Understanding Electric Potential Energy in Quantum Mechanics

1. Stable equilibrium with other constraints Earnshaw’s theorem A point charge cannot be in stable equilibrium in electrostatic field of other charges (except right on top of another charge – e.g. in the middle of a distributed charge) Atom – system of charges with only Coulombic forces in play. According to Earhshaw’s theorem, charges in atom must move However, planetary model of atom doesn’t work Only quantum mechanics explains the existence of an atom

2. Electric Potential Energy Concepts of work, potential energy and conservation of energy For a conservative force, work can always be expressed in terms of potential energy difference Energy Theorem For conservative forces in play, total energy of the system is conserved

3. Potential energy Uincreases as the test charge q0 moves in the direction opposite to the electric force : it decreases as it moves in the same direction as the force acting on the charge

4. Electric Potential Energy of Two Point Charges

5. Electric potential energy of two point charges

6. a-particle positron What is the speed at the distance ? What is the speed at infinity? Suppose, we have an electron instead of positron. What kind of motion we would expect? Example: Conservation of energy with electric forces A positron moves away from an a – particle Conservation of energy principle

7. Electric Potential Energy of the System of Charges Potential energy of a test charge q0 in the presence of other charges Potential energy of the system of charges (energy required to assembly them together) Potential energy difference can be equivalently described as a work done by external force required to move charges into the certain geometry (closer or farther apart). External force now is opposite to the electrostatic force

8. Electric Potential Energy of System • The potential energy of a system of two point charges • If more than two charges are present, sum the energies of every pair of two charges that are present to get the total potential energy

9. Electric potential is electric potential energy per unit charge Finding potential (a scalar) is often much easier than the field (which is a vector). Afterwards, we can find field from a potential Units of potential are Volts [V] 1 Volt=1Joule/Coulomb If an electric charge is moved by the electric field, the work done by the field Potential difference if often called voltage

10. Two equivalent interpretations of voltage: • Vab is the potential of a with respect to b, equals the work done • by the electric force when a UNIT charge moves from a to b. • 2. Vab is the potential of a with respect to b, equals the work that must • be done to move a UNIT charge slowly from b to a against the • electric force. Potential due to the point charges Potential due to a continuous distribution of charge Finding Electric Potential through Electric Field

11. Some Useful Electric Potentials • For a uniform electric field • For a point charge • For a series of point charges

12. Potential of a point charge Moving along the E-field lines means moving in the direction of decreasing V. As a charge is moved by the field, it loses potential energy, whereas if the charge is moved by the external forces against the E-field, it acquires potential energy

13. Negative charges are a potential minimum Positive charges are a potential maximum

14. Positive Electric Charge Facts • For a positive source charge • Electric field points away from a positive source charge • Electric potential is a maximum • A positive object charge gains potential energy as it moves toward the source • A negative object charge loses potential energy as it moves toward the source

15. Negative Electric Charge Facts • For a negative source charge • Electric field points toward a negative source charge • Electric potential is a minimum • A positive object charge loses potential energy as it moves toward the source • A negative object charge gains potential energy as it moves toward the source

16. Electron Volts Electron volts – units of energy 1 eV – energy a positron (charge +e) receives when it goes through the potential difference Vab =1 V Unit: 1 Volt= 1 Joule/Coulomb (V=J/C) Field: N/C=V/m 1 eV= 1.6 x 10-19 J Just as the electric field is the electric force per unit charge, the electrostatic potential is the potential energy per unit charge.

17. Clicker question There is a 12 V potential difference between the positive and negative ends of a set of jumper cables, which are a short distance apart. An electron at the negative end ready to jump to the positive end has a certain amount of potential energy. On what quantities does this electrical potential energy depend? a. the distance and the potential difference between the ends of the cables b. the distance and the charge on the electron c. the potential difference and the charge d. the potential difference, charge, and distance

18. Assume that two of the electrons at the negative terminal have attached themselves to a nearby neutral atom. There is now a negative ion with a charge at this terminal. What are the electric potential and electric potential energy of the negative ion relative to the electron? a. The electric potential and the electric potential energy are both twice as much. b. The electric potential is twice as much and the electric potential energy is the same. c. The electric potential is the same and the electric potential energy is twice as much. d. The electric potential and the electric potential energy are both the same. e. The electric potential is the same and the electric potential energy is increased by the mass ratio of the oxygen ion to the electron.

19. Examples A small particle has a charge -5.0 mC and mass 2*10-4 kg. It moves from point A, where the electric potential is fa =200 V and its speed is V0=5 m/s, to point B, where electric potential is fb =800 V. What is the speed at point B? Is it moving faster or slower at B than at A? E A B F In Bohr’s model of a hydrogen atom, an electron is considered moving around a stationary proton in a circle of radius r. Find electron’s speed; obtain expression for electron’s energy; find total energy.

20. Calculating Potential from E field • To calculate potential function from E field

21. When calculating potential due to charge distribution, we calculate potential explicitly if the exact distribution is known. If we know the electric field as a function of position, we integrate the field. Generally, in electrostatics it is easier to calculate a potential (scalar) and then find electric field (vector). In certain situation, Gauss’s law and symmetry consideration allow for direct field calculations. Moreover, if applicable, use energy approach rather than calculating forces directly (dynamic approach) Example: Solid conducting sphere Outside: Potential of the point charge Inside: E=0, V=const

22. Potential of Charged Isolated Conductor • The excess charge on an isolated conductor will distribute itself so all points of the conductor are the same potential (inside and surface). • The surface charge density (and E) is high where the radius of curvature is small and the surface is convex • At sharp points or edges  (and thus external E) may reach high values. • The potential in a cavity in a conductor is the same as the potential throughout the conductor and its surface

23. At the sharp tip (r tends to zero), large electric field is present even for small charges. Corona – glow of air due to gas discharge near the sharp tip. Voltage breakdown of the air Lightning rod – has blunt end to allow larger charge built-up – higher probability of a lightning strike