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22. Electric Potential

22. Electric Potential. Topics. Recap: Potential Energy Electric Potential Difference The Volt and the Electronvolt The Potential of a Point Charge Potential Difference from Superposition Electric Field from Electric Potential Charged Conductors. Recap: Potential Energy.

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22. Electric Potential

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  1. 22. Electric Potential

  2. Topics • Recap: Potential Energy • Electric Potential Difference • The Volt and the Electronvolt • The Potential of a Point Charge • Potential Difference from Superposition • Electric Field from Electric Potential • Charged Conductors

  3. Recap: Potential Energy The work done by a force along a path from point A to point B is defined by

  4. Recap: Potential Energy If the force is conservative we can define the potential energy difference as follows: The potential energy difference DU depends only on the end points A and B, that is, it is independent of the path taken between A and B.

  5. Electric Potential Difference

  6. Electric Potential Difference Suppose a charge +q is moved from A to B against a uniform electric field. The change in potential energy is The potential energy is increased.

  7. Electric Potential Difference The electric potential difference between two points A and B is defined as the energy per unit charge: A B For the special case of a uniform electric field, we can write:

  8. The Volt and the Electronvolt The electric potential difference between two points is such an important idea that it is given its own unit: the volt (V). If a charge moves over an electric potential difference of DV volts, the potential energy changes by DU = q DV.Example: A 12-V car battery does 12 J of work in moving 1 C of charge from one battery terminal to the other. A B

  9. The Volt and the Electronvolt Some Electric Potential Differences Between arm and leg 1 mV Across cell membrane 80 mV Car battery 12 V Electric outlet 100 – 240 V Between power line and ground 365 kV Between base of thundercloud 100 MV and ground

  10. The Volt and the Electronvolt For molecular and atomic systems, it is usually more convenient to measure energy in electronvolts (eV). (This is not an SI unit!) One electronvolt is the energy gained by a particle carrying one elementary charge when it moves through a potential difference of 1 volt. Example: the ionization energy of hydrogen is 13.6 eV. Since the value of an elementary charge is 1.6 x 10-19 C, 1 eV is 1.6 x 10-19 J

  11. Example: A Power Line A long straight power-line wire, of radius r = 1.0 cm with charge density l = 2.6 mC/m, is at a height h = 22 m above the ground. What is the potential difference DV between the cable and the ground, assuming that the electric field is approximately that of a line charge? h DV

  12. Example: A Power Line We found that the electric field of an infinitely long line charge is given by where the unit vector is perpendicular to, and points away from, the wire. The potential difference is h DV that is, –360 kV

  13. Potential of a Point Charge

  14. The Potential of a Point Charge Consider a positive point charge. The change in electric potential between points A and B is given by This shows that if a positive charge moves from A to B the potential energy decreases

  15. The Potential of a Point Charge Although only changes in potential are physically relevant, it is often convenient to choose the location of the zero of the potential. For a car battery, this is typically the car’s chassis; for an electrical outlet it is the ground. For an isolated point charge, it is convenient to choose the potential to be zero at infinity

  16. The Potential of a Point Charge The potential difference between two points A and B from a point charge can be re-written as When rA = infinity the last term vanishes. We are free to choose V(A) as we please, e.g., V(A) = 0.

  17. The Potential of a Point Charge With this choice, the potential of a point charge becomes Remember, however, that only differences in this number are physically relevant because we can always add to it an arbitrary constant without altering the physics

  18. Potential Differences using Superposition

  19. Electric Potential of a Collection of Charges The potential at a given point is the sum of the electric potentials due to every point charge - + + - - + +

  20. Electric Potential for a Charge Distribution The electric potential for a charge distribution is given by a formula similar to that for an electric field But unlike the electric field the electric potential is a scalar

  21. x Example – A Charged Ring Note: for a fixed point P, the distance r is constant as we integrate around the ring. Therefore,

  22. x2 x Example – A Charged Disk The potential for a ring of charge is Therefore, for a disk we can write

  23. x2 x Example – A Charged Disk The charge on a ring of radius a is where s is the surface charge density. Therefore,

  24. x2 x Example – A Charged Disk After performing the integral we obtain

  25. Equipotentials If one draws a surface through all points with the same potential, one obtains an equipotential. Here, for example, are some equipotentials for a dipole

  26. Electric Field from Electric Potential

  27. Electric Field fromElectric Potential Since one can compute the electric field from the potential V using the gradient: This is the gradient in Cartesian coordinates

  28. Example – Field of A Charged Disk We found the potential of a charged disk to be From this we can compute the electric field in the x direction from the negative gradient Close to the disk the field is ±2pks as we found using Gauss’s law.

  29. R2 R1 Charged Conductors The surface of a conductor in electrostatic equilibrium is an equipotential. Therefore, if one connects a charged spherical conductor to a neutral one, via a thin wire, the charge will migrate rapidly through the entire system until the potential on the surface of the system is the same everywhere.

  30. R2 R1 Charged Conductors If the spheres are very far apart, the distribution of charge on each will be essentially unaffected by the charge distribution on the other. In this case, the spheres act like point charges with potentials V1 = kQ1/R1 V2 = kQ2/R2 kQ1/R1 and kQ2/R2 Since V1 = V2, the smaller sphere will have the higher surface charge density

  31. R Dielectric Breakdown of Air near Conductors Air suffers dielectric breakdown, and becomes a conductor, if the electric field in air exceeds Emax ~ 3 x 106 V/m (N/C) At the surface of a conductor the electric field is E = 4pks V/m. Therefore, if the surface charge density is large enough, an isolated conductor in air will trigger its dielectric breakdown.

  32. R Dielectric Breakdown of Air near Conductors The charge required to trigger dielectric breakdown can be estimated by equating Emax to E = 4pks V/m (with R in mm) Example: a 1 mm radius ball bearing with about 1/3 nC of charge is enough to cause the air to spark!

  33. Summary • Electric potential difference • Since only differences in potential matter, the location of the zero of the potential can be chosen as we wish • Electric potential of a point charge • If we chose the zero to be infinitely far from a point charge, we can write its potential as

  34. Summary • Equipotentials These are surfaces of equal potential difference. The surface of a conductor in equilibrium is an equipotential. • Electric field • It is equal to the negative gradient of the electric potential

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