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Essential University Physics Richard Wolfson 22 Electric Potential PowerPoint Lecture prepared by Richard Wolfson Slide 22-1

In this lecture you ll learn The concept of electric potential difference Including the meaning of the familiar term volt To calculate potential difference between two points in an electric field To calculate potential differences of charge distributions by summing or integrating over point charges The concept of equipotentials How charge distributes itself on conductors Slide 22-2

Electric potential difference The electric potential difference between two points describes the energy per unit charge involved in moving charge between those two points. Mathematically, VAB U AB q E dr A where V AB is the potential difference between points A and B, and U AB is the change in potential energy of a charge q moved between those points. Potential difference is a property of two points. Because the electrostatic field is conservative, it doesn t matter what path is taken between those points. In a uniform field, the potential difference becomes V E r AB B Slide 22-3

Clicker question What would happen to the potential difference between points A and B in the figure if the distance r were doubled? A. V would be doubled. B. V would be halved. C. V would be quadrupled D. V would be quartered. Slide 22-4

The volt and the electronvolt The unit of electric potential difference is the volt (V). 1 volt is 1 joule per coulomb (J/C). Example: A 9-V battery supplies 9 joules of energy to every coulomb of charge that passes through an external circuit connected between its two terminals. The volt is not a unit of energy, but of energy per charge that is, of electric potential difference. A related energy unit is the electronvolt (ev), defined as the energy gained by one elementary charge e falling through a potential difference of 1 volt. Therefore 1 ev is 1.6 10 19 J. Slide 22-6

Clicker question An alpha particle (charge 2e) moves through a 10-V potential difference. How much work, expressed in ev, is done on the alpha particle? A. 5 ev B. 10 ev C. 20 ev D. 40 ev Slide 22-7

Clicker question An alpha particle (charge 2e) moves through a 10-V potential difference. How much work, expressed in ev, is done on the alpha particle? A. 5 ev B. 10 ev C. 20 ev D. 40 ev Slide 22-8

Clicker question The figure shows three straight paths AB of the same length, each in a different electric field. Which one of the three has the largest potential difference between the two points? A. (a) B. (b) C. (c) Slide 22-9

Potential differences in the field of a point charge The point-charge field varies with position, so potential differences in the point-charge field must be found by integrating. The result is 1 1 VAB kq r A r B Taking the zero of potential at infinity gives V r V r kq r for the potential difference between infinity and any point a distance r from the point charge. Slide 22-11

Clicker question You measure a potential difference of 50 V between two points a distance 10 cm apart parallel to the field produced by a point charge. Suppose you move closer to the point charge. How will the potential difference over a closer 10-cm interval be different? A. The potential difference will remain the same. B. The potential difference will increase. C. The potential difference will decrease. D. We cannot find this without knowing how much closer we are. Slide 22-12

Potential difference of a charge distribution If the electric field of the charge distribution is known, potential differences can be found by integration as was done for the point charge on the preceding slide. If the distribution consists of point charges, potential differences can be found by summing point-charge potentials: For discrete point charges, V P where V(P) is the potential difference between infinity and a point P in the electric field of a distribution of point charges q 1, q 2, q 3, i kq i r i For a continuous charge distribution, k dq r. V P Slide 22-14

Discrete charges: the dipole potential The potential of an electric dipole follows from summing the potentials of its two equal but opposite point charges: For distances r large compared with the dipole spacing 2a, the result is V r, kpcos r 2 where p = 2aq is the dipole moment. A 3-D plot of the dipole potential shows a hill for the positive charge and a hole for the negative charge. Slide 22-15

Continuous distributions: a ring and a disk For a uniformly charged ring of total charge Q, integration gives the potential on the ring axis: k dq k kq r r x a dq 2 2 V x Integrating the potentials of charged rings gives the potential of a uniformly charged disk: 2kQ 2 2 V x x a x 2 a This result reduces to the infinitesheet potential close to the disk, and the point-charge potential far from the disk. Slide 22-16

Potential difference and the electric field Potential difference involves an integral over the electric field. So the field involves derivatives of the potential. Specifically, the component of the electric field in a given direction is the negative of the rate of change (the derivative) of potential in that direction. Then, given potential V (a scalar quantity) as a function of position, the electric field (a vector quantity) follows from V ˆ E i V ˆ j V kˆ x y z The derivatives here are partial derivatives, expressing the variation with respect to one variable alone. This approach may be used to find the field from the potential. Potential is often easier to calculate, since it s a scalar rather than a vector. Slide 22-17

Equipotentials An equipotential is a surface on which the potential is constant. In two-dimensional drawings, we represent equipotentials by curves similar to the contours of height on a map. The electric field is always perpendicular to the equipotentials. Equipotentials for a dipole: Slide 22-18

Clicker question The figure shows cross sections through two equipotential surfaces. In both diagrams the potential difference between adjacent equipotentials is the same. Which of these two could represent the field of a point charge? A. (a) B. (b) C. neither (a) nor (b) Slide 22-19

Charged conductors There s no electric field inside a conductor in electrostatic equilibrium. And even at the surface there s no field component parallel to the surface. Therefore it takes no work to move charge inside or on the surface of a conductor in electrostatic equilibrium. So a conductor in electrostatic equilibrium is an equipotential. That means equipotential surfaces near a charged conductor roughly follow the shape of the conductor surface. That generally makes the equipotentials closer, and therefore the electric field stronger and the charge density higher, where the conductor curves more sharply. Slide 22-21

Summary Electric potential difference describes the work per unit charge involved in moving charge between two points in an electric field: B V E dr The SI unit of electric potential is the volt (V), equal to 1 J/C. Electric potential always involves two points; to say the potential at a point is to assume a second reference point at which the potential is defined to be zero. Electric potential differences in the field of a point charge follow by integration: V P kq r, where the zero of potential is taken at infinity. This result may be summed or integrated to find the potentials of charge distributions. The electric field follows from differentiating the potential: Equipotentials are surfaces of constant potential. AB The electric field and the equipotential surfaces are always perpendicular. Equipotentials near a charged conductor approximate the shape of the conductor. A conductor in equilibrium is itself an equipotential. A V ˆ E i V ˆ j V kˆ x y z Slide 22-22