Capacitors, Multimeters, Circuit Analysis

Capacitor: Charging and Discharging ChargingDischarging

The capacitor in your set is similar to a large two-disk capacitor s D Capacitor: Construction and Symbols There is no connecting path through a capacitor

Capacitor: Discharge Electron current

Capacitor Discharge The fringe field of the capacitor plus the electric field of the charges on the surface of the wires drive current in a way to REDUCE the charge of the capacitor plates. Recall that we derived an expression for the fringe field by considering the superposition of two charged disks separated by a distance s.

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.

Fringe field of a capacitor rises until E=0 in a wire – static equilibrium. Fringe field opposes the flow of current! Capacitor: Charging

Ultimately, the fringe field of the capacitor and the field due to charges on the wire are such that E=0 inside the wire. At this point, i=0. Why does current ultimately stop flowing in the circuit?

Thick filamentThin filament Which light bulb will glow longer? Why? 1)Round is brighter  capacitor gets charged more? 2)Long bulb glows longer  capacitor gets charged more? The Effect of Different Light Bulbs

Use two different capacitors in the same circuit In the first moment, which capacitor will cause the bulb to produce more light? Which capacitor will make the light bulb glow longer? Fringe field: Effect of the Capacitor Disk Size

In the first moment, which capacitor will cause the bulb to produce more light? Fringe field: Which capacitor will make the light bulb glow longer? Effect of the Capacitor Disk Separation

In the first moment, which capacitor will cause the bulb to produce more light? Fringe field: Which capacitor will make the light bulb glow longer? Insulator Effect of Insulator in Capacitor

The capacitors shown are initially uncharged. When connected to identical circuits, after 0.01 s of charging: A)The fringe field of each capacitor is the same. B)The fringe field of the smaller capacitor is greater. C)The fringe field of the larger capacitor is greater. R1R1 s s R2R2

Consider two capacitors whose only difference is that capacitor number 1 has nothing between the plates, while capacitor number 2 has a layer of plastic in the gap. They are placed in two different circuits having similar batteries and bulbs in series with the Capacitor. In the first fraction of a second - A)The current decreases less rapidly in the circuit containing capacitor 1. B)The current decreases less rapidly in the circuit containing capacitor 2. C)The current is the same in both circuits.

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?

Reversing a Capacitor

Connecting Capacitor Parallel to a Battery

Circuits with Capacitors Isolated light bulb Two parallel capacitors Charging Discharging Experiments:

Charging time I Bulb Brightness time E cap Energy conservation Capacitor in a Circuit Do 19.X.7!

I 1 = I 2 + I 3 Charge conservation: I i > 0 for incoming I i < 0 for outgoing Capacitor transients: not a steady state! Cannot use Kirchhoff rule for a part of a capacitor (area 1 or 2) But can use for capacitor as a whole (area 3) The Current Node Rule in a Capacitor Circuit …in steady state

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)

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

A capacitor is formed by two rectangular plates 50 cm by 30 cm, and the gap between the plates is 0.25 mm. What is its capacitance? Exercise

s D No insulator:With insulator: A Capacitor With an Insulator Between the Plates

How much charge accumulates on one plate after charging by a circuit with two 1.5 V batteries? Q = 3 C How many electrons are there? N=(3 C)/( C)  Exercise

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 ammeter using your lab kit: Simple commercial ammeter Digital ammeters: uses semiconductor elements. ADC – analog-to-digital converter (Combination of comparator and DAC) Ammeter Design

Ammeter is inserted in series into a circuit – measured current flows through it. Remf A Process of measuring requires charges to do some work: Internal resistance r int A No ammeter: With ammeter: Internal resistance of an ammeter must be very small Ammeter Design: r int

Remf r int A R shunt IaIa I shunt I Using a shunt resistor one can reduce sensitivity of an ammeter Ammeter: Making it Less Sensitive

Is it correct connection? Exercise: Connecting Ammeter

 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

R1R1 R2R2 emf A B  V AB in absence of a voltmeter A r int  V AB in presence of a voltmeter Internal resistance of a voltmeter must be very large Voltmeter: Internal Resistance

A voltmeter has r int = 10 M  and is rated for a maximum voltage of 100 V. What is its ammeter part maximum current?  V=IR How much power is lost in this voltmeter? P =  VI = (100 V)(10  A) = 1 mW Exercise: Current Through a Voltmeter

0.100 A????? V Exercise: Multimeter

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

Assuming that a light bulb resistance is ~10  and independent of current, calculate time constant of this circuit. 1 F 10  RC=(10  )(1 F)=10 s How much power will dissipate in the light bulb after t=RC compared to initial power? if I reduces e times, then P reduces e 2 times Exercise: Time Constant

Question

A B C D

What is the final charge on a 1 F capacitor connected to a 1.5V battery through resistor 100  ? Can you apply the RC equations to the circuit below? Current in an RC circuit No! Resistance depends on current. Exercise

Find currents through resistors Loop 1Loop 2Loop 3 Loop 4 I1I1 I2I2 I3I3 I5I5 I4I4 loop 1: loop 2: loop 3: nodes: Five independent equations and five unknowns Exercise: A Complicated Resistive Circuit