April 1. Physics 272. Spring Prof. Philip von Doetinchem

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1 Physics 272 April 1 Spring Prof. Philip von Doetinchem Phys272 - Spring 14 - von Doetinchem - 164

2 Summary Gauss's law for electric fields (surface integral) Electric field is related to total charge in an enclosed surface Electric charges are sources of magnetic fields Gauss's law for magnetism (surface integral) No magnetic monopoles exist magnetic flux through closed surface is always zero Phys272 - Spring 14 - von Doetinchem - 165

3 Summary Ampere's law (line integral) Conducting and displacement current act as sources of magnetic fields Faraday's law (line integral) A changing magnetic field or magnetic flux induces an electric field Phys272 - Spring 14 - von Doetinchem - 166

4 Inductance Coiled pieces of wire behave differently in a circuit than just straight wire A changing current in a coil induces an emf in an adjacent coil A changing current also induces a n emf in that same coil If a coil carries a current energy is released when the current decreases automotive ignition system this energy was stored in the magnetic field Phys272 - Spring 14 - von Doetinchem - 167

5 Mutual Inductance We already studied: Magnetic interaction between two wires carrying steady currents current in one wire causes a magnetic field exert force on second wire Now: changing currents If current in coil 1 changes magnetic flux in coil 2 changes according to Faraday's law: emf is induced Phys272 - Spring 14 - von Doetinchem - 168

6 Mutual Inductance Flux through coil 2 changes: Change in flux is proportional to current in coil 1 mutual inductance Phys272 - Spring 14 - von Doetinchem - 169

7 Mutual Inductance Flux through coil 2 is directly proportional to current in coil 1 if both coils are in vacuum Mutual inductance only depends on geometry of coils and the orientation of the coils to each other Mutual inductance is also the same for the case when coil 2 carries the current and induces a current in coil 1: Careful: only time varying currents induce emf in second coil Induced emf is directly proportional to rate of change of the current Phys272 - Spring 14 - von Doetinchem - 170

8 Mutual Inductance Unit for mutual inductance Typical values: millihenry (mh) or microhenry ( H) Source: Joseph Henry ( ) Circuit design requires to suppress unwante mutual induction between nearby circuits e.g., place coils far apart from each other Can also be very useful: transformers Phys272 - Spring 14 - von Doetinchem - 171

9 Calculating mutual inductance Current in either coil causes a flux through the other coil No field outside of solenoid Coil 1 is powered Phys272 - Spring 14 - von Doetinchem - 172

10 Calculating mutual inductance Phys272 - Spring 14 - von Doetinchem - 173

11 Emf due to mutual inductance Substantial induced emf due to rapid change Emf is constant for this case because the current change happens at a constant rate Phys272 - Spring 14 - von Doetinchem - 174

12 Self-inductance and inductors So far: two circuits: Current in circuit 1 changes flux through circuit 2 changes emf is induced in circuit 2 Related effect occurs also in single isolated circuit Current in circuit sets up magnetic flux through the same circuit This flux changes when the current in the circuit changes Any circuit carrying a varying current has an emf induced by the variation of its own magnetic field self-induced emf Phys272 - Spring 14 - von Doetinchem - 175

13 Self-inductance and inductors Effect occurs in any circuit, but is enhanced for coils with many turns Self-inductance L is defined as: current changes in circuit magnetic flux changes self-induced emf after applying Faraday's law Phys272 - Spring 14 - von Doetinchem - 176

14 Inductors as circuit elements Circuit devices with a specific inductance are called inductors: Important component of modern electronics Purpose: oppose variations in the current through the circuit DC: maintain a steady current, despite fluctuations in applied emf AC: suppress variations that are too fast Phys272 - Spring 14 - von Doetinchem - 177

15 Inductors as circuit elements We have to modify Kirchhoff's rule when using inductors The magnetically induced electric field in the coils of the inductor is not conservative Use Faraday's law: In case of an open inductor i(t)=0, potential difference changes with current flow Potential difference across an inductor depends on the rate of change of the current Phys272 - Spring 14 - von Doetinchem - 178

16 Inductors as circuit elements Phys272 - Spring 14 - von Doetinchem - 179

17 Applications of inductors Inductors are useful to keep currents stable: When currents grow too high induced emf reduces current When currents go too low induced emf sustains current, prevents shut offs Self-inductance depends on size, shape, turns magnetic properties of the enclosed material ( 0 ) (ferromagnetic material makes a big difference) Phys272 - Spring 14 - von Doetinchem - 180

18 Applications of inductors Example: traffic light sensor automobiles contain steel driving a car over a current-carrying coil embedded in the street circuitry detects inductance change green light will be triggered Source: Phys272 - Spring 14 - von Doetinchem - 181

19 Calculating self-inductance What is the self-inductance of a toroidal solenoid? Cross section A, assume B uniform, N turns Phys272 - Spring 14 - von Doetinchem - 182

20 Calculating self-induced emf Current increases from 0 to 6A in 3 s Current is increasing magnetic field is increasing flux is increasing self-induced emf opposes the incoming current Phys272 - Spring 14 - von Doetinchem - 183

21 Magnetic-field energy Inductor carrying a current has energy stored in it This requires to input energy Energy input needed to establish final current in an inductor (zero resistance no energy dissipation) Phys272 - Spring 14 - von Doetinchem - 186

22 Magnetic-field energy When current decreases inductor as a source supplying 1/2LI2 to the external circuit Example: very sudden decrease in current by unplugging device from wall socket induced emf is large ionizes the air and creates arc Resistor vs. inductor: Energy is dissipated in resistor for any type of current, steady or varying in time Ideal inductor stores energy if current is increasing and releases energy when current is decreasing (no dissipation), steady current no change in energy Phys272 - Spring 14 - von Doetinchem - 187

23 Magnetic energy density Energy is stored in magnetic field in inductor like the energy is stored in the electric field of a capacitor Ideal toroidal solenoid (assume uniform magnetic field) Volume of toroidal solenoid: Energy density: Phys272 - Spring 14 - von Doetinchem - 188

24 Magnetic energy density Energy is stored in magnetic field in inductor like the energy is stored in the electric field of a capacitor Energy density in terms of magnetic field: This is also the correct expression for any magneticfield configuration in a material with constant permeability Phys272 - Spring 14 - von Doetinchem - 189

25 Magnetic energy density Example: ignition system in gasoline-powered cars Primary coil produces a strong magnetic field with N=250 Secondary coil surrounds primary coil with N=25,000 Interrupt current in primary coil magnetic field drops very high emf is induced in the second coil and causes spark plug to fire energy stored in magnetic field causes a short powerful pulse Phys272 - Spring 14 - von Doetinchem - 190

26 Coronal mass ejection Source: NASA Goddard Space Flight Center https://www.youtube.com/watch?v=ytqykrcemrs Magnetic field of the sun is constantly changing Release of stored magnetic energy can cause coronal mass ejection of billion tons of material Phys272 - Spring 14 - von Doetinchem - 191

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