Physics 212 Lecture 22 Physics 212 Lecture 21

Physics 212 Lecture 22 Physics 212 Lecture 21, Slide 1

Music Who is the Artist? A) B) C) D) E) Billy Joel Boz Scaggs Mark Knopfler Donald Fagen Michael Mc. Donald 1976 1994 Theme of the week? Tower of Power, Boz Scaggs – San Francisco sounds of the ’ 70 s

Your Comments “I got chills. This is incredible stuff!” “Fun stuff! but kinda hard. ” “The derivation of the wave equation lost me ” “This prelecture was so horrible. Maybe it was just me, but I’m not sure that was English being spoken. Please clarify these concepts that seemingly came out of nowhere. “ Wide Range: We will try to make clear, at least the BIG IDEAS “I got the modified Ampere’s law, but you lost me at one-dimensional waves; I’ve never EVER seen that equation before. I don’t understand what we just derived; why did we differentiate one with respect to time and the other with respect to z? ? This all went by so fast” The solutions of the wave equation are the “All of the derivations and second derivatives became important part a little jumbled. I feel like this isn’t a very hard topic, just the way it’s presented is a little overwhelming. ” “If I were Gauss, Ampere or Faraday I’d be mad that Maxwell just came along and claimed all the formulas for himself. ” “Woah, whoah! Lecture slide speed limit: 3. 00 x 10^8 m/s!” 05 Physics 212 Lecture 21, Slide 3

What We Knew Before Prelecture 22 06 Physics 212 Lecture 21, Slide 5

After Prelecture 22: Modify Ampere’s Law 10 Physics 212 Lecture 21, Slide 6

Displacement Current: Generalization 08 Real Current: Charge Q passes through area A in time t: Displacement Current: Electric flux through area A changes in time Free space Physics 212 Lecture 21, Slide 7

Calculation Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. S C V Ra At time t 1, what is the magnetic field B 1 at a radius r (point d) in between the plates of the capacitor? ●d I 1 r R Q 1 • Conceptual and Strategic Analysis • Charge Q 1 creates electric field between the plates of C • Charge Q 1 changing in time gives rise to a changing electric flux between the plates • Changing electric flux gives rise to a displacement current I D in between the plates • Apply (modified) Ampere’s law using ID (what does Amperian loop look like) to determine B 10 Physics 212 Lecture 21, Slide 8

Calculation Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. c● ●d r I 1 S C V Ra r R Q 1 Compare the magnitudes of the B fields at points c and d. (A) Bc < Bd (B) Bc = Bd What is the difference? Apply (modified) Ampere’s Law point c: I(enclosed) = I 1 11 r X (C) Bc > Bd r R point d: ID(enclosed) < I 1 Physics 212 Lecture 21, Slide 9

Calculation Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. ●d I 1 E S C V Ra r R Q 1 What is the magnitude of the electric field between the plates? (A) (B) Q 1 13 (C) (D) R Physics 212 Lecture 21, Slide 10

Calculation Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. ●d I 1 E S C V Ra r R Q 1 What is the electric flux through a circle of radius r in between the plates? (What does the Amperian loop look like? ) (A) (B) r (C) (D) R Physics 212 Lecture 21, Slide 11

Calculation Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. ●d I 1 E S C V Ra r R Q 1 What is the displacement current enclosed by circle of radius r ? (A) (B) r 17 (C) (D) R Physics 212 Lecture 21, Slide 12

Calculation Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. ●d I 1 E S C V Ra r R Q 1 What is the magnitude of the B field at radius r ? (A) (B) (C) (D) Ampere’s Law: r 19 R Physics 212 Lecture 21, Slide 13

Checkpoint 1 b At time t=0 the switch in the circuit shown below is closed. Points A and B lie inside the capacitor; A is at the center and B is toward the outer edge. A Compare the magnitudes of the magnetic fields at points A and B just after the switch is closed A. BA < BB B. BA = BB C. BA > BB “B has more flux enclosed, has a greater electric flux, and has greater B-field. ” “the electric field is the same, so the magnetic field should also be the same. ” “A is right in the center of the electric field, while B is further away, so A will be larger” 21 Physics 212 Lecture 21, Slide 14

Checkpoint 1 b At time t=0 the switch in the circuit shown below is closed. Points A and B lie inside the capacitor; A is at the center and B is toward the outer edge. A Compare the magnitudes of the magnetic fields at points A and B just after the switch is closed A. BA < BB B. BA = BB C. BA > BB From the calculation we just did: 21 Physics 212 Lecture 21, Slide 15

Checkpoint 1 a At time t=0 the switch in the circuit shown below is closed. Points A and B lie inside the capacitor; A is at the center and B is toward the outer edge. A After the switch is closed, there will be a magnetic field at point A which is proportional to the current in the circuit: A. True B. False “There is displacement current between the plates, this creates a magnetic field as well” “larger magnetic field will be produced by larger current. ” “at A, the B field is zero” Physics 212 Lecture 21, Slide 16

Checkpoint 1 a At time t=0 the switch in the circuit shown below is closed. Points A and B lie inside the capacitor; A is at the center and B is toward the outer edge. A After the switch is closed, there will be a magnetic field at point A which is proportional to the current in the circuit: A. True B. False B is proportional to I but At A, B = 0 !! Physics 212 Lecture 21, Slide 17

Follow-Up Switch S has been open a long time when at t = 0, it is closed. Capacitor C has circular plates of radius R. At time t = t 1, a current I 1 flows in the circuit and the capacitor carries charge Q 1. S C V Ra What is the time dependence of the magnetic field B at a radius r between the plates of the capacitor? (A) (B) (C) B at fixed r is proportional to the current I Close switch: VC =0 I = V/Ra (maximum) I exponentially decays with time constant t = Ra. C 25 Physics 212 Lecture 21, Slide 18

Follow-Up 2 Suppose you were able to charge a capacitor with constant current (does not change in time). Does a B field exist in between the plates of the capacitor? (A) YES (B) NO Constant current Q increases linearly with time Therefore E increases linearly with time ( E = Q/(Ae 0) d. E/dt is not zero Displacement current is not zero B is not zero ! Physics 212 Lecture 21, Slide 19

We learned about waves in Physics 211 30 Physics 212 Lecture 21, Slide 20

“How can light move at the same velocity in any inertial frame of reference? That's really trippy. ” see PHYS 225 33 Physics 212 Lecture 21, Slide 21

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Checkpoint 2 a An electromagnetic plane-wave is traveling in the +z direction. The illustration below shows this wave at some instant in time. Points A, B and C have the same z coordinate. Ex = E 0 sin(kz-wt) Compare the magnitudes of the electric fields at points A and B A. EA < EB B. EA = EB C. EA > EB “A is always greater because it has 2 magnetic fields acting on it instead of just 1 like the other points. “ “points A and B are located in the electric field wave, so both would be influenced by the electric field. However, point C is located in the magnetic field and not the electric field and as such, has 0 electric field. “ “E = E 0 sin (kz - wt): E depends only on z coordinate for constant t. z coordinate is same for A, B, C. “ 40 Physics 212 Lecture 21, Slide 24

Checkpoint 2 a An electromagnetic plane-wave is traveling in the +z direction. The illustration below shows this wave at some instant in time. Points A, B and C have the same z coordinate. Ex = E 0 sin(kz-wt) Compare the magnitudes of the electric fields at points A and B A. EA < EB B. EA = EB C. EA > EB E = E 0 sin (kz - wt): E depends only on z coordinate for constant t. z coordinate is same for A, B, C. 40 Physics 212 Lecture 21, Slide 25

Checkpoint 2 b An electromagnetic plane-wave is traveling in the +z direction. The illustration below shows this wave at some instant in time. Points A, B and C have the same z coordinate. Ex = E 0 sin(kz-wt) Compare the magnitudes of the electric fields at points A and C A. EA < EC B. EA = EC C. EA > EC “A is always greater because it has 2 magnetic fields acting on it instead of just 1 like the other points. “ “points A and B are located in the electric field wave, so both would be influenced by the electric field. However, point C is located in the magnetic field and not the electric field and as such, has 0 electric field. “ “E = E 0 sin (kz - wt): E depends only on z coordinate for constant t. z coordinate is same for A, B, C. “ 45 Physics 212 Lecture 21, Slide 26

Checkpoint 2 b An electromagnetic plane-wave is traveling in the +z direction. The illustration below shows this wave at some instant in time. Points A, B and C have the same z coordinate. Ex = E 0 sin(kz-wt) Compare the magnitudes of the electric fields at points A and C A. EA < EC B. EA = EC C. EA > EC E = E 0 sin (kz - wt): E depends only on z coordinate for constant t. z coordinate is same for A, B, C. 45 Physics 212 Lecture 21, Slide 27

Follow-up Ex = E 0 sin(kz-wt) Consider a point (x, y, z) at time t when Ex is negative and has its maximum value. At (x, y, z) at time t, what is By? A) B) C) D) 45 By is positive and has its maximum value By is negative and has its maximum value By is zero We do not have enough information Physics 212 Lecture 21, Slide 28