magnetization M of a small volume of matter is the sum (a
`vector sum) of the magnetic dipole moments in the small volume divided by
`that volume. M is measured in units of amperes per metre. The degree of
`induced magnetization is given by the magnetic susceptibility of the material
`χ , which is commonly defined by the equation
`m
`
`The field H is called the magnetic intensity and, like M, is measured in units
`of amperes per metre. (It is sometimes also called the magnetic field, but the
`symbol H is unambiguous.) The definition of H is
`
`Magnetization effects in matter are discussed in some detail below. The
`permeability µ is often used for ferromagnetic materials such as iron that have
`a large magnetic susceptibility dependent on the field and the previous
`magnetic state of the sample; permeability is defined by the equation B = µH.
`From equations (8) and (9), it follows that µ = µ (1 + χ ).
`0
`m
`
`The effect of ferromagnetic materials in increasing the magnetic field
`produced by current loops is quite large. Figure 10 illustrates a toroidal
`winding of conducting wire around a ring of iron that has a small gap. The
`magnetic field inside a toroidal winding similar to the one illustrated in Figure
`10 but without the iron ring is given by B = µ Ni/2πr, where r is the distance
`0
`from the axis of the toroid, N is the number of turns, and i is the current in the
`wire. The value of B for r = 0.1 metre, N = 100, and i = 10 amperes is only
`0.002 tesla—about 50 times the magnetic field at Earth’s surface. If the same
`toroid is wound around an iron ring with no gap, the magnetic field inside the
`iron is larger by a factor equal to µ/µ , where µ is the magnetic permeability
`0
`of the iron. For low-carbon iron in these conditions, µ = 8,000µ . The
`0
`magnetic field in the iron is then 1.6 tesla. In a typical electromagnet, iron is
`used to increase the field in a small region, such as the narrow gap in the iron
`ring illustrated in Figure 10. If the gap is 1 cm wide, the field in that gap is
`about 0.12 tesla, a 60-fold increase relative to the 0.002-tesla field in the
`toroid when no iron is used. This factor is typically given by the ratio of the
`circumference of the toroid to the gap in the ferromagnetic material. The
`maximum value of B as the gap becomes very small is of course the 1.6 tesla
`obtained above when there is no gap.
`
`#
`
`electromagnet
`Figure 10: An electromagnet made of a toroidal winding around an iron ring that has a small gap (see
`text).
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`1 2
`
`The energy density in a magnetic field is given in the absence of matter by
`
`2
`B /µ ; it is measured in units of joules per cubic metre. The total magnetic
`0
`energy can be obtained by integrating the energy density over all space. The
`direction of the magnetic force can be deduced in many situations by studying
`distribution of the magnetic field lines; motion is favoured in the direction
`that tends to decrease the volume of space where the magnetic field is strong.
`This can be understood because the magnitude of B is squared in the energy
`density. Figure 11 shows some lines of the B field for two circular current
`loops with currents in opposite directions.
`
`#
`
`magnetic field of two current loops
`Figure 11: Magnetic field B of two current loops with currents in opposite directions (see text).
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`Because Figure 11 is a two-dimensional representation of a three-dimensional
`field, the spacing between the lines reflects the strength of the field only
`qualitatively. The high values of B between the two loops of the figure show
`that there is a large energy density in that region and separating the loops
`would reduce the energy. As discussed above, this is one more way of looking
`at the source of repulsion between these two loops. Figure 12 shows the B
`field for two loops with currents in the same direction. The force between the
`loops is attractive, and the distance separating them is equal to the loop
`radius. The result is that the B field in the central region between the two
`loops is homogeneous to a remarkably high degree. Such a configuration is
`called a Helmholtz coil. By carefully orienting and adjusting the current in a
`large Helmholtz coil, it is often possible to cancel an external magnetic field
`(such as Earth’s magnetic field) in a region of space where experiments
`require the absence of all external magnetic fields.
`
`#
`
`magnetic field of two current loops
`Figure 12: Magnetic field B of two current loops with currents in the same direction (see text).
`Image: Courtesy of the Department of Physics and Astronomy, Michigan State University
`
`Frank Neville H. Robinson
`
`Eustace E. Suckling
`
`Edwin Kashy
`
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`Apple v. GUI Global Products
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`Apple v. GUI Global Products
`IPR2021-00471 GUI Ex. 2017
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