Circuit Elements

Conventional current: Widely known as Ohm’s law Resistance of a long wire: Units: Ohm,  George Ohm ( ) Resistance Resistance combines conductivity and geometry!

Microscopic Macroscopic Can we write V=IR ? Microscopic and Macroscopic View Current flows in response to a  V

L=5 mm A = mm 2 Conductivity of Carbon:  = (A/m 2 )/(V/m) What is its resistance R? (V/A) What would be the current through this resistor if connected to a 1.5 V battery? Exercise: Carbon Resistor

Mobility of electrons: depends on temperature Conductivity and resistance depend on temperature. Conductivity may also depend on the magnitude of current. Constant and Varying Conductivity

Ohmic resistor: resistor made of ohmic material Ohmic materials: materials in which conductivity  is independent of the amount of current flowing through not a function of current Examples of ohmic materials: metal, carbon (at constant T!) Ohmic Resistors

Tungsten: mobility at room temperature is larger than at ‘glowing’ temperature (~3000 K) V-A dependence: 3 V100 mA 1.5 V 80 mA 0.05 V 6 mA R 30  19  8  VV I Is a Light Bulb an Ohmic Resistor?

Metals, mobile electrons: slightest  V produces current. If electrons were bound – we would need to apply some field to free some of them in order for current to flow. Metals do not behave like this! Semiconductors: n depends exponentially on E Conductivity depends exponentially on E Conductivity rises (resistance drops) with rising temperature Semiconductors

Capacitors |  V|=Q/C, function of time Batteries: double current, but |  V|  emf, hardly changes has limited validity! Ohmic when R is indep- pendent of I! Conventional symbols: Nonohmic Circuit Elements Semiconductors

 V batt +  V 1 +  V 2 +  V 3 = 0 emf - R 1 I - R 2 I - R 3 I = 0 emf = R 1 I + R 2 I + R 3 I emf = (R 1 + R 2 + R 3 ) I emf = R equivalent I, where R equivalent = R 1 + R 2 + R 3 Series Resistance

Know R, find  V 1,2 Solution: 1) Find current: 2) Find voltage: 3) Check: Exercise: Voltage Divider R1R1 R2R2 V1V1 V2V2 emf

I = I 1 + I 2 + I 3 Parallel Resistance

R 1 = 30  R 2 = 10  What is the equivalent resistance? What is the total current? Alternative way: Two Light Bulbs in Parallel

What would you expect if one is unscrewed? Two Light Bulbs in Parallel A)The single bulb is brighter B)No difference C)The single bulb is dimmer

Current: charges are moving  work is done Work = change in electric potential energy of charges Power = work per unit time: I Power for any kind of circuit component: Work and Power in a Circuit Units:

emf R Know  V, find P Know I, find P In practice: need to know P to select right size resistor – capable of dissipating thermal energy created by current. Power Dissipated by a Resistor What is the power output of the battery?

Electric field in a capacitor: E s +Q -Q In general: Definition of capacitance: Capacitance Capacitance of a parallel- plate capacitor: Capacitance

Michael Faraday ( ) Units: C/V, Farads (F)

This 1 Farad capacitor is equivalent to a large two-disk capacitor s=1 mm D How large would it be? D ~ 10 km (6 miles) Exercise

Alternative approach: Energy density: Energy: Energy Stored in a Capacitor

Capacitor: Charging and Discharging ChargingDischarging

Positive and negative charges are attracted to each other: how can they leave the plates? Fringe field is not zero! How is Discharging Possible? Electrons in the wire near the negative plate feel a force that moves them away from the negative plate. Electrons near the positive plate are attracted towards it.

Initial moment: brighter? Will it glow longer? Parallel Capacitors Fringe field: Capacitors in parallel effectively increase A

Will it glow at all? How do electrons flow through the bulb? An Isolated Light Bulb Why do we show charges near bulb as - on the left and + on the right?

Ammeter: measures current I Voltmeter: measures voltage difference  V Ohmmeter: measures resistance R Ammeters, Voltmeters and Ohmmeters

0.150 Connecting ammeter: Conventional current must flow into the ‘+’ terminal and emerge from the ‘-’ terminal to result in positive reading. Using an Ammeter

Simple commercial ammeter Ammeter Design Want tiny resistance in coil so current isn’t affected What happens if not connected correctly?

 V AB – add a series resistor to ammeter Measure I and convert to  V AB =IR Connecting Voltmeter: Higher potential must be connected to the ‘+’ socket and lower one to the ‘-’ socket to result in positive reading. Voltmeter Voltmeters measure potential difference

R How would you measure R? A Ohmmeter Ammeter with a small voltage source

Initial situation: Q=0 Q and I are changing in time Quantitative Analysis of an RC Circuit

Current in an RC circuit What is I 0 ? Current in an RC circuit RC Circuit: Current

What about charge Q? Current in an RC circuit RC Circuit: Charge and Voltage

Current in an RC circuit Charge in an RC circuit Voltage in an RC circuit RC Circuit: Summary

Current in an RC circuit When time t = RC, the current I drops by a factor of e. RC is the ‘time constant’ of an RC circuit. The RC Time Constant A rough measurement of how long it takes to reach final equilibrium

What is the value of RC? About 9 seconds

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