Energy in a Magnetic Field

11 Key excerpts on "Energy in a Magnetic Field"

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  eBook - ePub

  Electromagnetics Explained

  A Handbook for Wireless/ RF, EMC, and High-Speed Electronics

  • Ron Schmitt(Author)
  • 2002(Publication Date)
  • Newnes

    (Publisher)

  Chapter 6 , you will learn more about the vector potential when we discuss quantum physics.

  MAGNETIC MATERIALS

  Diamagnetism

  In Chapter 2 , you learned that different materials behave differently in electric fields. You learned about conductors and dielectrics. Electric fields induce reactions in materials. In conductors, charges separate and nullify the field within the conductor. In dielectrics, atoms or molecules rotate or polarize to reduce the field. Magnetic fields also induce reactions in materials. However, since there are no magnet charges, there is no such thing as a “magnetic conductor.” All materials react to magnetic fields similarly to the way dielectrics react to electric fields. To be precise, magnetic materials usually interact with an external magnetic field via dipole rotations at the atomic level. For a simple explanation, you can think of an atom as a dense positive nucleus with light electrons orbiting the nucleus, an arrangement reminiscent of the planets orbiting the sun in the solar system. Another similar situation is that of a person swinging a ball at the end of a string. In each situation, the object is held in orbit by a force that points toward the orbit center. This type of force is called a centripetal force. The force is conveyed by electricity, gravity, or the string tension, respectively, for the three situations. Referring to Figure 3.17 and using the cross product right hand rule, you find that the force due to the external magnetic field points inward, adding to the centripetal force. The increase in speed increases the electron’s magnetic field, which is opposite to the external field. The net effect is that the orbiting electron tends to cancel part of the external field. Just as the free electron rotates in opposition to a magnetic field, the orbiting electron changes to oppose the magnetic field. This effect is called diamagnetism

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  Components, Laws and Concepts of Electromagnetism

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 Magnetic Field A magnetic field is a field of force produced by a magnetic object or particle, or by a changing electric field an d is detected by the force it exerts on other magnetic materials and moving electric charges. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field. The complex mathematics underlying the magnetic field of an object is usually illustrated using magnetic field lines. These lines are strictly a mathematical concept and do not exist physically. Nonetheless, certain physical phenomena, such as the alignment of iron filings in a magneti c field, produces lines in a similar pattern to the imaginary magnetic field lines of the object. Magnets exert forces and torques on each other through the magnetic fields they create. Electric currents and moving electric charges produce magnetic fields. Even the magnetic field of a magnetic material can be modeled as being due to moving electric charges. Magnetic fields also exert forces on moving electric charges. The magnetic fields within and due to magnetic materials can be quite complicated and is d escribed using two separate fields which can be both called a magnetic field : a magnetic B field and a magnetic H field. Energy is needed to create a magnetic field. This energy can be reclaimed when the field is destroyed and, therefore, can be considered as being stored in the magnetic field. The value of this energy depends on the values of both B and H . An electric field is a field created by an electric charge and such fields are intimately related to magnetic fields; a changing magnetic field genera tes an electric field and a changing electric field produces a magnetic field. The full relationship between the electric and magnetic fields, and the currents and charges that create them, is described by the set of Maxwell's equations.

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  eBook - ePub

  Electrical Engineering: Know It All

  • Clive Maxfield, John Bird, Tim Williams, Walt Kester, Dan Bensky(Authors)
  • 2011(Publication Date)
  • Newnes

    (Publisher)

  Figure 21.16 . As you can see, its main characteristic is that it points in a direction parallel to the current, and it decays in magnitude as the distance to the current increases.

  Figure 21.16 A plot of the magnetic vector potential surrounding a current-carrying wire

  The magnetic vector potential is much harder to understand than voltage, the electric potential. However, I will sketch out some of its characteristics. The magnetic field stores energy just as the electric field stores energy. In some situations the vector potential can be interpreted as the potential momentum of a charge. In fact, the units of the vector potential are those of momentum per charge. When Maxwell developed his theory of electromagnetism, he called the vector magnetic potential the electrodynamic momentum because it can be used to calculate the total momentum or total kinetic energy of a system of charged particles and their electromagnetic fields.

  21.5 Magnetic Materials

  21.5.1 Diamagnetism

  In Chapter 20 , you learned that different materials behave differently in electric fields. You learned about conductors and dielectrics. Electric fields induce reactions in materials. In conductors, charges separate and nullify the field within the conductor. In dielectrics, atoms or molecules rotate or polarize to reduce the field. Magnetic fields also induce reactions in materials. However, since there are no magnet charges, there is no such thing as a “magnetic conductor.” All materials react to magnetic fields similarly to the way dielectrics react to electric fields. To be precise, magnetic materials usually interact with an external magnetic field via dipole rotations at the atomic level. For a simple explanation, you can think of an atom as a dense positive nucleus with light electrons orbiting the nucleus, an arrangement reminiscent of the planets orbiting the sun in the solar system. Another similar situation is that of a person swinging a ball at the end of a string. In each situation, the object is held in orbit by a force that points toward the orbit center. This type of force is called a centripetal force. The force is conveyed by electricity, gravity, or the string tension, respectively, for the three situations. Referring to Figure 21.17 and using the cross product right hand rule, you find that the force due to the external magnetic field points inward, adding to the centripetal force. The increase in speed increases the electron’s magnetic field, which is opposite to the external field. The net effect is that the orbiting electron tends to cancel part of the external field. Just as the free electron rotates in opposition to a magnetic field, the orbiting electron changes to oppose the magnetic field. This effect is called diamagnetism

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  Matter and Interactions, Volume 2

  Electric and Magnetic Interactions

  • Ruth W. Chabay, Bruce A. Sherwood(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  QUESTION Where is there a decrease of energy in the surroundings? Evidently the energy stored in the fields surrounding the two particles must decrease. Clearly, the electric field at any location in space does change as the positions of the particles change. The electric field in the region between the particles gets larger, but the electric field everywhere else in space decreases (since E dipole is proportional to s, the distance between the particles). It would be a somewhat daunting task to integrate E 2 over the volume of the Universe, with the additional complication that close to a charged particle E approaches infinity. However, we do not actually need to do this integral to figure out the change in energy of the electric field throughout space. Since ∆(Field energy) +∆K positron +∆K electron = 0 then ∆(Field energy) = -2(∆K electron ) In this example, the principle of conservation of energy leads us directly to the idea that energy must be stored in electric fields, since there is no other way to account for the decrease of energy in the surroundings. If we had chosen the electron plus the positron as our system, we would have found that ∆U el is equal to -2(∆K electron ). The change in potential energy for the two-particle system is the same as the change in the field energy. Evidently in a multiparticle system we can either consider a change in potential energy or a change in field energy (but not both); the quantities are equal. The idea of energy stored in fields is a general one. It is not only electric fields that carry energy, but magnetic fields and gravitational fields as well. 658 Chapter 16 Electric Potential 16.11 *POTENTIAL OF DISTRIBUTED CHARGES Potential Along the Axis of a Uniformly Charged Disk R z r Δr Figure 16.53 A ring of radius r and width ∆r makes a contribution V ring to the potential of the disk. Consider a disk of radius R (area A = πR 2 ) with charge Q uniformly distributed over its surface.

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  Magnetism in Medicine

  A Handbook

  • Wilfried Andrä, Hannes Nowak, Wilfried Andrä, Hannes Nowak(Authors)
  • 2007(Publication Date)
  • Wiley-VCH

    (Publisher)

  Magnetic Dipole Interaction (Demagnetizing) Energy Another important energy contribution occurs due to the dipolar interaction of magnetic moments: any magnetic dipole creates the magnetic field (Eq. 1.34); if another dipole is placed into this field, then it possesses an energy according to Eq. (1.37). The energy of a system of two dipoles can be written either as an energy of the first dipole m 1 in the field of the second one at the location point of the first h 21 , or vice versa: E ¼ m 1 h 21 ¼ m 2 h 12 ð1:56Þ Rewriting this expression in a symmetrical form E ¼  1 2 ðm 1 h 21 þ m 2 h 12 Þ ð1:57Þ 1.2.4 Magnetic Field in Condensed Matter: Special Topics 47 we can immediately generalize it to a system of many dipoles: E ¼  1 2 X i m i h i ð1:58Þ where h i means the dipole field created at the location of the i-th dipole by all other dipoles. What we need now is the continuous version of Eq. (1.58) – that is, the energy of the magnetic dipolar interaction which exists between various parts of a ferromag- netic body if the magnetization configuration of a body MðrÞ is known. According to the rules explained in Section 1.2.3, for the transition to the condensed matter case exact microscopic quantities appearing in Eq. (1.58) should be replaced as fol- lows: h i ! B i and m i ! M i DV i . Here, we have subdivided the ferromagnetic body into a (finite) number of small parts with volumes DV i , so that the index i refers now not to the point dipole [as in Eq. (1.58)] but to such a small part of a body. Passing to a continuous limit, we obtain as a generalization of Eq. (1.58) an inte- gral expression E dip ¼  1 2 ð V MB dV ð1:59Þ For further use it is more convenient to rewrite the equation using the field H. Substituting the expression B ¼ H þ 4pM (see Eq. 1.43) into Eq.

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  eBook - ePub

  Energy Medicine - E-Book

  The Scientific Basis

  • James L. Oschman(Author)
  • 2015(Publication Date)
  • Churchill Livingstone

    (Publisher)

  Much of the discussion that follows will concern the behaviour of electrons and other charged particles. We shall see that when a charge moves, magnetic fields are produced. And we will also see that the opposite is true: Magnetic fields alter the motions of nearby charges. These principles are profoundly important for energy medicine. Many of the techniques used in energy medicine look like New-Age hocus-pocus until they are viewed through the discerning eyes of the physicist and biophysicist. Hence this chapter is an introduction to the worldview from these perspectives.

  It is necessary to look at the nature of these particles in some detail because many of the seemingly remarkable phenomena in energy medicine will remain inexplicable without reference to the behaviour of these fundamental particles. What is being presented in this section goes deep – to the fundamental properties of all matter in the universe. Readers may wish to postpone reading this section until their curiosity has built up from the concepts presented in later sections of the book, and they feel motivated to dig deeper.

  Understanding the basic physics and quantum aspects of charged particles is a continuing challenge for scientists from every field of inquiry. Many of the fundamental questions have not been answered completely. Other questions have several answers, with great minds disagreeing about which is correct. This must not stop us from looking at the pieces of the puzzle we understand or think we understand. As we look at these concepts, we must keep in mind that all of the information is subject to change as more discoveries are made. Some of the more extraordinary conclusions that have been reached are not agreed upon by all physicists. For example, there is abundant evidence for non-locality, i.e., at a fundamental level, all of the particles in the universe are dependent on and continuously interacting with all of the other particles in the universe. All parts are continuously in relationship and communication. Some physicists and quantum physicists regard this concept as preposterous; others observe phenomena every day in their laboratories that can only be explained in this way.

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  eBook - PDF

  Quantum Mechanics

  A Modern Development

  • Leslie E Ballentine(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  312 Ch. 11: Charged Particle in a Magnetic Field In interpreting this expression, it should be remembered that, in spite of the apparent complexity, the sum of the first four terms is just the kinetic energy, MV 2 . Sometimes the first term is described as the kinetic energy, and the next three terms are described as paramagnetic and diamag-netic corrections. That is not correct, and indeed the individual terms have no distinct physical significance because they are not separately invariant under gauge transformations. For many purposes, it is preferable not to expand the quadratic term of the Hamiltonian, but rather to write it more compactly as Gauge transformations The electric and magnetic fields are not changed by the transformation (11.2) of the potentials. On the basis of our previous experience, we may anticipate that there will be a corresponding transformation of the state function that will, at most, transform it into a physically equivalent state function. Since the squared modulus, |$(x,£)| 2 , is significant as a probability density, this implies that only the phase of the complex function *(x, t) can be affected by the transformation. (This is similar to the Galilei transformations, which were studied in Sec. 4.3.) The Schrodinger equation, ^ ( ? v -! A ) 2 * + ^ * = i 4 * ' (u.i7) is unchanged by the combined substitutions: A -> A ' = A + V x , (11.18a) 1 9 Y 0_>0' = 0___^ (1L18b) * -> # ' = tfc'fo^* , (11.18c) where x = x(x, t) is an arbitrary scalar function. It is this set of transforma-tions, rather than (11.2), which is called a gauge transformation in quantum mechanics. That the transformed equation ■M^-H 2 *'^'*'^*' (nir) 11.2 Quantum Theory 313 is equivalent to the original (11.17) can be demonstrated in two steps. First, on the right hand side of (11.17') the time derivative of the phase factor from (11.18c) exactly compensates for the extra term introduced on the left hand side by the scalar potential (11.18b).

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  The Energy of Nature

  • E. C. Pielou(Author)
  • 2008(Publication Date)
  • University of Chicago Press

    (Publisher)

  17 That electric charge is measured in terms of electric current, which is meas-ured in terms of magnetic force, gives some idea of the order in which differ-ent topics were developed. The knowledge that flowing electric charge (a current) creates a magnetic field leads to the suspicion that a moving magnetic field might create a current. It does. The most convenient way to make it happen is to move a conductor (a wire or a coil of wire) through a stationary magnetic field; this causes an elec-tric current to flow through the conductor.The kinetic energy provided to the conductor by whatever is moving it becomes converted to electrical energy. This is how an electrical generator (sometimes called a dynamo) works. The close relation between electric and magnetic forces should now be clear. 185 e l e c t r o m a g n e t i c e n e r g y N S N S a b Figure 16.7. (a) The observed field of force of a bar magnet. (b) The (inferred) complete field of force. All lines of force are closed loops passing, for part of their length, through the solid iron of the magnet. A motionless electric charge (an electrostatic charge) creates an electrostatic field, while a stream of electric charges (an electric current) creates a magnetic field. The discovery that electric fields and magnetic fields are actually two manifestations of a single phenomenon, now known as an electromagnetic field, was one of the greatest scientific advances of the nineteenth century. We return to the topic in chapter 18. Before doing so, we look at the magnetic field of the whole earth, which enfolds all of us, everywhere and all the time. The Earth as a Magnet A hiker using a compass is benefiting from the fact that the whole earth is a magnet. The compass needle, because it is a magnet, spontaneously aligns it-self with the earth’s magnetic field.

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  Electromagnetism for Engineers

  • Andrew J. Flewitt(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  It should be noted that the expression for inductance (Eq. (3.38)) is very similar in form to that for capacitance (Eq. (2.25)). The capacitor is an energy storage device (Section 2.3) as an electric field fills the volume of space between the conductors (see Section 1.3). Similarly, energy is stored in inductors as there is a magnetic field filling the volume within the coil, which is given by U = 1 2 LI 2 (3.41) Reference Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Annalen der Physik 322 (10): 891–921. 51 4 Magnetic Fields in Materials 4.1 The Interaction of Magnetic Fields with Matter In Chapter 3, we confined ourselves to considering the properties of magnetic fields in free space. However, it is very common for magnetic fields to interact with matter which itself consists of atoms and, depending on its nature, molecules. In this chapter, we will consider this interaction in detail. We will begin by considering a simple case of an ideal linear magnetic material to give us some basic relationships between key quantities that are used to describe magnetism before considering some real magnetic materials, includ- ing diamagnetic materials, ferromagnetic materials and paramagnetic materials. We will conclude with a discussion on magnetic circuits and electromagnets. 4.2 The Magnetization of Ideal Linear Magnetic Materials Electrons exist in well-defined orbitals around atoms, as determined by quantum mechan- ics, and these behave like small loops of current. Quantum mechanics also shows us that electrons themselves have an intrinsic angular momentum that is normally called spin, and which can be thought of as the electron spinning around on own axis. This will also produce a small loop of current. Therefore, each microscopic constituent of matter, whether individ- ual atoms or molecules, can be considered to have a net current loop associated with it.

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  eBook - ePub

  Physics

  From Natural Philosophy to the Enigma of Dark Matter

  • Anne Rooney(Author)
  • 2020(Publication Date)
  • Arcturus

    (Publisher)

  CHAPTER 4

  Energy fields and forces

  When a force acts to move a mass it seems obvious to us that energy is involved. So it may seem surprising that for all the consideration of forces since antiquity, energy was largely neglected by the early natural philosophers. The concept of energy is relatively new, emerging only in the 17th century. Indeed, the term ‘energy’ (from the Greek energia , coined by Aristotle) was only introduced with its modern meaning in 1807 by the genius and polymath Thomas Young (of the double-slit experiment). The most obvious forms of energy are light and heat, both of which come for free from the Sun. Humankind has also harnessed chemical energy (released by burning fuels), the gravitational energy of a falling body, the kinetic energy of wind and moving water, and, latterly, electric and nuclear energy.

  The conservation of energy

  Just as matter is conserved, being neither created nor destroyed, so energy too is conserved. It may be converted from one form to another – and this is how we harness energy to do useful work – but that energy is never actually spent. Galileo noticed that a pendulum converts gravitational potential energy into kinetic energy or the energy of movement. When the pendulum bob is at the high point of its swing it is momentarily still, and has maximum potential energy. This is converted into kinetic energy as the bob moves, and the bob regains potential energy as it climbs up on the other side of its swing.

  ‘There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy.

  It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.’

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  eBook - ePub

  Electromagnetism

  • I. S. Grant, W. R. Phillips(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  −3 , much less than that in large magnetic fields.

  A proof of Equation (6.35) which has general validity will be giVen m section 6.3.6.

  6.3.5 Non-linear media

  Equation (6.35) is not valid if non-linear media like iron are present. If they are present the magnetic field B is not proportional to the currents which give rise to the field. Equation (6.26) however remains valid for that part of the work done (in changing the magnetic fluxes through the circuits) which goes to increase the stored magnetic energy.

  Consider the simple situation in which a circuit is wound over a piece of magnetic material. Let the circuit have negligible resistance so that heating losses can be ignored. The work d

  Wb

  done by a battery to increase the flux through the circuit by an amount dΦ in time dt is given by Equation (6.26) .

  (6.36)

  where I is the current in the circuit. By writing the flux Φ as ∫s B·dS, where S is any surface whose perimeter is the circuit it can be shown that

  (6.37)

  where the volume integral is taken over all space. (The proof of this equation is given in section 6.3.6.) If the relation between the fields B and H is known, Equation (6.37) can be integrated to obtain the total work done W b in establishing a final field B 0 .

  (6.38)

  When the field is reduced to zero again, the total work done is zero if the relation between the fields B and H is single valued, since then

  In such a situation, as we discussed in the previous section, the work done against induced e.m.f.s whilst the field is being established can be considered to be stored in the field reversibly; the energy can be regained when the field is returned to zero. However, with a piece of ferromagnetic material like iron inside the circuit, the work done to establish the field cannot all be regained, and some energy is dissipated as heat in the iron. If the iron is taken around a complete hysteresis cycle (i.e. the current through the circuit, initially at a certain value, is decreased through zero to the same value in the opposite direction and then returned to its original condition), the energy dissipated as heat in unit volume of the iron can be related to the area under the hysteresis curve* . The curve labelled abcdefa on Figure 6.10 is the major hysteresis curve for commercial iron. If unit volume of commercial iron were taken around this cycle the current sources have to do an amount of work equal to the integral around the cycle. This energy is about 350 joules m−3 as can be estimated from the figure. Iron cores are used in transformers in order to obtain maximum flux linkage between primary and secondary windings. The fields in a mains transformer core are taken through a hysteresis cycle 50 times every second. Transformers are not used under conditions where the iron is near saturation. A typical situation is one in which the fields follow the minor hysteresis loop labelled a'b'c'd'e'f'a' on Figure 6.10 , and the area under this loop is about one sixth of the area under the major hysteresis loop. If 1000 cm3 of commercial iron is used in a transformer which has currents flowing at 50 Hz, the power dissipation is about (350 × ) × 50 × 10−3 W, i.e. about 3 watts. Hysteresis losses are proportional to frequency, since current sources haveto do an amount of work

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