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Transcript
Superposition Electric Field From a Dipole ~ field Superposition and Dipole E PHYS 272 - David Blasing Tuesday June 17th, 2014 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Our “Road Map” 2/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Brief Review of Last Lecture Electric Field Created by a Point Charge ~ p.c. = E kq r̂ . |~r |2 Coulomb’s Law: Force Between Two “Point” Charges ~ 12 = F 1 q1 q2 4π0 |~r12 |2 r̂12 ~ 21 = and F 1 q2 q1 4π0 |~r21 |2 r̂21 3/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Principle of Superposition Definition: Principal of Superposition The net electric field at a location is the vector sum of every electric field made there by all the other charged particles around. 4/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Principle of Superposition Definition: Principal of Superposition The net electric field at a location is the vector sum of every electric field made there by all the other charged particles around. Notes: 1 2 3 P ~ net = P E ~ Mathematically expressed, E qi i = i qi 1 4π0 |~ri |2 r̂i ~ field that a single charged particle creates is not The E affected at all by the presense of other charged particles or nearby electric fields. ~ net is The sum runs over all charges qi not present where E being calculated 4/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Superposition Example 5/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere ~ net is related to F ~ net E Net Electric Force on a charge Q ~ net , on any amount of charge Q at a location is The net force, F ~ net QE 6/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere ~ net is related to F ~ net E Net Electric Force on a charge Q ~ net , on any amount of charge Q at a location is The net force, F ~ net QE Notes: ~ net is the net electric field created at Q’s lcation, but not 1 E including any electric field from Q itself 6/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Clicker Question 1 ~ net at the location of q3 if |q2 | ≈ 2|q1 | ? What is the direction of E 7/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Clicker Question 1 ~ net at the location of q3 ? What is the direction of E 8/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Clicker Question 2 9/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Clicker Question 3 10/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Clicker Question 4 11/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere Clicker Question 5 12/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere ~ from a uniformly charged sphere E 13/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere ~ from a uniformly charged sphere E ~ from a uniformly charged sphere E ~ sphere = for r > R, E 1 Q 4π0 r 2 r̂ ~ sphere = 0 for r < R, E 13/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere ~ from a Uniformly Charged Sphere E 1 ~ sphere = Note: for r > R, E 1 Q 4π0 r 2 r̂ . Look familiar? 14/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Definition ~ E from a Uniformly Charged Sphere ~ from a Uniformly Charged Sphere E 1 2 3 ~ sphere = Note: for r > R, E 1 Q 4π0 r 2 r̂ . Look familiar? ~ field that a uniformly charged sphere creates outside its The E radius is identical to that of a point charge at the sphere’s center (with charge equal to the sphere’s total charge). The charged sphere responds to applied electric fields the same way as a point charge at its center would 14/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ dip and the electric dipole moment vector E 15/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ dip and the electric dipole moment vector E Definition: Dipole Moment Vector ~p ~p = q~s where q is the magnitude of both the dipole’s charges, ~s is the positive charge’s position relative to the negative charge. 15/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ field of a dipole at other locations E In lab you will recreate this plot and much more 16/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ field of a dipole at other locations E In lab you will recreate this plot and much more Here we will do two points analytically: 1 2 Location 1 at a distance r along the dipole axis Location 2 at a distance y along the perpendicular axis 16/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 1 Now we will apply the principal of superposition to get the electric field of a dipole. 2 Symmetry is generally your friend. Math is often simpler when you preserve or take advantage of a physical symmetry. 17/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 1 Now we will apply the principal of superposition to get the electric field of a dipole. 2 Symmetry is generally your friend. Math is often simpler when you preserve or take advantage of a physical symmetry. 3 There are two lines of symmetry for a dipole, and we are going to derive simpler formulas along each of those lines. 17/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 1 Now we will apply the principal of superposition to get the electric field of a dipole. 2 Symmetry is generally your friend. Math is often simpler when you preserve or take advantage of a physical symmetry. 3 There are two lines of symmetry for a dipole, and we are going to derive simpler formulas along each of those lines. 4 The following is valid only along those symmetry lines 17/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 18/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 19/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 20/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 21/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 22/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E 22/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Parallel to the Axis Location 1: E ~ fields created by charge distributions do not generally have the E 1 distance dependence of the point charges constituting them r2 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 22/32 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 23/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 24/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 25/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 26/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 27/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 28/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Location 2: Perpendicular to the Axis 29/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Field Summary Dipole E ~ Far From the Dipole, On or Perpendicular to its Axis E ~ ⊥| = |E 1 p 4π0 y 3 ~ axis | = |E 1 2p 4π0 r 3 30/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Field Summary Dipole E ~ Far From the Dipole, On or Perpendicular to its Axis E ~ ⊥| = |E 1 p 4π0 y 3 ~ axis | = |E 1 2p 4π0 r 3 Notes: ~ axis | = 2 ∗ |E ~ ⊥ | at the same distance 1 |E ~ ~ 2 I presented E ⊥ and Eaxis because they simplified nicely and demonstrated superposition 3 y is the “perpendicular distance” from the center, r is the “parallel distance” from the center 30/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ ~ Field Summary Dipole E ~ Far From the Dipole, On or Perpendicular to its Axis E ~ ⊥| = |E 1 p 4π0 y 3 ~ axis | = |E 1 2p 4π0 r 3 Notes: ~ axis | = 2 ∗ |E ~ ⊥ | at the same distance 1 |E ~ ~ 2 I presented E ⊥ and Eaxis because they simplified nicely and demonstrated superposition 3 y is the “perpendicular distance” from the center, r is the “parallel distance” from the center 4 These results are valid when y , r >> s and the observation location is perpendicular and exactly in the middle of the dipole’s axis, or parallel and on the dipole’s axis respectively ~ net at an arbitraty position just superpose (i.e. add) 5 To get E ~ + = 1 q+2 r̂+ to E ~ − = 1 q− E 4π0 r 4π0 r 2 r̂− + PHYS 272 - David Blasing − Matter and Interactions: 14.5-14.6 30/32 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Group Question 1 ~ field, What is the net force? Dipole in a uniform E 31/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Group Question 1 ~ field, What is the net force? Dipole in a uniform E The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so ~ net = F ~+ + F ~− = qE ~ − qE ~ =0 F 31/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Group Question 1 ~ field, What is the net force? Dipole in a uniform E The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so ~ net = F ~+ + F ~− = qE ~ − qE ~ =0 F So what could a dipole be used to measure? 31/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Group Question 1 ~ field, What is the net force? Dipole in a uniform E The force on the positive is equal in magnitude but opposite in direction of the force on the negative, so ~ net = F ~+ + F ~− = qE ~ − qE ~ =0 F So what could a dipole be used to measure? A dipole would ~ field. So they can measure experience a force in a non-uniform E ~ an E field’s uniformity PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 31/32 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Group Question 2 ~ field Dipole in a uniform E ~ app field torques the dipole. What additional piece of This E information might a dipole measure? 32/32 PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 Superposition Electric Field From a Dipole Electric Dipole: Two Close Opposite Charges ~ Eaxis and ~ E⊥ Group Question 2 ~ field Dipole in a uniform E ~ app field torques the dipole. What additional piece of This E information might a dipole measure? ~ , so one could Torque aligns the dipole to the direction of E measure Ê PHYS 272 - David Blasing Matter and Interactions: 14.5-14.6 32/32
