Physics
Potentials
Potentials in physics refer to scalar or vector fields that represent the potential energy associated with a force or the electric potential in an electric field. Scalar potentials are used to describe conservative forces, while vector potentials are used to describe non-conservative forces. In both cases, potentials provide a way to analyze and understand the behavior of physical systems.
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11 Key excerpts on "Potentials"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Repeating this procedure we find that an electric potential is set up at every point in the rod’s electric field. In fact, every charged object sets up electric potential V at points throughout its electric field. If we happen to place a particle with, say, charge q at a point where we know the pre-existing V, we can immedi- ately find the potential energy of the configuration: (electric potential energy) = (particle’s charge) ( electric potential energy unit charge ) , or U = qV, (24-3) where q can be positive or negative. 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 associ- ated with the force between the baseball and Earth). However, because only the baseball noticeably moves (its motion does not noticeably affect Earth), we might assign the gravitational 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. 24-2 is the joule per coulomb. This combination occurs so often that a special unit, the volt (abbrevi- ated V), is used to represent it.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Repeating this procedure we find that an electric potential is set up at every point in the rod’s electric field. In fact, every charged object sets up electric potential V at points throughout its electric field. If we happen to place a particle with, say, charge q at a point where we know the pre-existing V, we can immedi- ately find the potential energy of the configuration: (electric potential energy) = (particle’s charge) ( electric potential energy ______________________ unit charge ) , or U = qV, (24.1.3) where q can be positive or negative. 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 associ- ated with the force between the baseball and Earth). However, because only the baseball noticeably moves (its motion does not noticeably affect Earth), we might assign the gravitational 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.
- 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 - PDF- Akira Tonomura(Author)
- 1998(Publication Date)
- World Scientific(Publisher)
Chapter 10 VECTOR Potentials, REAL OR NOT? Aharonov and Bohm asserted that the AB effect was due to vector Potentials. What are vector Potentials? In the 1960s we learned at universities that vector Potentials were merely a mathematical tool with which electromagnetic problems could be solved easily. Now, however, vector Potentials are regarded as the most fundamental physical quantity in unified theories of all fundamental forces in nature. There, vector Potentials are called gauge fields. In this chapter, I would like to give you a physical image of vector Potentials and the history of vector Potentials and then to tell you the story of our efforts until the physical reality or significance of vector Potentials was established. What are Vector Potentials? Faraday introduced the concept of magnetic lines of force. The density of magnetic lines of force is called a magnetic field, or magnetic flux density. This quantity has not only a magnitude but also a direction. A physical quantity having both magnitude and direction is called a vector and is conventionally written with a bold and italic letter, such as B. The vector B has the direction of a magnetic line of force and has a magnitude proportional to the density of magnetic lines. What then are vector Potentials? The vector potential A is mathematically related to magnetic field B as follows: B = rot A. (15) If you are not familiar with mathematics, you may be puzzled at this equation. But please don't hesitate. It is only a representation of the relation between two quantities. Here rot is the abbreviation of rotation. The meaning of this equation is as follows. If a distribution of A has a rotational component like a vortex, then the magnitude of B is given by the strength of the vortex and the direction of B is along a line perpendicular to the vortex plane. The vector B cannot yet be determined, since the line has two opposite directions.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
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. DANIEL MIHAILESCU/AFP/Getty Images/NewsCom CHAPTER 19 Electric Potential Energy and the Electric Potential 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 photograph 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. 19.1 Potential Energy In Chapter 18 we discussed the electrostatic force that two point charges exert on each other, the magnitude of which is F = k|q 1 ||q 2 |/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 = 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 asso- ciated with a conservative force. Thus, an electric potential energy exists that is analogous to the gravitational potential energy. To set the stage for a discussion of the electric poten- tial energy, let’s review some of the important aspects of the gravitational counterpart.
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 - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 19 Electric potential energy and the electric potential 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. INTRODUCTION 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 photograph 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. Source: DANIEL MIHAILESCU / AFP / Getty Images / NewsCom 19.1 Potential energy LEARNING OBJECTIVE 19.1 Define electrical potential energy. FIGURE 19.1 Gravity exerts a force, F = m g, on the basketball of mass m. Work is done by the gravitational force as the ball falls from A to B. h A F = mg F = mg h B A B In chapter 18 we discussed the electrostatic force that two point charges exert on each other, the magni- tude of which is F = k|q 1 ||q 2 |/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 = Gm 1 m 2 /r 2 , according to Newton’s law of univer- sal gravitation (see section 4.7). Both of these forces are conservative and, as section 6.4 explains, a poten- tial energy can be associated with a conservative force.
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Usually we will refer to electric potential simply as “poten- tial.” Using the relation between work and potential energy given in Eq. 28-2, we can write the definition of potential difference as (28-11) where W ab is the work done by the electrostatic force ex- erted by q on q 0 when the test charge moves from a to b. By defining a suitable choice of the reference point of potential energy (such as U a 0 for an infinite initial sepa- ration of the charges), we obtained in the previous section an expression (Eq. 28-7) for the potential energy of a par- ticular configuration rather than the change in potential en- ergy for a change in configuration. We can do the same for electric potential. Only differences in potential have physi- cal significance, so we are free to choose the zero point and its reference value at our convenience. When the potential is taken to be zero at points that are infinitely far from q, the electric potential is (28-12) In a complicated arrangement of many charges, the po- tential V may be positive, negative, or zero. The potential at a point near an isolated positive charge is positive. If we were to move a positive test charge from infinity to that point, the charge would move from a location where V 0 to a location where V 0. Thus V 0 and (according to Eq. 28-9) U 0, indicating that the electric force on the test charge has done negative work. Similarly, the potential at a point near an isolated negative charge is negative; the electric force does positive work when we move a positive test charge from infinity to that point. If the potential is zero at a point, no net work is done by the electric force as the test charge moves in from infinity to that point, although the test charge may pass through re- gions where it experiences attractive or repulsive electric forces. A potential of zero at a point does not necessarily mean that the electric force is zero at that point.
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 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. Figure 19.1, which is essentially Figure 6.10, shows a basketball of mass m falling from point A to point B. The gravitational force, m g B , is the only force acting on the ball, where g is the magnitude of the acceleration due to gravity.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
19.1 Electric Potential Energy: Potential Difference When a free positive charge q is accelerated by an electric field, such as shown in Figure 19.2, it is given kinetic energy. The process is analogous to an object being accelerated by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge q by the electric field in this process, so that we may develop a definition of electric potential energy. Figure 19.2 A charge accelerated by an electric field is analogous to a mass going down a hill. In both cases potential energy is converted to another form. Work is done by a force, but since this force is conservative, we can write W = –ΔPE . The electrostatic or Coulomb force is conservative, which means that the work done on q is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the potential energy (because it depends only on position) than to calculate the work directly. We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, ΔPE , is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, W = –ΔPE . For example, work W done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative ΔPE . There must be a minus sign in front of ΔPE to make W positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point. Potential Energy W = –ΔPE . For example, work W done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative ΔPE.
eBook - PDF- Harold Jeffreys, Bertha Jeffreys(Authors)
- 1999(Publication Date)
- Cambridge University Press(Publisher)
Chapter 6 POTENTIAL THEORY *But all that moveth doth Mutation love.* 8PEN8BE, The Faerie Queenc, Bk. 7 6*01. 1/r as a solution of V 2 = 0. Let x t be the coordinates of a point P and r its distance from the origin. Then dx t r drXrJdXi r 3 ' K) d 2 1 _ d x i _ 1 3^. 3^.2^ (2) dX{dx k r dx k r 3 r 3 dx k r 5 _Sx i x k -r 2 S ik Now put A; = i and apply the summation convention; since Wi = r 2 , 8 U = 3, a 2 i dXi dx.- r = °> (3) so that 1/r is a solution of Laplace's equation, V 2 ^ = 0, except at the origin. It follows at once that if ^ are the coordinates of another point Q, and (summed), then d 2 1 n except at x i — £ t . Note that ^ = — — - , a result that will be needed repeatedly. Further, if we take n points Q 1 ...Q n and denote their coordinates by £, u , and the various distances Q 8 P by r 8 , then V 2 S J = 0, (5) where a 8 are any constants, except when any of the r 8 is 0, that is, when P coincides with any of the points Q 8 . Differentiation is of course understood to be with regard to n the coordinates of P. Hence with this restriction any function of the form 2 ajr 8 is a solution of Laplace's equation. Now the gravitational potential due to a distribution of particles is of this form. So is the electrostatic potential due to a set of point charges. Hence both satisfy Laplace's equation. This equation arises also in the hydrodynamics of an incompressible fluid. For if u t is the velocity at P(x t ) the condition that the mass within any closed surface is constant requires that % i is a solenoidal vector; and for any circuit capable of being filled 200 Potential due to particles 6-01 by a cap occupied wholly by fluid that has never passed near a solid boundary the cir- culation \u i dx i round it is practically zero, so that to a good approximation u t is also an irrotational vector.
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