Module M2-1 Electrical Engineering. Natural phenomena and engineering applications. Topics. After this tutorial, you will be able to

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1 Module M2-1 Electrical Engineering LECTURE 7 TME-VRYNG FELD MXWELL EQUTON 1 Topics Natural phenomena and engineering applications of time-varying magnetic fields Faraday s law Lenz s law Maxwell s equations 2 EPTEMER 29, 2016 fter this tutorial, you will be able to understand the natural phenomena and engineering applications of time-varying field use Faraday s law to determine the electromotive force () and current 3 For the case where the surface is stationary. ut the magnetic field is time-varying use Lenz s law to determine the direction of current recognize the differential and integral forms of Maxwell s equations Natural phenomena and engineering applications 4

2 fter Oersted s discovery, Faraday hypothesized that magnetic field can produce current 5 Michael Faraday is an English scientist 6 Oersted s discovery (1820) Current Magnetic field???? born 1791, died 1867 His life overlaps with the reign of King Ramas V famous for his work in electromagnetics The unit farad is named after him Faraday works intermittently about 10 years and discovers that magnetic field can produce current 7 Joseph Henry is an merican scientist 8 Oersted s discovery (1820) Current Magnetic field Farday s discovery (1831) Joseph Henry also discovered similar results, independently of Faraday and around the same time ut Faraday published the results first born 1797, died 1878 His life overlaps with the reign of King Ramas V famous for his work in electromagnetics The unit henry is named after him

3 teady magnetic field will not produce current 9 The key idea to produce current is the change in magnetic flux 10 Galvanometer (predecessor of ammeter) small coil loop of a wire (at rest) N large coil liquid battery bar magnet (at rest) ammeter pparatuses in one of Faraday s experiments current is detected when we move the small coil (continued) 11 Electromagnetic induction: time-varying magnetic flux produces current 12 wire ammeter Modern replication of one of Faraday s experiments current is detected momentarily when we turn on the switch turn off the switch VDO 71: steady magnetic field does not produce current, but moving bar magnet and moving electromagnet produces current as shown by an ammeter; Duration: 2 min (trimmed, 3:27-5:27) VDO 68: induced powers a fan and light bulbs; Duration: 1:44 min

4 We have seen the definition of a vector field Time-varying field 13 vector field ~F is a mapping (a function) from position (x, y, z) into a vector n rectangular coordinate system, a vector field takes the form ~F(x, y, z) =f(x, y, z) b i g(x, y, z) b j h(x, y, z) b k 14 ~F functions of position (x, y, z) We interpret ~F(2, 3, 1), for example, as a vector quality associated with point (2, 3, 1) vector field vector field could be uniform ~F 15 is uniform if it takes the form ~F(x, y, z) = (constant) b i (constant) b j (constant) b k Examples of uniform vector fields: ~F(x, y, z) = b i b j k b ~U(x, y, z) = 10 b i 1 b j 4k b The vector field may depend on time Definition: time-varying vector field ~F is a mapping from position (x, y, z) and time t into a vector time-varying vector field takes the form ~F(x, y, z, t) =f(x, y, z, t) b i g(x, y, z, t) b j h(x, y, z, t) b k 16 ~V(x, y, z) =2 b i 3 b j functions of position (x, y, z) and time t We interpret ~F(2, 3, 1,t= 10), for example, as a vector quality associated with point (2, 3, 1) at time t=10

5 For each time, a vector field ~F(x, y, z, t). could be uniform 17 Definition: time-varying vector field ~F is uniform for each time instance if it takes the form ~F(x, y, z, t) =f(t) b i g(t) b j h(t) k b functions time t For each time t, the vector field ~F(t) is a constant vector, not depending on the position (x, y, z) Does the vector field depend on position? No (a uniform field) Yes (non-uniform field) n summary, here are the forms of vector fields for the 4 possible cases 18 Does the vector field depend on time t? No (a static or a steady field) ~F = (constant) b i (constant) b j (constant) b k ~F = f(x, y, z) b i g(x, y, z) b j h(x, y, z) b k Yes (a time-varying field) ~F = f(t) b i g(t) b j h(t) b k ~F = f(x, y, z, t) b i g(x, y, z, t) b j h(x, y, z, t) b k Recall: The magnetic flux Φ passing through an area of the loop is... Faraday s law Magnetic flux Z = d ~ Faraday discovers that a change in current produces

6 Faraday found that a voltage difference occurs across the galvanometer This voltage difference is called the electromotive force (), V and is caused by the change in magnetic flux 21 (continued) 22 The induced in a closed conducting loop of N turns is V = N d dt = N d dt Z d ~ (volt) VDO 73: The larger the number N of turns the larger the induced current Duration: 2:10 min (trimmed 0:04-2:14) To gain insight into the, consider a simple case of uniform magnetic field 23 (continued) may be induced by varying the magnetic field. 24 urface area surface normal vector. The magnetic flux is Hence, the is To have a non-zero, we vary these with time: the magnetic field the surface area the angle between ~ and the surface = R ~ d ~ = cos (uniform ) d( cos ) V = N dt 0 <

7 (continued) may be induced by varying the surface area 25 (continued) may be induced by varying the angle θ 26 ~ rea = rea 0 < = = 90 n general, from Faraday s law, there are 3 ways to generate the (and hence the current) 1. Keep the loop unchanged with time 27 (continued) 2. Keep steady (unchanged with time) 28 Change the magnetic field with time The induced is called transformer, We will study this case only Move the loop with a time-varying surface area (relative to the normal component of ) The induced is called motional, We will not study this case V m

8 3. Moving the loop (continued) 29 tationary loop in a time-varying magnetic field Change with time 30 We will not study this case Faraday s law for this case becomes We will focus on the simplest case 32 surface normal vector (a unit vector perpendicular to the surface) = N d dt Z Z d = N d ~ (transformer ) ammeter a loop of wire (only the shaded area is exposed to a changing magnetic field) uppose is uniform for each time instance: = f(t) b i g(t) b j h(t) b k uppose the surface lies on a plane

9 The transformer for the simplest case equals = f(t) b i g(t) b j h(t) b k Here is a derivation of the formula in the previous slide = = N N d ~ Z 34 = f(t) b i g(t) b j h(t) b k h f 0 (t) b i g 0 (t) b j h 0 (t) k b i d = Number of turns h N f 0 (t) b i g 0 (t) b j h 0 (t) k b i area of each loop a unit vector perpendicular to the surface the term does not depend on the position (x, y, z) h = N f 0 (t) b i g 0 (t) b j h 0 (t) k b i Z d = (surface area) Here is a convention of the direction of. and the polarity of 35 f you define downward, the polarity of the transformer reverses 36 V tr 1. Point the thumb of your right hand in the direction of V tr 2. The four fingers pass across the opening from the terminal of V tr to the terminal

10 n equivalent circuit is the bottom one 37 n this course, we will assume that the internal resistance is negligible: R i = 0 Ω 38 resistance R R (t) R i (internal resistance of the loops) R (t) The equivalent circuit further simplifies, as shown above Voltage difference and current are related by Ohm s law 39 Concept Question 40 point R point point R =2 point Choose a direction of Ohm s law states that Without magnetic field, the current flows from high voltage to low voltage Potential difference V V =1 ม ค าเท าใด Voltage at the arrow tail Voltage at the arrow head = R ก. 2 V n the above picture, this is the voltage at point n the above picture, this is the voltage at point ข. 2 V

11 Concept Question 41 way to think of a negative current is to reverse its the direction and sign 42 V =3V R =1 V = 1V = 4 กระแส ในร ปม ค าเท าใด is the same as ก. 4 ข. 4 =4 Concept Question (continued) olution V tr = 10 V กระแสไหลในท ศใด หาก ก. ตามเข มนาฬ กา ข. ทวนตามเข มนาฬ กา = 10 V จาก terminal และ จะได ว า = V ด งน น V V > 0 = 10 (เป นบวก) บร เวณท ไม เปล ยนแปลง (นอกบร เวณแรเงา) กระแสไหลจาก high voltage ไป low voltage (จ ด ไป จ ด ) กระแสไหล V

12 Concept Question Concept Question V tr R =2 กระแสไหลในท ศใด หาก ก. ตามเข มนาฬ กา ข. ทวนตามเข มนาฬ กา = 10 V จากเคร องหมาย terminal และ ได ว า = V 10 (ต ดลบ) V กระแสไหล ก. 5 ข. 5 ม ค าเท าใด หาก = 10 V (continued) olution Concept Question R =2 V tr = 10 V 47 ร ปเด ยวก น R =2 V R =2 48 จาก terminal และ - ได ว า Ohm s law: V จ ดร ป Ohm s law ได ว า V = R = 10 V = V V V กระแสไหล ก. 5 ข. 5 ม ค าเท าใด หาก = 10 V

13 You will be asked to use Faraday s law for only this type of problems 49 = f(t) b i g(t) b j h(t) b k Here are typical steps to find the and current 1. Choose any of the two orientations for ~n 2. Use right hand to label the and terminals of 3. Use Faraday s law find 50 Given a loop or surface Given a time-varying, uniform magnetic field Find the or find the induced current t each time t, 4. Determine the direction of current from the sign of V tr Recall that in the region of steady magnetic field, the current flows from high voltage to low voltage 5. Determine the magnitude of current from Ohm s law Lenz law 51 Lenz is a Russian physicist 52 Heinrich Lenz (born 1804, died 1865) His life overlaps with the reign of King Ramas - V Famous for Lenz s law

14 Lenz s law is useful in determining the direction of the induced current Lenz s law: The current in the loop is always in a direction that opposes the change in magnetic flux Φ(t) that produced f the loop is stationary (the case for us), the current in the loop is always in a direction that opposes the change in the magnetic field that produced 53 Recall: current produces the magnetic field in the direction of your right hand > 0 54 Or curl the four fingers in the direction of ind ind The thumb points in the direction of in the loop The current that flows counterclockwise produces the magnetic field ind pointing out of the slide (continued) 55 > 0 ind Or curl the four fingers in the direction of The thumb points in the direction of ind in the loop f increases, the induced current is in the direction that decreases in the loops ind 56 magnetic field produced by the inducted current The current that flows clockwise produces the magnetic field pointing out of the slide ind uppose increasing is increasing is out of the slide and its magnitude is Lenz s law says that the current > 0 must flows in the clockwise direction

15 f decreases, the induced current is in the direction that increases in the loops ind 57 magnetic field produced by the inducted current Concept Question กระแสจะม ท ศใด เม อขนาดของ เพ มข น 58 ind is decreasing uppose is out of the slide and its magnitude is increasing Lenz s law says that the current > 0 must flows in the counterclockwise direction induced magnetic field ind ช ออกจาก กระดาษเพ อต อต านการเปล ยนแปลงของ ~ 0 > ก. ตามเข มนาฬ กา ข. ทวนเข มนาฬ กา Lenz s law can be observed in nature Magnet falling through a hollow conductor; an inducted magnetic field opposes the external magnetic field VDO 70: Eddy current (trimmed 13:11-14:44, 1:24 min) VDO 69: comparison of falling speed (1:38 min) 59 Maxwell s equations 60 #70 #69

16 lthough Faraday discovers electromagnetic induction, he did not put it in a formula we saw man who mathematizes the formula is James Clerk Maxwell, a cottish mathematical physicist born 1831, died 1879 His life overlaps with the reign of King Rama -V Well known for Maxwell s equations 61 Maxwell s equations are 4 equations Maxwell s equations describe the interplay among electric field, magnetic field, current, and charge They were published around They were usually written in either a differential form or an integral form We will not use Maxwell s equations in the full form 62 Here are Maxwell s equations Reference Differential form ntegral form Gauss s law r ~ D = v ~D d ~ = Q enc Faraday s law (for a stationary surface ) Gauss s law for magnetism mpere s law r ~ E = r ~ =0 r ~ H = ~ ~ volume charge density (C/m 3 ) C C ~E d ~ L = d ~ =0 d ~ Z ~H d L D = d ~ Here are new symbols in this lecture ymbol Name Unit bbreviation V V m v electromotive force () volt V transformer volt V motional volt V 64 volume charge density coulomb/meter 3 C/m 3 This symbol is called the divergence of This symbol is called the curl of ~H

17 ummary Natural phenomena and engineering applications electromagnetic induction Faraday s law a change in the magnetic flux produces current statement of Faraday s law electromotive force () the loop is stationary but is time-varying Lenz s law statement of Lenz s law Examples Maxwell s equations in differential form in integral form 65