Physics
Electric Potential due to a Point Charge
The electric potential due to a point charge is a measure of the potential energy per unit charge at any point in the electric field created by the charge. It is calculated using the formula V = kQ/r, where V is the electric potential, k is Coulomb's constant, Q is the charge, and r is the distance from the charge. The electric potential is a scalar quantity and is measured in volts.
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10 Key excerpts on "Electric Potential due to a Point Charge"
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.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(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- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Positive charges create positive potential, while negative charges create negative poten- tial. Electric potential is a scalar quantity, so the total electric potential due to a collec- tion of charges is simply the algebraic sum of the electric potentials due to the individual charges. This is illustrated in Example 19.3.1. Example 19.3.1 The Electric Potential Created by Two Point Charges Two point charges, µ = + q 5.62 C 1 and µ = − q 3.14 C 2 , are at two corners of an equilat- eral triangle of side length = l 0.658 m, as shown. I N T E R A C T I V E F E A T U R E Electric Potential of Point Charges | 525 q 1 = +5.62 μC q 2 = -3.14 μC l = 0.658 m l = 0.658 m P (a) What is the electric potential at point P (the empty corner) due to these two charges? (b) How much work is done by the electric force when a third charge µ = + q 8.34 C 3 is moved from very far away to point P? Identify The electric potential created by a point charge can be calculated using Equation 19.3.1. The electric potential at point P is the algebraic sum of the potential at P due to q 1 and the potential at P due to q 2 . Charges q 1 and q 2 exert electric forces on charge q 3 as it is moved from very far away to point P. These electric forces do work. Plan (a) The electric potential at point P is the sum of the electric potential at P due to each charge, which can be calculated from Equation 19.3.1. (b) The work done by the electric force as a charge moves from point to point can be calculated using = − ∆ W q V 0 (Equation 19.1.3), where = q q 0 3 and ∆V is the potential difference experienced by q 3 as it moves from very far away to point P.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
(The default assumption in the absence of other information is that the test charge is positive.) We briefly defined a field for gravity, but gravity is always attractive, whereas the electric force can be either attractive or repulsive. Therefore, although potential energy is perfectly adequate in a gravitational system, it is convenient to define a quantity that allows us to calculate the work on a charge independent of the magnitude of the charge. Calculating the work directly may be difficult, since W = F → · d → and the direction and magnitude of F → can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that because F → = q E → , the work, and hence ΔU, is proportional to the test charge q. To have a physical quantity that is independent of test charge, we define electric potential V (or simply potential, since electric is understood) to be the potential energy per unit charge: Electric Potential The electric potential energy per unit charge is (7.4) V = U q . Since U is proportional to q, the dependence on q cancels. Thus, V does not depend on q. The change in potential energy ΔU is crucial, so we are concerned with the difference in potential or potential difference ΔV between two points, where Chapter 7 | Electric Potential 293 ΔV = V B − V A = ΔU q . Electric Potential Difference The electric potential difference between points A and B, V B − V A , is defined to be the change in potential energy of a charge q moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta. 1 V = 1 J/C The familiar term voltage is the common name for electric potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example, every battery has two terminals, and its voltage is the potential difference between them.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The quantity EPE/q 0 is the electric potential energy per unit charge and is an important concept in electricity. It is called the electric potential or, simply, the potential and is referred to with the symbol V, as in Equation 19.3. DEFINITION OF ELECTRIC POTENTIAL The electric potential V at a given point is the electric potential energy EPE of a small test charge q 0 situated at that point divided by the charge itself: V = EPE q 0 (19.3) SI Unit of Electric Potential: joule/coulomb = volt (V) The SI unit of electric potential is a joule per coulomb, a quantity known as a volt. The name honors Alessandro Volta (1745–1827), who invented the voltaic pile, the forerunner of the battery. In spite of the similarity in names, the electric potential energy EPE and the electric potential V are not the same. The electric potential energy, as its name implies, is an energy and, therefore, is measured in joules. In contrast, the electric potential is an energy per unit charge and is measured in joules per coulomb, or volts. Initial gravitational, potential energy, GPE A { Final gravitational potential energy, GPE B { h A F = mg F = mg h B A B 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. q 0 F = q 0 E q 0 F = q 0 E A B − − − − − − − − − − − − − − + + + + + + + + + + + + + + INTERACTIVE FIGURE 19.2 Because of the electric field E → , an electric force, F → = q 0 E → , is exerted on a positive test charge +q 0 . Work is done by the force as the charge moves from A to B. 19.2 The Electric Potential Difference 525 We can now relate the work W AB done by the electric force when a charge q 0 moves from A to B to the potential difference V B − V A between the points.
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
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. The SI unit of potential that follows from Eq. 28-9 is the joule per coulomb. This combination is given the name of volt (V): (28-13) The common name of “voltage” is often used for the poten- tial at a point, and we often speak of “voltage difference” instead of potential difference. When you touch the two probes of a voltmeter to two points in an electric circuit, you are measuring the voltage difference or potential differ- ence (in volts) between those points. We have already discussed that the electric force is con- servative, and so the potential energy difference when a test charge is moved between any two points depends only on the locations of the points and not on the path taken to move 1 volt 1 joule/coulomb. V U q 0 . V W ab q 0 , 28-3 Electric Potential 639 from one point to the other. Equation 28-9 therefore sug- gests that the potential difference is similarly path indepen- dent: the potential difference between any two points in an electric field is independent of the path through which the test charge moves in traveling from one point to the other. For any arbitrary potential difference V, no matter what the arrangement of charges that produces it, we can write Eq. 28-9 as (28-14) This equation indicates that when any charge q moves be- tween two points whose potential difference is V, the sys- tem experiences a change in potential energy U given by Eq. 28-14. The potential difference V is set up by other charges that are fixed at rest, so that the motion of q does not change V.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
578 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 = 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 associated with a conservative force. Thus, an electric potential energy exists that is analogous to the gravitational potential energy. To Lightning permeates the sky around the ash plume above the Puyehue-Cordon Caulle volcano in south-central Chile in 2011. Unlike normal lightning associated with rain clouds, where static charges are produced by colliding ice particles, volcanic lightning, or “dirty thunderstorms,” can result from frictional charging between colliding ash and dust particles. The natural convective thermal currents in the hot ash cloud aid in the separation of charges. This creates extremely high differences in voltage, or potential, between different parts of the dust cloud or between the cloud and the ground. If the voltage difference is sufficiently large, the insulating properties of the air break down, and it conducts electricity in spectacular fashion. The electric potential, and its relationship to charge, will be one of the topics we study in this chapter. Carlos Gutierrez/Reuters 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.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The quantity EPE/q 0 is the electric potential energy per unit charge and is an important concept in electricity. It is called the electric potential or, simply, the potential and is referred to with the symbol V, as in Equation 19.3. Definition of Electric Potential The electric potential V at a given point is the electric potential energy EPE of a small test charge q 0 situated at that point divided by the charge itself: V 5 EPE q 0 (19.3) SI Unit of Electric Potential: joule/coulomb 5 volt (V) The SI unit of electric potential is a joule per coulomb, a quantity known as a volt. The name honors Alessandro Volta (1745–1827), who invented the voltaic pile, the forerunner of the battery. In spite of the similarity in names, the electric potential energy EPE and the electric potential V are not the same. The electric potential energy, as its name implies, is an energy and, therefore, is measured in joules. In contrast, the electric potential is an energy per unit charge and is measured in joules per coulomb, or volts. We can now relate the work W AB done by the electric force when a charge q 0 moves from A to B to the potential difference V B 2 V A between the points. Combining Equations 19.2 and 19.3, we have: V B 2 V A 5 EPE B q 0 2 EPE A q 0 5 2W AB q 0 (19.4) Often, the “delta” notation is used to express the difference (final value minus initial value) in potentials and in potential energies: DV 5 V B 2 V A and D(EPE) 5 EPE B 2 EPE A . In terms of this notation, Equation 19.4 takes the following more compact form: DV 5 D (EPE) q 0 5 2W AB q 0 (19.4) q 0 F = q 0 E q 0 F = q 0 E A B − − − − − − − − − − − − − − + + + + + + + + + + + + + + B B Figure 19.2 Because of the electric field E B , an electric force, F B 5 q 0 E B , is exerted on a positive test charge 1q 0 . Work is done by the force as the charge moves from A to B.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The quantity EPE/q 0 is the electric potential energy per unit charge and is an important concept in electricity. It is called the electric potential or, simply, the potential and is referred to with the symbol V, as in Equation 19.3. Definition of Electric Potential The electric potential V at a given point is the electric potential energy EPE of a small test charge q 0 situated at that point divided by the charge itself: V 5 EPE q 0 (19.3) SI Unit of Electric Potential: joule/coulomb 5 volt (V) The SI unit of electric potential is a joule per coulomb, a quantity known as a volt. The name honors Alessandro Volta (1745–1827), who invented the voltaic pile, the forerunner of the battery. In spite of the similarity in names, the electric potential energy EPE and the electric potential V are not the same. The electric potential energy, as its name implies, is an energy and, therefore, is measured in joules. In contrast, the electric potential is an energy per unit charge and is measured in joules per coulomb, or volts. We can now relate the work W AB done by the electric force when a charge q 0 moves from A to B to the potential difference V B 2 V A between the points. Combining Equations 19.2 and 19.3, we have V B 2 V A 5 EPE B q 0 2 EPE A q 0 5 2W AB q 0 (19.4) Often, the “delta” notation is used to express the difference (final value minus initial value) in potentials and in potential energies: DV 5 V B 2 V A and D(EPE) 5 EPE B 2 EPE A . In terms of this notation, Equation 19.4 takes the following, more compact form: DV 5 D (EPE) q 0 5 2W AB q 0 (19.4) q 0 F = q 0 E q 0 F = q 0 E A B − − − − − − − − − − − − − − + + + + + + + + + + + + + + B B Figure 19.2 Because of the electric field E B , an electric force, F B 5 q 0 E B , is exerted on a positive test charge 1q 0 . Work is done by the force as the charge moves from A to B.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
The electric field is no longer zero, however, because the arrange- ment is no longer symmetric. A net field is now directed toward the charge distribution. Instructional video is available at the website www.wiley.com After reading this module, you should be able to . . . 24.4.1 Calculate the potential V at any given point due to an electric dipole, in terms of the magnitude p of the dipole moment or the product of the charge separation d and the magnitude q of either charge. 24.4.2 For an electric dipole, identify the locations of positive potential, negative potential, and zero potential. 24.4.3 Compare the decrease in potential with increasing distance for a single charged particle and an electric dipole. 24.4 POTENTIAL DUE TO AN ELECTRIC DIPOLE KEY IDEA 1. At a distance r from an electric dipole with dipole moment magnitude p = qd, the electric potential of the dipole is V = 1 ____ 4πε 0 p cos θ _______ r 2 for r ⪢ d; the angle θ lies between the dipole moment vector and a line extending from the dipole midpoint to the point of measurement. LEARNING OBJECTIVES Potential Due to an Electric Dipole Now let us apply Eq. 24.3.6 to an electric dipole to find the potential at an arbitrary point P in Fig. 24.4.1a. At P, the positively charged particle (at distance r (+) ) sets up potential V (+) and the negatively charged particle (at distance r (–) ) sets up potential V (–) . Then the net potential at P is given by Eq. 24.3.6 as V = ∑ i=1 2 V i = V (+) + V (−) = 1 ____ 4πε 0 ( q ___ r (+) + −q ___ r (−) ) = q ____ 4πε 0 r (−) − r (+) ________ r (−) r (+) . (24.4.1) Naturally occurring dipoles—such as those possessed by many molecules— are quite small; so we are usually interested only in points that are relatively far from the dipole, such that r ⪢ d, where d is the distance between the charges and r is the distance from the dipole’s midpoint to P.
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