NYJC EJC RLC Circuits Notes

Page 1 of 26 9814 (20 26) H3 Physics H305 RLC Circuits - Notes H3 Topic 5 - RLC Circuits Content • Inductance • Dielectrics and ferromagnetic materials • Energy in an inductor • Circuits with capacitors and inductors Learning Outcomes Candidates should be able to: (a) define self-inductance as the ratio of the e.m.f. induced in an electrical circuit / component to the rate of change of current causing it and use ( /)V L d dt= I to solve problems (b) show an understanding that mutual inductance is the tendency of an electrical circuit / component to oppose a change in the current in a nearby electrical circuit / component (c) show a qualitative understanding that dielectric materials enhance capacitance, and that dielectric breakdown can occur when the electric field is sufficiently strong (knowledge of the quantitative modification of electric fields in matter through the permittivity is not required) (d) show a qualitative understanding that ferromagnetic materials enhance inductance and that this enhancement is non- linear especially near saturation (knowledge of the quantitative modification of magnetic fields in matter through the permeability is not required) (e) derive, by considering work done on charges, the expression for potential energy stored in an inductor, U = LI2/2, and use this to solve problems (f) solve problems using the formulae for the combined inductance of two or more inductors in series and in parallel (g) solve problems involving circuits with resistors, inductors, and sources of constant e.m.f. (includes solving first-order differential equations) [RL series circuits with constant e.m.f. source] (h) solve problems involving circuits with inductors and capacitors only (includes solving second- order differential equations) [LC series circuits without e.m.f. source] (i) solve problems involving circuits with resistors, inductors and capacitors only (candidates are not expected to solve the general second- order differential equations, though they can be asked to verify and use particular solutions). [RLC series circuits without e.m.f. source] 5.0 Introduction Capacitors and inductors are electrical components that store energy. Together with resistors, they are the key components in modern electrical devices. Depending on the intended purpose of the circuit, these components are used in a variety of combinations and configurations so as to influence how the electric current in the circuit (and potential difference across components) vary with time.

Page 2 of 26 9814 (20 26) H3 Physics H305 RLC Circuits - Notes 5.1 Effects of Dielectric Materials in a Capacitor In H2 Physics, we learned about capacitor and capacitance. A capacitor is essentially two conductors (usually plates or sheets) separated by a gap (or an insulator). The insulator in the gap between the plates is called the dielectric. It prevents charges from flowing across the plates of the capacitor. While the dielectric can simply be air, dielectric materials such as paper, paper soaked in mineral oil, plastic or mica are commonly used in practical capacitors which help increases capacitance: 1. They physically separate the plates and prevent electrical contact . This allows the plates to be closer to each other. 2. They have a higher dielectric constant than air. This allows more charge to be stored on the plates for the same potential difference between them. 3. They have higher dielectric strength than air. This allows a higher p.d. to be applied across the capacitor all else being constant (before dielectric breakdown, see next Section). When a p.d. is applied across the dielectric, the molecules of the dielectric become polarised by the electric field setup between the plates. The positive ends are induced towards the negative plate and vice versa. This implies that within the dielectric, the electric field is opposite in direction to the electric field originally set up between the plates, causing the resultant electric field to decrease. Polarised dielectric molecules setting up a reverse electric field ER The reduction of the resultant electric field reduces the p.d. across the plates. If the capacitor is still connected to the same battery, more charges flow into the capacitor until the p.d. across the capacitor increases to match the e.m.f. of the battery. + + + + + + + + + - - - - - - - - - +q -q - + - + - + - + - + - + ER Eq polarised molecule

Page 3 of 26 9814 (20 26) H3 Physics H305 RLC Circuits - Notes A capacitor with a vacuum between the conductors A capacitor with a dielectric between the conductors, where Q’ > Q The amount of additional charge that a capacitor with a dielectric can store depends on the latter’s dielectric constant. More details in Annex A. When the externally-applied electric field between the plates is greater than the dielectric strength of the dielectric material, dielectric breakdown occurs. When dielectric breakdown occurs, there will be permanent physical and/ or chemical changes. The capacitor can no longer store charge as it will have a permanent conducting path etched within. Hence, each practical capacitor will have a maximum p.d. that it can be operated under, this limiting the maximum amount of charge that it can store. + + + + + - - - - - +Q -Q εbatt V = εbatt + + + + + + + + + - - - - - - - - - +Q’ -Q’ εbatt V = εbatt A permanent path (in black) etched into the solid dielectric after dielectric breakdown has occurred Dielectric breakdown is the failure of a dielectric to withstand the electric field between the two plates.

Page 4 of 26 9814 (20 26) H3 Physics H305 RLC Circuits - Notes 5.2 Inductors An inductor is a conductor that is wound up into a large coil. The most common form of inductors come in the form of solenoids and toroids. An assortment of inductors A Solenoid A Toroid Like a capacitor, inductors also store energy and are used in circuits to control the time-variation of current through a circuit or the potential difference across other components The circuit symbol of a inductor is . 5.2.1 Inductance of an Inductor Mutual Inductance We build on our knowledge from Electromagnetic Induction: mutual induction is the phenomenon in which an e.m.f. is induced in a n inductor when the current in a neighbouring inductor changes. Consider 2 coils placed next to each other with their axis aligned. When a current I1 flows through coil 1, it produces a magnetic field that links with coil 2. As I1 changes, the magnetic flux linkage through coil 2 changes and in accordance to Faraday’s Law, an e.m.f. ε2 is induced in coil 2. This induced e.m.f. is proportional to the rate of change of current I1. 1 2 1 2 ∝ = d dt dM dt ε ε I I where M is the proportionality constant called the mutual inductance between the 2 inductors. coil 1 coil 2 B-Field due to current in coil 1 I1 ε2 (induced) Mutual Inductance between two inductors is the tendency of one inductor to oppose a change in the current in the other circuit / component.

Page 5 of 26 9814 (20 26) H3 Physics H305 RLC Circuits - Notes Self-Inductance Self-induction is when an e.m.f. is induced in an inductor when the current flowing in it (i.e. the same inductor) changes. Consider a circuit consisting of a switch S, a resistor R and an e.m.f. source ε. At the instant switch S is closed, the current in the circuit increases. This increase in current produces a changing magnetic flux that links the circuit, which in turn leads to a self- induced e.m.f. in the circuit. By Lenz’s Law, the direction of this self -induced e.m.f. will establish a magnetic field that opposes the original change in magnetic flux linkage through it , i.e