Physics 2102 Jonathan Dowling Lecture 28 MON 23 MAR Ch 30 10 11 Inductors Inductance QuickTime and a decompressor are needed to see this picture Nikolai Tesla EXAM 03 6PM THU 02 APR 2009 The exam will cover Ch 28 second half through Ch 32 1 3 displacement current and Maxwell s equations The exam will be based on Final Day to Drop Course FRI 27 HW08 HW11 Inductance Consider a solenoid of length l that has N loops of N area A each and n windings per unit length A current l i flows through the solenoid and generates a uniform r B l magnetic field B 0 ni inside the solenoid The solenoid magnetic flux is B NBA 2 2 L 0n l A 0 N l l A 0N 2 l A The total number of turns N nl B 0 n 2 l A i The result we got for the special case of the solenoid is true for any inductor B Li Here L is a constant known as the inductance of the solenoid The inductance depends on the geometry of the particular inductor Inductanceof theSolenoid B 0n2l Ai For the solenoid L 0n2l A i i RL Circuit Fluxing up an Inductor How does the current in the circuit change with time i t di iR E L 0 dt E t t i t 1 e R i t Fast Small E R ime constant of RL circuit L R Slow Large t RL Circuit Fluxing UP QuickTime and a Animation decompressor are needed to see this picture L R i 0 0 E i 0 L VL 0 E E i R i 0 VL 0 E E i t 1 e t i t e t R L VL t Li t Ee t Fluxing Down an Inductor The switch is at a for a long time until the inductor is charged Then the switch is closed to b i What is the current in the circuit di LoopiR rule L around 0 the new i t circuit dt E R E Rt L E t i t e e R R Exponential defluxing t Inductors Energy Recall that capacitors store energy in an electric field Inductors store energy in a magnetic field di E iR L dt di 2 iE i R Li dt i P iV i 2 R 2 d Li 2 iE i R dt 2 Power delivered by power dissipated by R battery d dt energy stored in L Inductors Energy Li 2 UB 2 Magnetic Potential Energy UB Stored in an Inductor di P Li dt Magnetic Power Returned from Fluxing Down Inductor to Circuit or Put into Fluxing Up Inductor from Circuit Example The switch has been in position a for a long time It is now moved to position b without breaking the circuit What is the total energy dissipated by the resistor until the circuit reaches equilibrium 10 9 V 10 H When switch has been in position a for long time current through inductor 9V 10 0 9A Energy stored in inductor 0 5 10H 0 9A 2 4 05 J When inductor defluxes through the resistor Energy Density of aMagnetic Field Consider the solenoid of length l and loop area A that has n windings per unit length The solenoid r B l B2 uB 2 0 carries a current i that generates a uniform magnetic field B 0 ni inside the solenoid The magnetic field outside the solenoid is approximately zero 1 2 0 n 2 Al i 2 The energy U B stored by the inductor is equal to Li 2 2 This energy is stored in the empty space where the magnetic field is present U We define as energy density uB B where V is the volume inside V 0n2 Al i 2 0n2i 2 02n2i 2 B2 the solenoid The density uB 2Al 2 2 0 2 0 This result even though it was derived for the special case of a uniform magnetic field holds true in general 3 Units u B J m Energy Density in E and B Fields 0E uE 2 2 2 B uB 2 0 The Energy Density of the Earth s Magnetic Field Protects us from the Solar Wind
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