Electrostatics so far

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2 Electrostatics so far F = Electric Force b/n q and q : qq 1 2 kq Electric Field E due to q : E = 1 1 r 2 kq q r q e = 1.6 x10-19 C k = 9 x 10 9 Nm 2 /C 2

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4 Tesla Envy zeqz4efqa&feature=related

5 The Simple Case Uniform Electric Fields (Parallel Plates)

6 Infinite Charged Plane E = σ 2ε 0

7 Parallel Plates: In the center we assume they are infinite and that the field is constant and Uniform! E = σ ε 0

8 Uniform Infinite Parallel Plates E = σ ε 0 IMPORTANT!!!! The electric field is uniform, therefore the force has equal magnitude at any distance from the plates. Therefore, the acceleration is constant between the plates. Fig 23-25, p.726

9 What Direction are the Electric Field Lines?

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12 The Electric Field does work on the charge to move it from A to B, converting Potential Energy into Kinetic Energy, same as with the gravitational field. Conservation of Energy and the Work-Energy Theorem apply: W =Δ KE = ΔPE AB Δ PE < 0 - -

13 Electric Potential Energy -ΔPE = Work = Force x distance = (q 0 E) x d = q 0 Ed (J) d Looks like mgh! (Uniform Field Only)

14 Electric Potential The Electric Potential Energy per unit charge between A and B is called the Electric Potential Difference between A and B. For a uniform field: Δ V = V V B A =ΔPE/ q0 = qed/ q = Ed 0 0 Δ V = Ed VAB = Ed (Uniform Field Only)

15 Electric Potential High Potential VAB = Ed Unit for Electric Potential is the VOLT: 1 V = J/C = N m /C Electric Potential is a SCALAR! Only DIFFERENCES in potential between two points can be measured. There is no potential for a single point! Electric Potential is NOT Potential Energy! + charges move from hi to low potential (think gravity water fall) - charges move from low to hi potential (think antigravity water pump) Low Potential

16 Electron Volts What is the energy gained by an electron when it moves across a potential difference of 1 volt? Δ PE =ΔVq = = 19 1 V(1.6x10 C) 19 1 J / C(1.6x10 C) = 1.6x10 19 J ev = x J Note: 5eV electron breaks organic bonds in molecules. Nuclear decay energies ~ 1MeV

17 Example What is the final speed of a free electron accelerated from rest through a potential difference of 1 V? 100V? What is the electric field? What is the energy gained by an electron when it moves across a potential difference of 1 volt?

18 Example What is the final speed of a free electron accelerated from rest through a potential difference of 100V? Δ KE = ΔPE V =ΔPE q / e = ΔKE / qe = 1 /( ) 2 2 mv q v = 2Vq m 6 v= 5.93x10 m/ s

19 Example What is the electric field? ΔV Δ V = Ed => E = ( V / m) d What is the force on the electron? ΔV F = Eqe = q d What is the work done on the electron? W = qδ V =Δ KE = ΔPE e

20 Potential Difference in a Uniform Field V V =Δ V = Ed B A Electric field lines always point in the direction of decreasing electric potential When the electric field is directed downward, point B is at a lower potential than point A When a positive test charge moves from A to B, the charge-field system loses potential energy

21 Parallel Plate Capacitors The charge stored in a capacitor is proportional to the potential difference across the plates: 12V Q = CV Q is the charge on either plate. Capacitance: C = Q/ V 1 farad = 1 C/ V (Typical capacitors: C~ pf or nf) Energy: E = CV = Q / C 12V Net charge is zero!

22 Dielectrics A dieletric is an insulator that increases the capacitance. C = κε d 0 A Reduced E field prevents breakdown & discharge between plates.

23 Dielectrics The dieletric constant is the ratio of the field magnitude without and with the dielectric. κ = E 0 E κ κ κ Air Paper Water ~1 = 3.7 = 80 Reduced E field prevents breakdown & discharge between plates.

24 Dielectrics If the field becomes too great, the dielectric breaks down and becomes a conductor and the plates discharge. Reduced E field prevents breakdown & discharge between plates.

25 Problem What is the maximum voltage that can be sustained between 2 parallel plates separated by 2.5 cm of dry air? Dry air supports max field strength (dielectric strength CH 26)of 3x 10 6 V/m. V = = = Ed (3x10 6 V)(.025 m) 7.5x10 4 V = 75kV More than this and the air breaks down and becomes a conductor. LIGHTENING!

26 Electric Field

27 Equal Distance Equal Force A B C

28 Equipotential Lines A B C

29 Where is the Potential Highest? A, B, C? A B C

30 Potential Due to Point Charges

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32 E and V for a Point Charge E is the FORCE FIELD V is the ENERGY FIELD The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines

33 Electric Potential Energy & Potential Difference As a + particle moves from A to B, its electric potential energy changes by Δ U = U U = W B A positive test charge will gain kinetic energy and lose potential energy as it rolls down the electric hill from A to B. A Define the Potential Energy per unit charge as the Potential Difference, In general: V V ΔU / q B A 0

34 Gravitational Potential Energy for the Earth Graph of the gravitational potential energy U versus r for an object above the Earth s surface The potential energy goes to zero as r approaches infinity. The potential energy is negative because the force is attractive and we chose the potential energy to be zero at infinite separation An external agent must do positive work to increase the separation between two objects The work done by the external agent produces an increase in the gravitational potential energy as the particles are separated U becomes less negative The absolute value of the potential energy can be thought of as the binding energy If an external agent applies a force larger than the binding energy, the excess energy will be in the form of kinetic energy of the particles when they are at infinite separation GM Em Ur () = r

35 Bohr Orbital Binding Energy for Single Electron Atoms E n Z = 13.6eV n 2 2 Ground State: n = 1 First Excited: n = 2 2 nd Excited: n = eV is the energy of the H ground state. 2. Negative because it is the Binding Energy and work must be done on the atom (by a photon) to ionize it. 3. A 13.6eV photon must be absorbed to ionize the ground state

36 The total potential due to a distribution of charge is equal to the algebraic sum of all the potentials (not vector sum!) The sign of the charge matters!!! Potential due to a Point Charge The potential a distance r from a point charge is given by: V() r = Q k r V is the potential difference between r and infinity where the potential is taken to be zero (the ground is at infinity.) r V gives the amount of energy per unit charge that it takes to bring a test charge from infinity to r.

37 PROBLEM What is the electric potential 1.2m from a point charge Q = + 4x10-8 C. How does the potential change if the charge is positive or negative? Q Point Charge: V( r) = k r V( r) = 9.00x Nm 410 x C C V( r) = 300 Nm C 2 1.2m r V( r) 300V = (Note: If Q is negative, V is negative!)

38 Electric Potential Summary Electric Potential: V = PE q Q Point Charge: V( r) = k r 0 Uniform Field: VAB = Ed Electric Potential Difference: Electric Potential Energy: Δ V =ΔV V = Δ PE =Δ Vq = W B 0 A W q AB 0 AB

39 E Compared to V The electric potential is a function of r The electric field is a function of r 2 The effect of a charge on the space surrounding it: The charge sets up a vector electric field which is related to the force The charge sets up a scalar potential which is related to the energy

40 Electric Potential of a Dipole The graph shows the potential (y-axis) of an electric dipole The steep slope between the charges represents the strong electric field in this region

41 E and V for a Dipole The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines

42 Problem Determine the electric potential at point A & B due to a dipole A proton is placed at point A and released. What is the change in potential energy of the proton between A and B? What is the velocity of the proton at B?

43 U for System of Static Charges The potential energy of a system of charges is the energy needed to assemble them from infinity where U = 0. For two charges it is: For three charges: U = k e qq r U k qq qq qq = e + + r12 r13 r23 +: repulsive -: attractive

44 You Try Problem Determine the electric potential energy of the triangular group. That is, find the work done to bring them from infinity to the triangular arrangement. Is there a place interior to the group where the potential is zero? The side of the triangle is 0.50m. U qq qq qq = ke + + r12 r13 r23

45 Electric Potential of a Point The electric potential in the plane around a single point charge is shown The red line shows the 1/r nature of the potential V = q ke r Charge

46 You Try: Electric Potential Problem At which of the empty corners A or B, is the potential greater? If the side of the square is L=0.25 m, find the total electric potential at A and B. V q q = ke + r r

47 Consider two charged particles The electric potential due to several point charges is the sum of the potentials due to each individual charge, taking the potential at infinity equal to zero and r is the distance from the charge to the point P: V Multiple Charges = k e i q r i i The potential energy of the system is U = k e qq r

48 V Due to a Charged Conductor The potential difference between A and B is also zero The charge density is high where the radius of curvature is small And low where the radius of curvature is large The electric field is large near the convex points having small radii of curvature and reaches very high values at sharp points

49 Corona Discharge If the electric field near a conductor is sufficiently strong, electrons resulting from random ionizations of air molecules near the conductor accelerate away from their parent molecules These electrons can ionize additional molecules near the conductor This creates more free electrons The corona discharge is the glow that results from the recombination of these free electrons with the ionized air molecules The ionization and corona discharge are most likely to occur near very sharp points Low energy corona discharge surrounding a circular conductor.

50 Corona Discharge Corona Discharge Supersonic Free-Jet (CDSFJ) to generate excited-state molecular nitrogen for growing III-N semiconductors..

51 Corona Discharge Corona discharge into free air

52 Jacob s Ladder WARNING! Exposure to an arc-producing device can pose health hazards. In a closed space such as a classroom or home, the continuous arc formation of an open-air Jacob's Ladder will ionize oxygen and nitrogen, which then reforms into reactive molecules such as ozone and nitric oxide. These free radicals can be damaging to the mucous membranes of people near the spark gap. Time exposure

53 Van de Graaff Generator Charge is delivered continuously to a high-potential electrode by means of a moving belt of insulating material The high-voltage electrode is a hollow metal dome mounted on an insulated column Large potentials can be developed by repeated trips of the belt Protons accelerated through such large potentials receive enough energy to initiate nuclear reactions

54 Problem-Solving Strategies Conceptualize Think about the individual charges or the charge distribution Imagine the type of potential that would be created Appeal to any symmetry in the arrangement of the charges Categorize Group of individual charges or a continuous distribution?

55 Problem-Solving Strategies, 2 Analyze General Scalar quantity, so no components Use algebraic sum in the superposition principle Only changes in electric potential are significant Define V = 0 at a point infinitely far away from the charges If the charge distribution extends to infinity, then choose some other arbitrary point as a reference point

56 Problem-Solving Strategies, 3 Analyze, cont If a group of individual charges is given Use the superposition principle and the algebraic sum If a continuous charge distribution is given Use integrals for evaluating the total potential at some point Each element of the charge distribution is treated as a point charge If the electric field is given Start with the definition of the electric potential Find the field from Gauss Law (or some other process) if needed

57 Problem-Solving Strategies, final Finalize Check to see if the expression for the electric potential is consistent with your mental representation Does the final expression reflect any symmetry? Image varying parameters to see if the mathematical results change in a reasonable way