Ampere's Law
Ampere's Law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop. It provides a mathematical relationship between the magnetic field and the electric current that produces it, similar to how Gauss's Law relates electric fields to electric charges. Ampere's Law is a fundamental principle in the study of electromagnetism.
8 Key excerpts on "Ampere's Law"
eBook - ePub- Wen Geyi(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...In this case we have where ε r = 1 + χ e = ε/ε 0 is a dimensionless number, called relative permittivity. Note that (1.5) holds in the dielectric. 1.1.2 Ampère’s Law There is no evidence that magnetic charges or magnetic monopoles exist. The source of the magnetic field is the moving charge or current. Ampère’s law asserts that the force that a current element J 2 dV 2 exerts on a current element J 1 dV 1 in vacuum is (1.10) where R is the distance between the two current elements, u R is the unit vector pointing from current element J 2 dV 2 to current element J 1 dV 1, and μ 0 = 4π x 10 –7 is the permeability in vacuum. Equation (1.10) can be written as where d B is defined as the magnetic induction intensity produced by the current element J 2 dV 2 By superposition, the magnetic induction intensity generated by an arbitrary current distribution J is (1.11) This is called the Biot-Savart law, named after the French physicists Jean-Baptiste Biot (1774–1862) and Felix Savart (1791–1841). Equation (1.11) may be written as where A is known as the vector potential defined by Thus (1.12) This is called Gauss’s law for magnetism, which says that the magnetic flux through any closed surface S is zero Taking the rotation of magnetic induction intensity and using ∇ 2 (1/ R) = —4 πδ (R) and ∇ · J = 0 yields (1.13) This is the differential form of Ampère’s law. Example 1.2: Consider a small circular loop of radius a that carries current I. The center of the loop is chosen as the origin of the spherical coordinate system as shown in Figure 1.2. The vector potential is given by Figure 1.2 Small circular loop where u l is the unit vector along current flow and l stands for the loop. Due to the symmetry, the vector potential is independent of the angle ϕ of the field point P. Making use of the following identity where S is the area bounded by the loop l, the vector potential can be written as If the loop is very small, we can let R′ ≈ R...
eBook - ePubEnergy Medicine - E-Book
The Scientific Basis
- James L. Oschman(Author)
- 2015(Publication Date)
- Churchill Livingstone(Publisher)
...One reason Ampère’s law is so important for energy medicine is that it explains the origin of the biomagnetic field surrounding the human body. As will be described in Chapter 5. See Figure 5.17, the various organs in the body generate electrical currents that flow through the tissues and therefore generate magnetic fields both within and around the body. Figure 2.5 André-Marie Ampère (1775–1836). The strongest electrical field is produced by the heart and generates a current that flows through the circulatory system, which is a good conductor. In accord with Ampère’s law, this current produces the strongest biomagnetic field of the body (Figure 2.6). This point is important because of the skepticism about energy fields around the body that are described and used by various complementary and alternative medicine (CAM) practitioners. The laws of physics require the production of a biomagnetic field around the body as a result of the electrical activity of the heart and other organs. In Chapter 8 we will see the methods that have been used to measure these biomagnetic fields. Figure 2.6 The strongest electrical field is produced by the heart and generates an electrical field that is conducted throughout the body via the circulatory system (A), which is a good conductor. The electrical field of the heart is recorded in the electrocardiogram (B). In accord with Ampère’s law, this current produces the strongest biomagnetic field of any organ, and the field is radiated into the space surrounding the body (C). Modern devices can record the biomagnetic field of the heart, which is called a magnetocardiogram (D). Biomagnetic fields are discussed in detail in Chapter 8. Electricity from Magnetism: Faraday’s Law of Induction About 11 years after Ørsted’s important discovery in Denmark, another important finding took place simultaneously in England and America...
- G. Jagadeeswar Reddy, T. Jayachandra Prasad(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...differential length, and the sine of the angle lying between the element and a line connecting the element to the point P where the field is desired, (b) Inversely proportional to the square of the distance from the differential element to the point P. (c) Directly proportional to the constant of the medium (µ) and (d) Directed normal to the plane containing the differential element and the line drawn from the filament to the point P. B = ∫ μ I d l sin θ 4 π r 2 Wb/m 2 6. State Ampere’s law for a magnetic field. The Ampere’s law states that the line integral of H around a single closed path is equal to the current enclosed. It can also be stated as the line integral of B around a single closed path is equal to the permeability of the medium times the current enclosed. ∫ H ⋅ d l = I ∫ B ⋅ d l = μ I 7. What is the force between two current carrying conductors? The force between the two conductors carrying current I 1 and I 2 separated by. a distance r is given by F = μ 0 I 1 I 2 2 π r N 8. Give the relation between magnetic flux density and magnetic field intensity. The permeability of any medium is the ratio of the magnetic flux density B to the magnetic field intensity H. μ = B H H/m 9. State the Gauss’s law for magnetic fields. The integral of the magnetic flux density B over a closed surface is zero. This is called the Gauss’s law for magnetic fields. ∮ B ⋅ d S = 0 where, dS is the normal component of the surface. 10. What is the torque on a current carrying loop? The torque, or moment, of a force is a vector whose magnitude is the product of the magnitudes of the vector force, the vector lever arm, and the sine of the angle between these two vectors...
eBook - ePubElectromagnetics Explained
A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
- Ron Schmitt(Author)
- 2002(Publication Date)
- Newnes(Publisher)
...This electric field manifests itself in the circuit as voltage. The opposing voltage persists until the current reaches its final steady-state value. The current, therefore, cannot change instantaneously, but continuously changes from zero to its final value over a period of time. Furthermore, while the current is increasing, a voltage drop exists across the wires. A voltage together with a current implies power loss. Although all real wires have resistive (heating) losses, you can ignore such losses for this example. The power loss encountered here actually corresponds to the power transferred into the magnetic field surrounding wires. Just as it takes energy to increase the speed of a car, it also takes energy to increase the speed of change in a circuit (i.e., the current). You can think of Lenz’s law as a way in which nature “balances its books.” Energy is always conserved, and Lenz’s law tells us how energy conservation is maintained with magnetic fields. FARADAY’S LAW Lenz’s law provides a qualitative understanding of how a changing magnetic field creates an electric field. Faraday’s law, proposed by 19th-century scientist Michael Faraday, describes this action qualitatively. For a solenoid with N turns and a cross sectional area A, Faraday’s law can be written as where dB/dt is the change in magnetic field per unit time and ν is the resulting voltage in the circuit. INDUCTORS At last it is time to learn about inductors. In contrast to the capacitor, which requires only one field (namely the electric field) to describe its operation, the inductor requires both fields to describe its operation even though it stores only magnetic energy. Here again is an inherent difference between the electric and magnetic fields. An inductor is a circuit element used to store magnetic energy. Typically, an inductor is created from several loops of wire stacked together to form a solenoid...
eBook - ePub- C R Robertson(Author)
- 2008(Publication Date)
- Routledge(Publisher)
...However, if it was refined and performed under controlled conditions, then it would yield the following results: The magnitude of the induced emf is directly proportional to the value of magnetic flux, the rate at which this flux links with the coil, and the number of turns on the coil. Expressed as an equation we have: Notes: 1 The symbol for the induced emf is shown as a lower-case letter e. This is because it is only present for the short interval of time during which there is relative movement taking place, and so has only a momentary value. 2 The term dΦ/d t is simply a mathematical means of stating ‘the rate of change of flux with time’. The combination N Φ/d t is often referred to as the ‘rate of change of flux linkages’. 3 The minus sign is a reminder that Lenz’s law applies. This law is described in the next section. 4 Equation (5.1) forms the basis for the definition of the unit of magnetic flux, the weber, thus: The weber is that magnetic flux which, linking a circuit of one turn, induces in it an emf of one volt when the flux is reduced to zero at a uniform rate in one second. In other words, 1 volt = 1 weber/second or 1 weber = 1 volt second. 5.2 Lenz’s Law This law states that the polarity of an induced emf is always such that it opposes the change which produced it. This is similar to the statement in mechanics, that for every force there is an opposite reaction. 5.3 Fleming’s Righthand Rule This is a convenient means of determining the polarity of an induced emf in a conductor. Also, provided that the conductor forms part of a complete circuit, it will indicate the direction of the resulting current flow. The first finger, the second finger and the thumb of the right hand are held out mutually at right angles to each other (like the three edges of a cube as shown in Fig. 5.3)...
eBook - ePubFields of Force
The Development of a World View from Faraday to Einstein.
- William Berkson(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...As mentioned above, the force between two current elements in Ampère’s theory is proportional to the strengths of the two currents and to the inverse square of the distance between the current elements—in analogy to Newton’s law of gravitation. The analogy to Newton breaks down because the force (a central force) is also a function of the angles between the two elements. In order to calculate the force between two currents (A and B) on Ampère’s theory, we must first add or integrate the effect of each current element of A on a particular element of B. If we thus work out the effect of A on each elment of B, and add the effects, we get the force of one current upon another. Ampère insisted on not applying his theory to open currents, as he was unable to carry out any experiments with them. Neumann thought that the induced currents must be some function of the already known ‘Ampèrian’ forces, and set out to find that function. 2 He was inspired in his idea by the theory of E. Lenz. Lenz had noticed that the forces due to induced currents were always such as to oppose the change in the forces existing before induction. 3 For example, if we have two like currents (which attract) and move them closer to each other, then the current induced will tend to decrease the attraction between the wires. By Ampère’s theory, the act of moving them closer together tends to increase the attraction. What Neumann discovered was that this mathematical function, giving the ‘electromotive’ forces causing the induced current, was the rate of change of the ‘potential’ of the force of one current upon the other. The ‘potential’ function was already a well-known function and the forces themselves were already calculable from Ampère’s theory...
eBook - ePub- Adrian Waygood(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...It follows, therefore, that if currents move through each of two, parallel conductors, then the resulting magnetic fields will react with each other, and produce a force between the two conductors. This force may be one of attraction or of repulsion, depending upon the relative directions of the electric currents in the conductors. If we apply Faraday’s properties of lines of magnetic flux, which we learnt from the chapter on magnetism (specifically, ‘parallel flux lines acting in the same direction repel other’, while ‘parallel flux lines acting in opposite directions cancel’) then, as illustrated in Figure 17.14, if the currents are drifting in opposite directions, the resulting force between them will be one of repulsion ; whereas if the currents are drifting in the same direction as each other, then the resulting force will be one of attraction. You will recall that the SI unit of electric current, the ampere, has been defined in terms of the force between two straight, parallel, current-carrying conductors. The above description explains the reason for this force. And these forces can be considerable. The exceptionally high fault currents that result from short-circuits in high-voltage power systems can actually cause severe distortion to any parallel conductors, such as busbars, carrying such currents. The forces between magnetic fields are very important in electrical engineering because, as we shall see, the operation of electric motors is entirely dependent upon these forces. Now let’s turn our attention to the behaviour of a current-carrying conductor, placed within a permanent magnetic field – such as that illustrated in Figure 17.15...
eBook - ePub- Donald W. McRobbie(Author)
- 2020(Publication Date)
- Wiley-Blackwell(Publisher)
...2 Let’s get physical: fields and forces BASIC LAWS OF MAGNETISM The fundamental laws of magnetism were summarized by Scottish physicist James Clerk Maxwell in four equations. These equations are not for the faint‐hearted nor for the mathematically challenged, but if you aspire to be an expert in MRI safety, then you should have a good understanding of their consequences. By comparison, if you did not understand Newton’s laws of gravitation or Einstein’s theory of relativity you would not become a rocket scientist. Maxwell’s equations underpin everything in electromagnetism: the biological effects of EM fields, interactions with implants, electromagnetic modeling of field exposures and specific absorption rate (SAR), projectiles and magnet safety, magnetic shielding, fringe field gradients, and acoustic noise. A full understanding requires some knowledge of vector calculus and differential equations (see Appendix 2) but for now we will not need this. Those aspiring to be MR Safety Experts should read this chapter in conjunction with Appendix 1. Understanding Maxwell’s Equations Maxwell’s equations are given in Appendix 1. Here we describe their main consequences for MRI safety. Electrical charge and electric fields Gauss’s Law (Maxwell’s first equation) describes how electrical charges produce static electric fields E. Electric fields start at a positive charge and are directed towards their conclusion at negative charges (Figure 2.1)...
