Unit 9: Electromagnetic Induction Magnetic flux Electromagnetic induction phenomena. Faraday’s law and Lenz’s law. Examples of Faraday’s and Lenz’s laws. Michael Faraday 1791-1867 Mutual inductance and self-inductance. The transformer. Stored energy on an inductor. Hall of Royal Society London. Today
Magnetic flux Given a surface element dS, magnetic flux through such surface element is defined as (inner product): d B d S d S Unit: Weber B Wb = T m 2 B d S If surface is finite (surface S): S Tipler, chapter 28.1
Faraday’s law Induced currents on a loop are due to electromotive forces depending on speed of change of magnetic flux passing throug the surface enclosed by the loop: d Faraday’s law Experimental law dt If R is the resistance of loop, a current will flow through the loop: i R Derivative of flux against time give us the modulus of induced electromotive force, but not the polarity. The polarity of emf comes from Lenz’s law. Tipler, chapter 28.2
Electromagnetic induction phenomena Electromagnetic induction phenomena are known from 1830 when Michael Faraday and Joseph Henry discovered that a varying magnetic field produces electric currents on a coil (or a loop). Current 0 Moving a magnet into a coil produces an electric current on coil We’ll see that a varying magnetic field isn’t the only way to induce a current. Most important are changes on magnetic flux Tipler, chapter 28, Introduction
Why could magnetic flux change? As magnetic flux through a loop surface comes from: cos B d S B dS S S Changes on flux can be due to: a) Changes on magnetic field, B b) Changes on surface of loop, S c) Changes on angle between B and S, Tipler, chapter 28.2
Why could magnetic flux change? a) Changes on magnetic field, B N S 0 0 I induced
Why could magnetic flux change? b) Changes on surface of loop, S v i I I produces a magnetic field through loop. When side of loop moves (speed v), magnetic flux changes and i appears.
Why could magnetic flux change? c) Changes on angle between B and S, A.C. Generator B S
Lenz’s law Polarity of induced emf ( ) and intensity of current are always opposites to changes produced on magnetic flux through loop. It’s very important to note that induced current is not opposite to flux, but to changes on flux. Induced current always tries to keep existing flux. Tipler, chapter 28.3
Examples. Applications of induced currents Generating an alternating current d NBS cos N B S wt NBSwsenwt dt B S Eddy currents. Induction heating Data reading on a magnetic disk head https://www.youtube.com/watch?v=RlvnTk3BJKU Tipler, chapter 28.5
Mutual Inductance If we have two different and close loops (or circuits) electric current along one of circuits (I 1 ) creates a magnetic field and a magnetic flux through the second circuit ( 21 ). The rate of 21 to I 1 is the mutual inductance coefficient between circuits 1 and 2 (M 21 ): 21 M M I 21 21 21 1 I due to I 1 1 B 1 I 1 1 2 B 1 Tipler, chapter 28.6
Mutual Inductance In the same way could be defined 12 M 12 I 2 B 2 B 2 I 2 1 2 due to I 2 It can be proved that M 12 =M 21 =M M is mutual inductance between circuits 1 and 2. Inductance unit: Henry (H) 1 Henry=1Wb/1A M Symbol of mutual inductance Tipler, chapter 28.6
Self-Inductance If we have only one loop or circuit (instead two) the rate of magnetic flux to the intensity is the self-inductance of such circuit. The devices having a high self-inductance are the coils, solenoids or inductors. On a coil with a current I, such current produces a magnetic field and a magnetic flux through coil. Rate of flux to I is called self-inductance of coil, L: I L LI I B On circuits it’s drawn with symbol: L Physical device is called “Inductor” Tipler, chapter 28.6
Example: Self-Inductance of a solenoid A solenoid with a current I, produces a magnetic field inside solenoid: N: turns N L B I 0 A: cross section l l Assuming B is uniform inside solenoid, magnetic flux through solenoid will be: 2 NI N IA 0 0 NBA N A l l 2 N A Thus: 0 L Self-inductance of a solenoid I l Tipler, chapter 28.6
Electromotive force on an self-inductance. d L dI According faraday’s law a emf appears: dt dt Polarity of induced e.m.f.: I L L I dI dI 0 L dt + - dt I L L I dI dI 0 L dt - + dt General rule: • dI/dt must have its own sign (>0 or <0 according I) The + terminal is that where current is entering by. • dI L For steady currents (D.C.) ε =0. In this case, the • dt inductor behaves as a short-circuit.
The transformer If current on primary isn’t constant, it produces a flux varying on time. This flux is completely driven by the ferromagnetic material, producing an induced emf on terminals of secondary. The magnetic field created on primary is driven by the ferromagnetic material and it creates a magnetic flux on secondary. It works due to electromagnetic induction and then it doesn’t work on D.C. Primary d Secondary dB 1 V N S N BS 1 1 1 1 1 dt dt V N winding winding 1 1 V N d dB 2 2 2 V N S N BS N 1 N 2 2 2 2 2 2 dt dt Voltage ratio or ratio V 2 ~ V 1 ~ of transformation equals the turns ratio S cross section of solenoids On an ideal transformer the power on primary equals the power on secondary In order to minimize Eddy currents the core is built with sheets isolated between them On an real transformer there are three type of losses: On windings: Joule heating i On core: Magnetic Histeresys and Eddy currents B
RL circuit. Stored energy on an inductor L I t 0 I 0 dI - + L IR 0 When switch is closed (t>0): dI g L dt dt L R ε g The intensity increases from 0 (on t=0) up to the steady t=0 current (dI/dt=0 and therefore ε L =0 on t= ). At this time the inductor behaves as a shortcircuit and I(t= )= g /R. g t I R x I If we multiply the equation of this circuit I times, dI the power consumed by the inductor plus the 2 LI I R I g power lost on resistor by Joule heating equals dt P R the generated power on the battery: P g P L Then, when intensity reachs its value I, the stored energy on inductor is: I dI 1 2 W P dt LI dt LIdI LI L L dt 2 0 1 LI W L 2 Stored energy on an inductor 2 Tipler, chapter 28.7
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