Physics
Electromagnetic Potential Definition
The electromagnetic potential is a concept in physics that describes the potential energy associated with electromagnetic fields. It is a fundamental quantity in the theory of electromagnetism and is used to describe the behavior of electric and magnetic fields. The electromagnetic potential is a key concept in understanding the interactions between charged particles and electromagnetic fields.
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8 Key excerpts on "Electromagnetic Potential Definition"
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Two Cautions. (1) The (now very old) decision to call V a potential was unfortunate because the term is easily confused with potential energy. Yes, the two quantities are related (that is the point here) but they are very different and not interchangeable. (2) Electric potential is a scalar, not a vector. (When you come to the homework problems, you will rejoice on this point.) Language. A potential energy is a property of a system (or configuration) of objects, but sometimes we can get away with assigning it to a single object. For example, the gravitational potential energy of a baseball hit to outfield is actually a potential energy of the baseball–Earth system (because it is associated with the force between the baseball and Earth). However, because only the baseball notice- ably moves (its motion does not noticeably affect Earth), we might assign the gravi- tational potential energy to it alone. In a similar way, if a charged particle is placed in an electric field and has no noticeable effect on the field (or the charged object that sets up the field), we usually assign the electric potential energy to the particle alone. Units. The SI unit for potential that follows from Eq. 24.1.2 is the joule per coulomb. This combination occurs so often that a special unit, the volt (abbrevi- ated V), is used to represent it. Thus, 1 volt = 1 joule per coulomb. With two unit conversions, we can now switch the unit for electric field from newtons per coulomb to a more conventional unit: 1 N/C = ( 1 N __ C ) ( 1 V _____ 1 J / C ) ( 1 J _______ 1 N ⋅ m ) = 1 V / m. The conversion factor in the second set of parentheses comes from our definition of volt given above; that in the third set of parentheses is derived from the defini- tion of the joule. From now on, we shall express values of the electric field in volts per meter rather than in newtons per coulomb. Motion Through an Electric Field Change in Electric Potential.
eBook - PDFPrinciples of Optics
Electromagnetic Theory of Propagation, Interference and Diffraction of Light
- Max Born, Emil Wolf(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
Eqs. (27) and (28) have a simple interpretation. Consider the electrostatic potential of two fixed electric charges — e and + e situated at points whose position vectors with respect to a fixed point Q 0 (r Q ) are — a and a respectively. The Coulomb potential φ {Β) due to the charges is (see Fig. 2.1): (30) 2.2] E L E C T R O M A G N E T I C P O T E N T I A L S A N D P O L A R I Z A T I O N 79 If a is sufficiently small, we may expand l/R 2 in powers of the components of a and obtain i - = i - + a . g r a d i - + . . . > (31) where the operator grad is taken with respect to the coordinates of the point at which the charge — e is situated. If we neglect the higher-order terms in (31), (30) becomes <£ = e a . g r a d -^ = e ^ . (32) Suppose now that a is gradually decreased whilst e is increased without limits in such a way that ea approaches a finite value p : Urn ea = p. (33) Α-»Ό Then R 1 approaches R and (32) becomes in the limit 0 = p . g r a d o ^ . (34) This expression is identical with (27) in the special case when p is independent of time. If the charges depend on time but at each instant of time differ only in sign, we have, in place of (30), e(t - R 2 /c) e(t - Rjc) φ = ^ j — (35) and obtain by the same limiting process <^ = [ p ] . g r a d 0 ^ + ^ i Ä . [ p ] , which is (27). Thus (23) may be interpreted as the scalar potential of a distribution of electric dipoles of moment Ρ per unit volume. The last term in (28) may be shown to be the magnetic potential arising from these dipoles. In a similar way (28) may be shown to be the vector potential of a magnetic dipole of moment m(t), such a dipole being equivalent, of course, to an infinitesimal closed circuit of area A normal to m, carrying the current* cm/A. Hence the first two terms in (24) may be interpreted as the vector potential of a distribution of magnetic dipoles of moment M per unit volume.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
In the medical diagnostic technique of electroencephalography, electrodes placed at various points on the head detect the small voltages that exist between the points. The cap worn by the young man in this photo- graph facilitates the placement of a number of electrodes, so that the voltages created by different regions of the brain as he plays a flipper pinball game can be measured simultaneously. The voltage between two points is another name for the difference in electric potential between the points, which is related to the concept of electric potential energy. Electric potential energy and electric potential are the central ideas in this chapter. 19 | Electric Potential Energy and the Electric Potential Chapter | 19 LEARNING OBJECTIVES After reading this module, you should be able to... 19.1 | Define electrical potential energy. 19.2 | Solve problems involving electric potential and electric potential energy. 19.3 | Calculate electric potential created by point charges. 19.4 | Relate equipotential surfaces to the electric field. 19.5 | Solve problems involving capacitors. 19.6 | Describe biomedical applications of electric potential. 19.1 | Potential Energy In Chapter 18 we discussed the electrostatic force that two point charges exert on each other, the magnitude of which is F 5 ku q 1 uu q 2 u /r 2 . The form of this equation is similar to the form for the gravitational force that two particles exert on each other, which is F 5 Gm 1 m 2 /r 2 , according to Newton’s law of universal gravitation (see Section 4.7). Both of these forces are conservative and, as Section 6.4 explains, a potential energy can be associated with a conservative force. Thus, an electric potential energy exists that is analogous to the gravitational potential energy. To set the stage for a discus- sion of the electric potential energy, let’s review some of the important aspects of the gravitational counterpart.
eBook - PDFWaves and Oscillations in Nature
An Introduction
- A Satya Narayanan, Swapan K Saha(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
However, this statement is not true for time-varying fields, in which case, Faraday’s law (see Section 2.3.1) applies. The potential difference (voltage) between two points, P 1 and P 2 , in an electric field, E , is defined as the work required to move a unit positive test charge from P 1 to P 2 , and is given by V 21 = − P 2 P 1 E · d l = E J . C − 1 orV (2.58) with E as the electromotive force (EMF), which is defined to be the limiting maximum potential difference between the terminals of a voltage generator (e.g., battery) as the current drawn from it is reduced towards zero. The potential difference is independent of the path taken from P 1 to P 2 , i.e., it depends on the endpoints. The negative sign in equation (2.58) indicates that the field does work on the positive charger in moving from P 1 to P 2 and there is a fall in potential. With the choice of zero potential, we introduce an electric potential function, V ( r ), in which r is the distance from the point-like charged object with charge Q V ( r ) = 1 4 π 0 Q r (2.59) In the presence of more than one point charge, the electric potential becomes the sum of potentials due to individual charges, that is, V ( r ) = 1 4 π 0 N j =1 q j r (2.60) 2.1.7.1 Deriving Electric Field from Electric Potential In electrostatics, to avoid the addition of vectors when applying Coulomb’s law (see Section 2.1.1) to get the electric field intensity at a point, we develop the concept of a potential and obtain the field as the gradient (see Appendix B.2.1) of this potential function since potential produced by an electric charge is a scalar quantity. In equation (2.57), we find the relation between the elec-tric field E and the electric potential V .
eBook - PDF- Alastair I. M. Rae, Jim Napolitano(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
Consider moving a charge q from point a to point b along some path C . The electric field E ( r ) exerts a force F ( r ) = q E ( r ) on the charge, so the field does an amount of work W = R C F · d l = R C q E · d l on the charge. It is easy to show that W is 34 Quantum Mechanics independent of the path C . If we choose two such paths C 1 and C 2 , then Z b a , C 1 q E · d l -Z b a , C 2 q E · d l = Z b a , C 1 q E · d l + Z a b , C 2 q E · d l = I C 1 + C 2 q E · d l = Z q ∇× E · d A = -Z q ∇×∇ V · d A = 0 (2.21) using Stokes’ theorem and (2.20), plus the fact that the curl of any gradient is zero. Therefore, W is the same for both paths, and we can define the electrostatic potential energy as U = -W = -Z C q E · d l = Z C q ∇ V · d l = qV ( r b ) -qV ( r a ) = q Δ V ab (2.22) where we refer to Δ V ab as the potential difference between points a and b . In the case of localized charge distributions, the energy change when a charge is moved to a position r from infinitely far away is U ( r ) = qV ( r ). Combining (2.20) with (2.19a) leads us to ∇ 2 V = -ρ 0 (2.23) which is called Poisson’s equation . In practice, one solves (2.23) for V ( r ), given a charge distribution ρ ( r ), and then uses (2.20) to find the electric field. There is a wide variety of techniques for solving Poisson’s equation, but we shall not consider these here. Now consider the potential associated with the magnetic field. In this case, we start with (2.19b), which holds even for time dependent fields. Since the divergence of any curl is also zero, we can write B = ∇× A (2.24) where A = A ( r , t ) is called the magnetic vector potential . In practice, it turns out that this vector potential is not especially useful in solving time-independent problems given a distribution of currents. It is, however, very useful for gaining insight into the nature of electromagnetic fields. For example, it can be used to generalize (2.20) to the case of time-dependent electric fields.
- B N Basu(Author)
- 1996(Publication Date)
- World Scientific(Publisher)
In an analogous way, the electrostatic potential energy is stored in a charged particle when it is moved against the force on the particle due to a static electric field. In the electrostatic case, the zero reference potential is considered to be at infinity. Thus the electrostatic potential at a point is the work done in bringing a point charge from infinity upto the point per unit charge. Quantitatively, if Q is the point charge which is imagined to be brought from infinity upto a point, and W is the work done in the process, then the electrostatic potential V at the point is given by V = W/Q . (2.4.1) The unit of potential, as can be appreciated from (2.4.1), is Joul/Coulomb which is put as volt as a practical unit. Example 2.4J_ Find the electrostatic potential due to a point charge at a distance from it. The force on a point charge Q at a distance r from the point charge q can be found using Coulomb's law (2.1.4). The Coulomb force may be regarded as constant over an infinitesimal distance. Then it is easy to find an expression for the element of work done in moving the point charge Q, against this force, through an element of distance. Integrating this expression one can then find the total work done W in moving the point charge Q all the way from infinity upto the point where the electrostatic potential V due to the point charge q is sought. Subsequently, V due to point charge q at a distance r can be found using (2.4.1) as (for the details of deduction, see appendix a2.4. ]): V = 47te 0 r (2.4.2) Example 2.4.2 Find the electrostatic potential due to a circular charge disc. Let the charge be sprayed uniformly over a plane with a circular boundary. The electric field due to this charge distribution at a perpendicular distance measured from the centre of the circle was found in Example 2.1.2 (see (2.1.17)).
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
In the case of the gravitational force, we showed in Section 14-6 that, when an object with mass m 2 moves from a distance r i from mass m 1 to a distance r f from m 1 , the potential energy difference is (28-3) This potential energy difference is associated with the en- tire system consisting of m 1 and m 2 , not with either object alone. Because of the similarity of the electrostatic and gravi- tational force laws, we can make the same conclusion about the electrostatic force that we did about the gravitational force: The electrostatic force is conservative, and therefore there is a potential energy associated with the configuration U Gm 1 m 2 1 r f 1 r i . F B U U f U i W if f i F B d s B , ELECTRIC POTENTIAL ENERGY AND POTENTIAL I n Chapters 11 through 13 we learned that methods based on energy concepts offered new insights in understanding mechanics and often provided simplifica- tions in solving mechanics problems. In Chapter 14 we used methods based on potential energy in situa- tions involving the gravitational force to determine such properties as the motions of satellites and planets. In this chapter we introduce the energy method to the study of electrostatics. We begin with electric po- tential energy, which we shall find can characterize an electrostatic force just as gravitational potential en- ergy can characterize a gravitational force. We then generalize to the concept of electric potential and show how to find the electric potential for various discrete and continuous charge distributions. CHAPTER 28 CHAPTER 28 (the relative locations of the objects) of a system in which electrostatic forces act. Why is this approach useful for electrostatic forces? In mechanics, we learned that there are two ways to analyze problems. One approach is based on force (a vector) and al- lows us to determine the position and velocity of an object at every point of its motion.
- Prabhakar H. Pathak, Robert J. Burkholder(Authors)
- 2021(Publication Date)
- Wiley-IEEE Press(Publisher)
(5.148) 5.5 Potentials and Fields of a Moving Point Charge Consider an ideal point charge of strength Q 0 , which moves along a prescribed trajectory as shown in Figure 5.11. The charge has a nonuniform velocity v at r ¤ 0 and at time t ¤ . It is of interest here to find the electromagnetic fields E , H produced by this moving point charge. The results obtained below for the ideal point charge can be readily extended via a superposition integral to deal with a classical charge distribution moving together with the same velocity. The charge in Figure 5.11 defines the following 186 ELECTROMAGNETIC POTENTIALS AND FIELDS OF SOURCES IN UNBOUNDED REGIONS Figure 5.11 Point charge Q 0 moving with a velocity v and producing the electromagnetic fields E , H which are observed at the point P . charge and current densities: L v ( r ¤ , t ¤ ) = Q 0 r ¤ -r ¤ 0 ( t ¤ ) J v ( r ¤ , t ¤ ) = Q 0 v ( t ¤ ) r ¤ -r ¤ 0 ( t ¤ ) M ; r ¤ ¸ V , (5.149) where, V is the volume of external space consisting of a homogeneous, isotropic medium ( , ) . The results developed below are of interest in predicting the interactions of high-speed charged parti-cles with dielectric media, as well as in some particle physics applications. One also notes that electro-magnetic radiation resulting from the emission of charged particles from the sun may cause disruptions of electronic communications systems. The starting point for evaluating the EM fields radiated by a moving charge is the use of the space-time integrals of (5.116) and (5.119), rather than just the spatial integrals in (5.128) and (5.129). The latter equations are actually obtained from the former (i.e., (5.116) and (5.119)) after evaluating the temporal integrals in closed form thus leaving only the spatial integrals over the source region.
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