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Physics 2112 Unit 22 Outline Displacement Current Maxwell’s Equations (Final Form) E&M Waves Electricity & Magnetism Lecture 22, Slide 1 Where we are now Qenc E dA o surface d loopE dl dt B dA Faraday’s Law Gauss’ Law B dA 0 surface B dl o I ENCL loop Gauss’ Law for B field Ampere’s Law Our equations so far….. Unit 22, Slide 2 Displacement Current 3 Points, all a distance r from axis of a current c carrying wire connected to capacitor 1 2 ● ● 3 ● r I1 | B1 || B3 | I o / 2r | B2 | 0 ?? Define “displacement current” such that: B dl o ( I ENCL I dis ) loop Electricity & Magnetism Lecture 22, Slide 3 Modify Ampere’s Law Q E 0 0 A EA Q 0 Q 0 dQ d 0 ID dt dt Electricity & Magnetism Lecture 22, Slide 4 Displacement Current Real Current: Charge Q passes through area A in time t: I dQ dt Displacement Current: Electric flux through area A changes in time ID 0 d E dt d E loopB dl o ( I ENCL o dt ) Electricity & Magnetism Lecture 22, Slide 5 Example 22.1 A parrallel plate capacitor has plates that are 2cm in diameter and 1mm apart. If the current into the capacitor is 0.5A, what is the magnetic field between the plates 0.5cm from the axis of the center of the plates? ●d I1 r 1cm 0.5cm R Q1 Conceptual Plan What is the magnetic field 3cm from the axis of the center of the plates? Use modified Ampere’s Law Strategic Plan Find electric flux contained within circle with radius of 0.5cm Find time rate of change of that flux Electricity & Magnetism Lecture 22, Slide 6 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 at the outer edge.. 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 A Electricity & Magnetism Lecture 22, Slide 7 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 at the outer edge.. 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 A Electricity & Magnetism Lecture 22, Slide 8 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 t1, a current I1 flows in the circuit and the capacitor carries charge Q1. What is the time dependence of the magnetic field B at a radius r between the plates of the capacitor? (A) A (B) B S C V Ra B1 0 I1 r 2 R 2 (C) C Electricity & Magnetism Lecture 22, Slide 9 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 Electricity & Magnetism Lecture 22, Slide 10 Final form Qenc E dA o surface Gauss’ Law B dA 0 surface Gauss’ Law for B field d loopE dl dt B dA Faraday’s Law d loopB dl o I ENCL o dt E dA Ampere’s Law Tada!..... Maxwell’s Equations Unit 22, Slide 11 Wave Equation Remember from 2111?? Remember this guy? Not the spring constant! Electricity & Magnetism Lecture 22, Slide 12 Some Calculations + + - d E dl B dA dt loop d dE * h (dx * h) B dt d E d dB 2 dx dt dx 2 d loopB dl o o dt E dA dB dE o o dx dt Electricity & Magnetism Lecture 22, Slide 13 Some Calculations 2 2 d E d E o o 2 2 dx dx A wave equation???? v 1 o o With a velocity of …? see PHYS 2115 Electricity & Magnetism Lecture 22, Slide 14 Electricity & Magnetism Lecture 22, Slide 15 Keep us warm Keep us safe The Sun Electricity & Magnetism Lecture 22, Slide 16 Example 22.2 An electromagnetic plane wave has a wavelength of 0.100nm. a) What is its wave number, k? b) What is its frequency? c) What portion of the electro-magnetic spectrum does it fall in? Electricity & Magnetism Lecture 22, Slide 17 Past Confusion Nothing is moving here. Arrows only represent strength of field. Actually a plane wave. Electricity & Magnetism Lecture 22, Slide 18 CheckPoint 2(A) Ex E0sin(kz wt) An electromagnetic plane wave is traveling in the +z direction. The illustration below shows this wave an some instant in time. Points A, B, and C have the same z coordinate. Compare the magnitudes of the electric field at points A and B. A. Ea < Eb B. Ea = Eb C. Ea > Eb E E0 sin (kz wt): E depends only on z coordinate for constant t. z coordinate is same for A, B, C. Electricity & Magnetism Lecture 22, Slide 19 CheckPoint 2(B) Ex E0sin(kz wt) An electromagnetic plane wave is traveling in the +z direction. The illustration below shows this wave an some instant in time. Points A, B, and C have the same z coordinate. Compare the magnitudes of the electric field at points A and C. A. Ea < Ec B. Ea = Ec C. Ea > Ec E E0 sin (kz wt): E depends only on z coordinate for constant t. z coordinate is same for A, B, C. Electricity & Magnetism Lecture 22, Slide 20