Electric Potential Differences: Essential Concepts & Calculations

1. 22 Essential University Physics Richard Wolfson Electric Potential

2. 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

3. 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,where ∆VAB is the potential difference between points A and B, and ∆UAB is the change in potential energy of a charge q moved between those points. • Potential difference is a propertyof 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

4. Clicker question • What would happen to the potential difference between points A and B in the figure if the distance were doubled? • would be doubled. • would be halved. • would be quadrupled • would be quartered.

5. Clicker question • What would happen to the potential difference between points A and B in the figure if the distance were doubled? • would be doubled. • would be halved. • would be quadrupled • would be quartered.

6. 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.

7. Clicker question • An alpha particle (charge ) moves through a 10-V potential difference. How much work, expressed in eV, is done on the alpha particle? • 5 eV • 10 eV • 20 eV • 40 eV

8. Clicker question • An alpha particle (charge ) moves through a 10-V potential difference. How much work, expressed in eV, is done on the alpha particle? • 5 eV • 10 eV • 20 eV • 40 eV

9. 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) • (b) • (c)

10. 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) • (b) • (c)

11. 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 • Taking the zero of potential at infinity givesfor the potential difference between infinity and any point a distance r from the point charge.

12. 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? • The potential difference will remain the same. • The potential difference will increase. • The potential difference will decrease. • We cannot find this without knowing how much closer we are.

13. 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? • The potential difference will remain the same. • The potential difference will increase. • The potential difference will decrease. • We cannot find this without knowing how much closer we are.

14. 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,where V(P) is the potential difference between infinity and a point P in the electric field of a distribution of point charges q1, q2, q3,… • For a continuous charge distribution,

15. x The potential is positive everywhere as charges are both positive EXAMPLE: A point charge q1 is at the origin, and a second point charge q2 is on the x-axis at x = a. Find the potential everywhere on the x-axis.

16. 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 iswhere 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.

17. Continuous distributions: a ring and a disk • For a uniformly charged ring of total charge Q, integration gives the potential on the ring axis: • Integrating the potentials of charged rings gives the potential of a uniformly charged disk: • This result reduces to the infinite-sheet potential close to the disk, and the point-charge potential far from the disk.

18. 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 fromThe 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.

19. 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:

20. 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) • (b) • neither (a) nor (b)

21. 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) • (b) • neither (a) nor (b)

22. 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.

23. Summary • Electric potential difference describes the work per unit charge involved in moving charge between two points in an electric field: • 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: 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. • 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.

24. Electric Potential Energy • CHECKPOINT: A proton moves from • point i to point f in a uniform • electric field directed as shown. • Does the electric field do • A. positive or • negative work on the proton? • Does the electric potential energy of the proton • increase or • decrease? Answers: B. the field does negative work A. the potential energy increases

25. A conducting sphere of radius a is surrounded by a concentric spherical shell of radius b. Both are initially uncharged. How much work does it take to transfer charge from one to the other until there is charge +Q on the inner sphere and –Q on the outer shell? • Hint: draw a graph of potential as a function of q, the charge transferred. Chapter 23 Problem 39 A conducting sphere of radius a is surrounded by a concentric spherical shell of radius b. Both are initially uncharged. How much work does it take to transfer charge from one to the other until there is charge +Q on the inner sphere and –Q on the outer shell? Hint: draw a graph of potential as a function of q, the charge transferred.