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Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Class 7Capacitor Problems and Resistance
Physics 106
Winter 2019
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Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Learning Outcomes
Last time we discussed:I CapacitanceI CapacitorsI Capacitors in series and parallel circuitsI Dielectrics
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Learning Outcomes
Today we will learn aboutI Capacitance ProblemsI Parallel-Plate CapacitorsI Capacitors in CircuitsI CurrentI ResistanceI Ohm’s Law
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Problems with Capacitorsin Circuits
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Problem-Solving Strategy
I Combine capacitors in seriesI Combine capacitors in parallelI Continue until one capacitor remainsI After finding Q and V , work back one step at a time
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Problem-Solving Strategy
I In series: the reciprocals of capacitance add, chargeis the same, voltages add
I In parallel: capacitance adds directly, voltage is thesame, charges add
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
Combine the twocapacitors in parallel
C24 = C2 + C4 = 2 + 4
= 6µF
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µ F capacitor.
Combine the twocapacitors in parallel
Ceq = C2 + C4 = 2 + 4
= 6µF
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.Combine the twocapacitors in series
1Ceq
=1
C3+
1C6
=13
+16
=12
Ceq = 2µF
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.Combine the twocapacitors in series
1Ceq
=1
C3+
1C6
=13
+16
=12
= 2µF
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
The voltage on thiscapacitor is 12 V and thecharge is Q = CV = 24µC.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
We made this capacitorout of two capacitors inseries.
Each original capacitorhas the same charge asthe equivalent capacitor.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
We made this capacitorout of two capacitors inseries.
Each original capacitorhas the same charge asthe equivalent capacitor.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
Now use V = Q/C to findthe voltages.
The voltages add to equalthe voltage of theequivalent capacitor.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
Now use V = Q/C to findthe voltages.
The voltages add to equalthe voltage of theequivalent capacitor.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
The right capacitor camefrom two capacitors inparallel.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
The right capacitor camefrom two capacitors inparallel.
The voltages are thesame and the chargesadd.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Sample Problem
I Find the charge on the 4 µF capacitor.
We find the charges fromQ = CV
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Current
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electric Current
I Electric current is the flow of chargesI We will usually consider the flow of electrons in
electrical circuits, but treat the current as if positivecharge carriers moved.
I Current is the rate at which charge past a point in awire
I =∆Q∆t
I The SI unit of current is Ampere (A)1 A = 1 C/ 1 s
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electric Current
I The direction of the current is the direction positivecharge would flow
I In actuality, electrons flow opposite the currentdirection
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Current and Drift Speed
I Chargedparticles movethrough aconductor ofcross-sectionalarea A
I n is the numberof chargecarriers perunit volume
I The number ofcharge carriersin the grayregion isnV = nA∆x
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Current and Drift Speed
I The total charge is the number of charge carrierstimes the charge per carrier, e:Q = (nA∆x)e
I The drift speed, vd , is the speed at which the carriersmove:vd = ∆x/∆t
I So: Q = (nAvd ∆t)eI The current is I = ∆Q/∆t = nevdA
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I With no voltage on a wire, free electrons moverapidly, but in random directions.
I The drift speed is much smaller than the averagespeed between collisions
I When a circuit is completed, the electric field travelswith a speed of about 2/3 the speed of light, causingelectrons throughout the wire to move almostinstananeously.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I In a wire, excess charge moves to the outside of thewire.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I In a wire, excess charge moves to the outside of thewire.
I Near the positive terminal of a battery, there is asmall deficit of electrons on the wire’s surface.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I In a wire, excess charge moves to the outside of thewire.
I Near the positive terminal of a battery, there is asmall deficit of electrons on the wire’s surface.
I Near the negative terminal, there is a small excess.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I This creates an electric field within the wire.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I This creates an electric field within the wire.I The electric field pushes electrons throughout the
volume of the wire toward the more positive direction.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Electrons in a Circuit
I This creates an electric field within the wire.I The electric field pushes electrons throughout the
volume of the wire toward the more positive direction.I This creates a current from higher to lower voltage.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Resistance
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Resistance in a Circuit
I The faster electrons flow in the wire, the greater isthe current.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Resistance in a Circuit
I It takes energy from the battery to push electronsonto the negative end of the wire and to pullelectrons from the positive end of the wire.
I Electrons in the wire lose this energy in colliding withatoms within the wire.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Resistance in a Circuit
I The energy loss in the wire sets an upper limit to thedrift speed of the electrons.
I In turn, the energy loss limits the current that flows inthe wire.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Resistance in a Circuit
I If the voltage across the wire is greater, the electronsmove faster.
I The more the wire opposes the flow of current, thegreater is the resistance of the wire.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Resistance
I Resistance is defined as the ratio of the voltageacross a conductor to the current flowing through it.
R ≡ VI
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Units of Resistance
I Units of resistance are ohms (Ω)I 1 Ω = 1 V / 1 A
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Ohm’s Law
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Georg Simon Ohm
I 1787 -1854I Formulated the
concept ofresistance
I Discovered theproportionalitybetweencurrent andvoltages
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Ohm’s Law
I Often, resistance remains constant over a widerange of applied voltages or currents
I This statement has become known as Ohm’s Law:
V = IR
I If resistance is constant in a material, it is said tobe ohmic.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Ohm’s Law
I For an ohmicmaterial, agraph of I vs.V is a straightline.
I The slope ofthe line is 1/R.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Ohm’s Law
I Non-ohmicmaterials arethose whoseresistancechanges withvoltage
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
I Many resistors are made of chunks of graphite mixedin a ceramic material
I From our understanding of how charges behave, let’spredict what happens when we connect blocks ofmaterial together.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Let’s connect two blocks end to end.I If the resistance of one block is 6 Ω, what is the
resistance of both blocks together?I If the current through one block is 2 A, what is the
current through the second block? through bothblocks?
I If the voltage across one block is 3 V, what is thevoltage across the second block? across both blockstogether?
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and ResistanceLet’s connect two blocks side by side.I If the resistance of one block is 6 Ω, what is the
resistance of both blocks together?I If the current through one block is 2 A, what is the
current through the second block? through bothblocks?
I If the voltage across one block is 3 V, what is thevoltage across the second block? across both blockstogether?
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.
I The current is the same through each resistor.I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.
I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks end to end is like adding resistors inseries.I The current is the same through each resistor.I The voltages across the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks side by side is like adding resistors inparallel.
I The voltage is the same across each resistor.I The curents through the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Blocks and Resistance
Combining blocks side by side is like adding resistors inparallel.I The voltage is the same across each resistor.I The curents through the resistors add.
Class 7
Physics 106
LearningOutcomes
CapacitorProblems
Current
Resistance
Ohm’s Law
Blocks andResistance
Light Bulbs
I Light bulbs act as resistors in a circuit.I A light bulb is brighter when more current flows
through it.
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