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Symmetry, Equipotentials and Energy

Symmetry, Equipotentials and Energy

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Symmetry, Equipotentials and Energy

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  1. Symmetry, Equipotentials and Energy Electric Potential

  2. What is an expert: three possibilities? • Knows answer without waking up • Has worked through most problems during a previous life • Is armed with a) logic b) intuition and c) ability to use math • Knows answer without waking up • Has worked through most problems during a previous life • Is armed with a) logic b) intuition and c) ability to use math You are all well on the way to being experts Enjoy and learn from the challenges

  3. Today… • Equipotentials and conductors • Electric Potential for other symmetries • A problem of infinities • We got V from E, now how can we get E from V • Calculate electric field of dipole from potential • Electric Potential Energy • of charge in external electric field • stored in the electric field itself (next time) • Appendix: • Calculate electric field of dipole from potential

  4. Preflight 6: Two spherical conductors are separated by a large distance. They each carry the same positive charge Q. Conductor A has a larger radius than conductor B. A B 2) Compare the potential at the surface of conductor A with the potential at the surface of conductor B. a) VA > VBb) VA = VBc) VA < VB

  5. Equipotential Potential from a charged sphere Last time… Er (where ) • The electric field of the charged sphere has spherical symmetry. • The potential depends only on the distance from the center of the sphere, as is expected from spherical symmetry. • Therefore, the potential is constant along a sphere which is concentric with the point charge. These surfaces are called equipotentials. • Notice that the electric field is perpendicular to the equipotential surface at all points.

  6. Equipotentials • Defined as: The locus of points with the same potential. • Example: for a point charge, the equipotentials are spheres centered on the charge. • The electric field is always perpendicular to an equipotential surface! Why?? Along an equipotential surface, there is NO change inV. Pick A and B on it Then, We can conclude then, that is zero. If the dot product of the field vector and the displacement vector is zero, then these two vectors are perpendicular, or the electric field is always perpendicular to the equipotential surface.

  7. + + + + + + + + + + + + + + Conductors • Claim The surface of a conductor is always an equipotential surface (in fact, the entire conductor is an equipotential). • Why?? If surface were not equipotential, there would be an electric field component parallel to the surface and the charges would move!!

  8. + + + + + + + + + + + + + + Conductors • Claim E-field lines are always locally perpendicular to a conducting surface • Why?? If the E-field had a component parallel to the surface, charges would move until the surface charge distribution acquired a pattern that had no component of E parallel to the surface. Remember the charge on the surface, together with all other charges, generates E-field.

  9. + + + + + + + + + + + + + + Conductors • Claim The local E-field magnitude at a point on the surface of a conductor is related to the local surface charge density by E=s/e0. (this is E right at the surface) • Why?? Just zoom in.

  10. Preflight 6: B A 3) The two conductors are now connected by a wire. How do the potentials at the conductor surfaces compare now ? a) VA > VB b) VA = VB c) VA< VB 4) What happens to the charge on conductor A after it is connected to conductor B ? a) QAincreases b) QAdecreases c) QAdoesn’t change

  11. r2 r1 Ball 1 Ball 2 Smaller r higherE closer to breakdown (a)Ball 1 (b)Ball 2 (c)Same Time Ex. High Voltage Terminals must be big! Followup Question: Two charged balls are each at the same potential V. Ball 2 is twice as large as ball 1. • As V is increased, which ball will induce dielectric (air) breakdown first?

  12. Other symmetries • What about cylindrical and planar symmtries? • E-fields are easily obtained using techniques from last two lectures – Gauss’ Law, etc • To get potential, integrate -E from reference point to any point (x,y,z). Where to start? • Charge distribution extends to infinity, so we can not pick rinfinity for “place” where V0. • Must pick some other point • This point and other points form an equipotential surface where V=0 • Discuss planar symmetry, then cylindrical symmetry

  13. planar symmerty s x V(x) x x=0 + for x>0 - for x<0 x=0 • Let’s assume s is positive • Pick x=0 plane to be V=0 • Integrate –E from this plane to get V(x) • Potential drops on both sides since E // dl on both sides

  14. planar symmerty -s s V(x) x x x=0 x=d V(x) x=0 x x=0 • With more planes, just use superposition • s at x=0, -s at x=d • Still pick x=0 plane to be V=0 • Second plane has offset wedge and opposite charge • Add potential graphs from both source planes to get final result

  15. cylindrical symmetry Segment of infinite line charge lC/m End on view a V=0 here • Consider an infinite line charge on the z-axis • Again, we can’t set V at infinity = 0, since line charge extends to infinity • Equipotential surfaces will be cylinders, V has cyl sym • Pick one of these cylinders (r=a) to be V=0

  16. cylindrical symmetry V(r) End on view a a r V=0 here Curve shows V(r) for l pos What happens when l neg? Still have zero at a, and curve flips through that point on x-axis…

  17. cylindrical symmetry • More complicated cylindrically symmetrical charge distributions can also be solved by superposition Vtotal(r) = V1(r)+ V2(r)+ V3(r)+ V4(r)+…. • But… • Must make sure you keep V(a)=0 for all of the potential functions you add up.

  18. + + + E=0 inside conducting shell. + + - - - - - + - + - - +q charge density induced on inner surface non-uniform. + - - + - + - + - - - + charge density induced on outer surface uniform + + + + Eoutside has spherical symmetry centered on spherical conducting shell. Charge on Conductors? • How is charge distributed on the surface of a conductor? • KEY: Must produce E=0 inside the conductor and E normal to the surface . Spherical example (with little off-center charge):

  19. Lecture 6, ACT 1 -q An uncharged spherical conductor has a weirdly shaped cavity carved out of it. Inside the cavity is a charge -q. How much charge is on the cavity wall? 1A (a) Less than< q(b) Exactly q(c) More than q 1B How is the charge distributed on the cavity wall? (a) Uniformly (b) More charge closer to –q (c) Less charge closer to -q 1C How is the charge distributed on the outside of the sphere? (a) Uniformly (b) More charge near the cavity (c) Less charge near the cavity

  20. Lecture 6, ACT 1 -q An uncharged spherical conductor has a weirdly shaped cavity carved out of it. Inside the cavity is a charge -q. How much charge is on the cavity wall? 1A (a) Less than< q(b) Exactly q(c) More than q By Gauss’ Law, since E=0 inside the conductor, the total charge on the inner wall must be q (and therefore -q must be on the outside surface of the conductor, since it has no net charge).

  21. Lecture 6, ACT 1 -q The induced charge will distribute itself nonuniformly to exactly cancel everywhere in the conductor. The surface charge density will be higher near the -q charge. 1B How is the charge distributed on the cavity wall? (a) Uniformly (b) More charge closer to -q (c) Less charge closer to -q

  22. Lecture 6, ACT 1 -q 1C How is the charge distributed on the outside of the sphere? (a) Uniformly (b) More charge near the cavity (c) Less charge near the cavity As in the previous example, the charge will be uniformly distributed (because the outer surface is symmetric). Outside the conductor the E field always points directly to the center of the sphere, regardless of the cavity or charge. Note: this is why your radio, cell phone, etc. won’t work inside a metal building!

  23. ConductorsversusInsulatorsCharges move to Charges cannotcancel electric field move at allin the conductor E=0  equipotential Charge distribution surface on insulatorunaffected byexternal fieldsAll charge on surface Charge can sit “inside”(Appendix B describes method of “images” to find the surface charge distribution on a conductor [only for your reading pleasure!])

  24. r L But: r S Smaller sphere has the larger surface charge density ! Þ Charge on Conductor Demo • How is the charge distributed on a non-spherical conductor?? Claim largest charge density at smallest radius of curvature. • 2 spheres, connected by a wire, “far” apart • Both at same potential

  25. Equipotential Example • Field lines more closely spaced near end with most curvature – higher E-field • Field lines ^ to surface near the surface (since surface is equipotential). • Near the surface, equipotentials have similar shape as surface. • Equipotentials will look more circular (spherical) at large r.

  26. Electric Dipole Equipotentials • First, let’s take a look at the equipotentials:

  27. Some fish have the ability to produce & detect electric fields • Navigation, object detection, communication with other electric fish • “Strongly electric fish” (eels) can stun their prey Electric Fish Black ghost knife fish Dipole-like equipotentials More info: Prof. Mark Nelson, Beckman Institute, UIUC -Electric current flows down the voltage gradient -An object brought close to the fish alters the pattern of current flow

  28. E from V? V+dV V • Expressed as a vector, E is the negative gradient of V • Cartesian coordinates: • Spherical coordinates: • We can obtain the electric field E from the potential V by inverting our previous relation between E andV:

  29. Preflight 6: This graph shows the electric potential at various points along thex-axis. 8) At which point(s) is the electric field zero? A B C D

  30. E from V: an Example • Consider the following electric potential: • Something for you to try: Can you use the dipole potential to obtain the dipole field? Try it in spherical coordinates ... you should get (see Appendix): • What electric field does this describe? ... expressing this as a vector:

  31. Lecture 6, ACT 2 2 The electric potential in a region of space is given by The x-component of the electric field Ex at x= 2 is (a)Ex = 0(b)Ex>0(c)Ex< 0

  32. Lecture 6, ACT 2 We know V(x) “everywhere” To obtain Ex “everywhere”, use 2 The electric potential in a region of space is given by The x-component of the electric field Ex at x= 2 is (a)Ex= 0(b)Ex> 0(c)Ex< 0

  33. The Bottom Line If we know the electric field E everywhere, Þ If we know the potential function V everywhere, allows us to calculate the electric field E everywhere • Units for Potential! 1 Joule/Coul = 1 VOLT allows us to calculate the potential function V everywhere (keep in mind, we often define VA = 0 at some convenient place)

  34. Therefore, a particle moving under the influence • of the Coulomb force is said to have an • electric potential energy defined by: this “q” is the “test charge” in other examples... Electric Potential Energy • The Coulomb force is a CONSERVATIVE force (i.e., the work done by it on a particle which moves around a closed path returning to its initial position is ZERO.) • The total energy (kinetic + electric potential) is then conserved for a charged particle moving under the influence of the Coulomb force.

  35. r q A Q • charge +q is brought to pt A, a distance r from Q. • charge +2q is brought to pt B, a distance 2r from Q. Q B 2r (b)UA= UB (c)UA> UB (a)UA< UB • Compare the potential energy of q (UA) to that of 2q(UB): 2q 3B • Suppose charge 2q has mass m and is released from rest from the above position (a distance 2r from Q). What is its velocity vf as it approaches r =¥? (c) (a) (b) Lecture 6, ACT 3 Two test charges are brought separately to the vicinity of a positive charge Q. 3A

  36. r Q q A Q 2q 2r B (b)UA= UB (c)UA> UB (a)UA< UB Lecture 6, ACT 3 • Two test charges are brought separately to the vicinity of positive charge Q. • charge +q is brought to pt A, a distance rfrom Q. • charge +2q is brought to pt B, a distance 2r from Q. • Compare the potential energy of q (UA) to that of 2q (UB): 3A • The potential energy of q is proportional to Qq/r. • The potential energy of 2q is proportional to Q(2q)/(2r). • Therefore, the potential energies UA and UB are EQUAL!!!

  37. 3B • Suppose charge 2q has mass m and is released from rest from the above position (a distance 2r from Q). What is its velocity vf as it approaches r =¥? (c) (a) (b) • What we have here is a little combination of 111 and 112. • The principle at work here is CONSERVATION OF ENERGY. • Initially: • The charge has no kinetic energy since it is at rest. • The charge does have potential energy (electric) = UB. • Finally: • The charge has no potential energy (Uµ 1/R) • The charge does have kinetic energy = KE Lecture 6, ACT 3

  38. Accelerators • Electrostatic: Van de Graaff • electrons ®100 keV ( 105 eV) • Electromagnetic: Fermilab • protons ®1TeV ( 1012 eV) Energy Units MKS: U = QV1 coul-volt = 1 joule for particles (e, p, ...) 1 eV = 1.6x10-19 joules

  39. Summary • Physically, DV is what counts • The place where V=0 is “arbitrary” (at infinity for sperically symmetric charges) • At some specific place for charge distributions that extend to infinity • Conductors are equipotentials and E is always perpendicular to the surface of a conductor • Find E from V: • Potential Energy • Next time  capacitors

  40. z r 1 +q r r 2 a q r2-r1 a -q • Rewrite this for special case r>>a: Þ Now we can use this potential to calculate the E field of a dipole(after drawing a picture) (remember how messy the direct calculation was?) Appendix A: Electric Dipole The potential is much easier to calculate than the field since it is an algebraic sum of 2 scalar terms.

  41. z r 1 +q r r 2 a q a -q • Calculate E in spherical coordinates: the dipole moment Þ Appendix A: Electric Dipole

  42. Appendix A: Dipole Field z +q r q a a -q

  43. (b) qmax= 45° (c) qmax= 90° (a) qmax= 0 • The expression for the electric field of a dipole (r >> a) is: • The polar component of E is maximum when sinq is maximum. • Therefore, Eq has its maximum value when q= 90°. z Sample Problem r • Consider the dipole shown at the right. • Fixr = r0>> a • Define qmax such that the polar component of the electric field has its maximum value (for r = r0). 1 +q r r 2 a q a -q What is qmax?

  44. - + - + - + - + + - - - - - - + + + + + + + + + + Appendix B: FYI: Induced charge distribution on conductorvia“method of images” • Consider a source charge brought close to a conductor: + • Charge distribution “induced” on conductor by source charge: • Induced charge distribution is “real” and sources E-field so that the total is zero inside conductor! • resulting E-field is sum of field from source charge and induced charge distribution • E-field is locally perpendicular to surface • With enough symmetry, can solve fors on conductor • how? Gauss’ Law

  45. - - - - - Appendix B: (FYI) Induced charge distribution on conductorvia“method of images” • Consider a source charge brought close to a planar conductor: • Charge distribution “induced” on conductor by source charge • conductor is equipotential • E-field is normal to surface • this is just like a dipole + - • Method of Images for a charge (distribution) near a flat conducting plane: • reflect the point charge through the surface and put a charge of opposite sign there • do this for all source charges • E-field at plane of symmetry - the conductor surface determines s.