Physics 2102 Gabriela Gonz lez Electric charge Electric force on other electric charges Electric field and electric potential Moving electric charges current Electronic circuit components batteries resistors capacitors Electric currents Magnetic field Magnetic force on moving charges Time varying magnetic field Electric Field More circuit components inductors All together Maxwell s equations Electromagnetic waves Matter waves Magnetic Flux B n A time varying magnetic flux creates an electric field which induces an EMF and a current if the edge of the surface is a conductor EMF C d B E ds dt Lenz s Law dA Notice that the electric field has closed field lines and is not pointing towards lower electric potential this is only true for fields produced by electric charges EMF C d B E ds dt Q A long solenoid has a circular cross section of radius R carrying a current i clockwise What s the direction and magnitude of the magnetic field produced inside the solenoid Ans B 0in out of the page The current through the solenoid is increasing at a steady rate di dt What will be the direction of the electric field lines produced Compute the variation of the electric field as a function of the distance r from the axis of the solenoid Ans from symmetry we know the magnitude of E depends only on r First let s look at r R C R Next let s look at r R magnetic field lines d B E ds dt d d B r 2 r 2 0in dt dt n di E r 0 r 2 dt E 2 r 0 n di R 2 E r 2 dt r electric field lines Inductors are with respect to the magnetic field what capacitors are with respect to the electric field They pack a lot of field in a small region Also the higher the current the higher the magnetic field they produce Capacitance how much potential for a given charge Q CV Inductance how much magnetic flux for a given current Li Using Faraday s law Joseph Henry 1799 1878 Solenoid of cross sectional area A length l total number of turns N turns per unit length n Field inside solenoid 0 n i Field outside 0 L inductance i Set up a single loop series circuit with a battery a resistor a solenoid and a switch Describe what happens when the switch is closed Key processes to understand What happens JUST AFTER the switch is closed What happens a LONG TIME after switch has been closed Key insights If a circuit is not broken one What happens in between cannot change the CURRENT in an inductor instantaneously If you wait long enough the current in an RL circuit stops changing Loop rule i i t Small L R E R Time constant of RL circuit L R Large L R The switch is in 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 Loop rule around the new circuit i t E R Exponential discharge In an RC circuit while charging In an RL circuit while charging Q CV and the loop rule mean emf Ldi dt and the loop rule mean charge increases from 0 to CE magnetic field increases from 0 to B current decreases from E R to 0 current increases from 0 to E R voltage across capacitor voltage across inductor increases from 0 to E decreases from E to 0 Recall that capacitors store energy in an electric field Inductors store energy in a magnetic field Power delivered by battery power dissipated by R energy stored in L i The current in a 10 H inductor is decreasing at a steady rate of 5 A s If the current is as shown at some instant in time what is the induced EMF a 50 V b 50 V i Current is decreasing Induced emf must be in a direction that OPPOSES this change So induced emf must be in same direction as current Magnitude 10 H 5 A s 50 V Immediately after the switch is closed what is the potential difference across the inductor a 0 V b 9 V c 0 9 V 10 9V 10 H Immediately after the switch current in circuit 0 So potential difference across the resistor 0 So the potential difference across the inductor E 9 V Immediately after the switch is closed what is 3V the current i through the 10 resistor a 0 375 A Immediately after switch is closed current b 0 3 A through inductor 0 c 0 Hence i 3 V 10 0 3 A 40 10 Long after the switch has been closed what is the current in the 40 resistor Long after switch is closed potential across inductor 0 a 0 375 A Hence current through 40 b 0 3 A resistor 3 V 40 0 075 A c 0 075 A 10 H 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 9V 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 discharges through the resistor all this stored energy is dissipated as heat 4 05 J Oscillators are very useful in practical applications for instance to keep time or to focus energy in a system All oscillators operate along the same principle they are systems that can store energy in more than one way and exchange it back and forth between the different storage possibilities For instance in pendulums and swings one exchanges energy between kinetic and potential form Newton s law F ma Capacitor initially charged Initially current is zero energy is all stored in the capacitor A current gets going energy gets split between the capacitor and the inductor Capacitor discharges completely yet current keeps going Energy is all in the inductor The magnetic field on the coil starts to collapse which will start to recharge the capacitor Finally we reach the same state we started with with opposite polarity and the cycle restarts the loop rule Compare with Analogy between electrical and mechanical oscillations The energy is constant and equal to what we started with Ideal LC circuit without resistance oscillations go on for ever LC 1 2 Real circuit has resistance dissipates energy oscillations die out or are damped Math is complicated Important points Frequency of oscillator shifts away from LC 1 2 Peak CHARGE decays with time constant 2L R For small damping peak ENERGY decays with time constant L R C R L i t In an RL circuit we can charge the inductor with a battery until there is a constant current or discharge the inductor through the resistor Time constant is L R An LC combination produces an electrical oscillator natural frequency of oscillator is 1 LC Total energy in circuit …
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