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Prozorov_21

# Prozorov_21 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 21 LC Circuit LRC Circuit Energy in E and B fields Energy stored in a capacitor 1 2 U CV 2 ++++++ E −−−−−− energy of the electric field Energy stored in a inductor U 1 LI 2 2 B energy of the magnetic field PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 2 RC circuits (review) Capacitor is initially charged with Q0. What happens when switch is closed? ++ C − − Current starts at R I 0 VC Q0 R RC Q0 t e And then decays: I t RC I Q Q0 RC U Q0 RC 2 Q0 2C t Current through R t Charge in capacitor PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University t Energy in capacitor 2011 12 October 3 The ideal LC circuit PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 4 oscillations Capacitor is initially charged with Q0. What happens when switch is closed? C ++ − − • Right after it is closed: I = 0. Then current builds up. L • Magnetic field is created in inductor while charge in capacitor is decreasing • Energy is being transferred form C to L (from electric to magnetic energy) As C discharges, current should decrease and stop, but inductor keeps it going, so at the end the capacitor is charged again but with the positive charge on the bottom plate. Then we start all over again… Oscillatory behavior! Note: No dissipation mechanism (no R) total energy is constant PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 5 LC circuit: math Kirchhoff: VC VL 0 Q dI L 0 C dt Q d 2Q L 2 0 C dt d 2Q 1 Q0 2 dt LC d 2Q 2Q 0 dt 2 with C 1 LC PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University L Solution: Q Q0 cos t 12 October 2011 6 mechanical analogue potential energy is transformed into kinetic and back d 2x F ma m 2 kx (Hook’s law) dt d 2x 2 x 0, k m dt 2 The same equation – harmonic oscillations with angular frequency x x t A cos t PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 7 PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 8 x x Q Q0 cos t VC I Q0 sin t Q Q0 cos t C C VL L 2 Q 2 Q0 UC cos2 t 2C 2C r1 0 , .. r1 1 2 VC LQ02 2 VL 1 2 UL LI 2 LQ022 sin2 t n 1.01 1 dI LQ0 2 cos t dt 2 LQ02 1 Q2 0 2 LC 2C Q02 sin2 t 2C 0 f( x ) 0 0, r1 .. r1 n 1.01 1 1 0 2 4 0 x 6 UL 6.28 Energy is continuously transformed: electric magnetic f( x ) 0.5 UC 00 0 2 4 6 PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University x t 12 October 2011 9 real inductor (with resistance) Then, our LC circuit becomes an RLC circuit… R C L PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 10 the LRC circuit PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 11 RLC circuit - damping R We’ve seen this already: Simple Harmonic Motion + Damping L C x 0, r1 .. r1 x R=0 n 0, r1 r1 .. r1 100 Q1 f( x ) 0 0 n R >0 n 1 10 f( x ) 0 0 Q 1 0 t5 1 10 0 x PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University t5 10 x 12 October 2011 12 damping in LRC circuit R VC VR VL 0 Q dI RI L 0 C dt I dI d 2I R L 2 0 C dt dt d 2 I R dI 1 I 0 2 dt L dt LC d 2I dI 2 2 0 I 0 dt 2 dt R 2L C - attenuation R 0 2 L LC - damping factor PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University L 1 1 1 12 October 2011 13 RLC circuit: math d 2I dI 2 2 0 I 0 dt 2 dt I I 0 e t cos(t ) - in under-dumped regime: 2 0 2 1 R2 2 0 1 2 LC 4 L obviously it only makes sense for 02 2 0 1 0 PHYS222 - Lecture 21 - Prof. Ruslan Prozorov - Iowa State University 12 October 2011 14 ...
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