2/13/07184 lecture 201 phy 184 spring 2007 lecture 20 title:

2/13/07 184 Lecture 20 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 20

Title:

2/13/07 184 Lecture 20 2

AnnouncementsAnnouncementsAnnouncementsAnnouncements

We hope to open the correction set tonight You will have a week to complete the problems

• You can re-do all the problems from the exam• You will receive 30% credit for the problems you missed

• To get credit, you must do all the problems in Corrections Set 1, not just the ones you missed

Homework Set 5 is due on Tuesday, February 20 at 8 am

2/13/07 184 Lecture 20 3

Kirchhoff’s Law, Multi-loop CircuitsKirchhoff’s Law, Multi-loop CircuitsKirchhoff’s Law, Multi-loop CircuitsKirchhoff’s Law, Multi-loop Circuits

One can create multi-loop circuits that cannot be resolved into simple circuits containing parallel or series resistors.

To handle these types of circuits, we must apply Kirchhoff’s Rules.

Kirchhoff’s Rules can be stated as• Kirchhoff’s Junction Rule

• The sum of the currents entering a junction must equal the sum of the currents leaving a junction

• Kirchhoff’s Loop Rule• The sum of voltage drops around a complete circuit loop

must sum to zero.

2/13/07 184 Lecture 20 4

Circuit Analysis ConventionsCircuit Analysis ConventionsCircuit Analysis ConventionsCircuit Analysis ConventionsElement Analysis Direction Current Direction Voltage Drop

iR

iR

iR

iR

Vemf

Vemf

Vemf

Vemf

i is the magnitude of the assumed current

2/13/07 184 Lecture 20 5

Multi-Loop CircuitsMulti-Loop CircuitsMulti-Loop CircuitsMulti-Loop Circuits

To analyze multi-loop circuits, we must apply both the Loop Rule and the Junction Rule.

To analyze a multi-loop circuit, identify complete loops and junction points in the circuit and apply Kirchhoff’s Rules to these parts of the circuit separately.

At each junction in a multi-loop circuit, the current flowing into the junction must equal the current flowing out of the circuit.

2/13/07 184 Lecture 20 6

Multi-Loop Circuits (2)Multi-Loop Circuits (2)Multi-Loop Circuits (2)Multi-Loop Circuits (2)

Assume we have a junction point a

We define a current i1 entering junction a and two currents i2 and i3 leaving junction a

Kirchhoff’s Junction Rule tells us that

i1 i2 i3

2/13/07 184 Lecture 20 7

Multi-Loop Circuits (3)Multi-Loop Circuits (3)Multi-Loop Circuits (3)Multi-Loop Circuits (3)

By analyzing the single loops in a multi-loop circuit with Kirchhoff’s Loop Rule and the junctions with Kirchhoff’s Junction Rule, we can obtain a system of coupled equations in several unknown variables.

These coupled equations can be solved in several ways• Solution with matrices and determinants• Direct substitution

Next: Example of a multi-loop circuit solved with Kirchhoff’s Rules

2/13/07 184 Lecture 20 8

The circuit here has three resistors, R1, R2, and R3 and two sources of emf, Vemf,1 and Vemf,2

This circuit cannot be resolved into simple series or parallel structures

To analyze this circuit, we need to assign currents flowing through the resistors.

We can choose the directions of these currents arbitrarily.

Example - Kirchhoff’s RulesExample - Kirchhoff’s RulesExample - Kirchhoff’s RulesExample - Kirchhoff’s Rules

2/13/07 184 Lecture 20 9

At junction b the incoming current must equal the outgoing current

At junction a we again equate the incoming current and the outgoing current

But this equation gives us thesame information as theprevious equation!

We need more informationto determine the three currents – 2 more independent equations

Example - Kirchhoff’s Laws (2)Example - Kirchhoff’s Laws (2)Example - Kirchhoff’s Laws (2)Example - Kirchhoff’s Laws (2)

i2 i1 i3

i1 i3 i2

2/13/07 184 Lecture 20 10

To get the other equations we must apply Kirchhoff’s Loop Rule.

This circuit has three loops.• Left

•R1, R2, Vemf,1

• Right

•R2, R3, Vemf,2

• Outer

•R1, R3, Vemf,1, Vemf,2


2/13/07 184 Lecture 20 11

Going around the left loop counterclockwise starting at point b we get

Going around the right loop clockwise starting at point b we get

Going around the outer loopclockwise startingat point b we get

But this equation gives us no newinformation!


i1R1 Vemf ,1 i2R2 0 i1R1 Vemf ,1 i2R2 0

i3R3 Vemf ,2 i2R2 0 i3R3 Vemf ,2 i2R2 0

i3R3 Vemf ,2 Vemf ,1 i1R1 0

2/13/07 184 Lecture 20 12

We now have three equations

And we have three unknowns i1, i2, and i3 We can solve these three equations in a variety of ways


i1 i3 i2 i1R1 Vemf ,1 i2R2 0 i3R3 Vemf ,2 i2R2 0

i1 (R2 R3)Vemf ,1 R2Vemf ,2R1R2 R1R3 R2R3

i2 R3Vemf ,1 R1Vemf ,2R1R2 R1R3 R2R3

i3 R2Vemf ,1 (R1 R2 )Vemf ,2R1R2 R1R3 R2R3

2/13/07 184 Lecture 20 13

Given is the multi-loop circuit on the right. Which of the following statements cannot be true:

A)

B)

C)

D)

Clicker QuestionClicker QuestionClicker QuestionClicker Question

2/13/07 184 Lecture 20 14

Given is the multi-loop circuit on the right. Which of the following statements cannot be true:

A)

B)

C)

D)


Junction rule

Not a loop!

Upper right loop

Left loop

2/13/07 184 Lecture 20 15

Ammeter and VoltmetersAmmeter and VoltmetersAmmeter and VoltmetersAmmeter and Voltmeters

A device used to measure current is called an ammeter A device used to measure voltage is called a voltmeter To measure the current, the ammeter must be placed

in the circuit in series To measure the voltage, the voltmeter must be wired in

parallel with the component across which the voltage is to be measured

Voltmeter in parallelHigh resistance

Ammeter in seriesLow resistance

2/13/07 184 Lecture 20 16

RC CircuitsRC CircuitsRC CircuitsRC Circuits

So far we have dealt with circuits containing sources of emf and resistors.

The currents in these circuits did not vary in time. Now we will study circuits that contain capacitors

as well as sources of emf and resistors. These circuits have currents that vary with time. Consider a circuit with

• a source of emf, Vemf,

• a resistor R, • a capacitor C

2/13/07 184 Lecture 20 17

RC Circuits (2)RC Circuits (2)RC Circuits (2)RC Circuits (2)

We then close the switch, and current begins to flow in the circuit, charging the capacitor.

The current is provided by thesource of emf, which maintainsa constant voltage.

When the capacitor is fully charged, no more current flows in the circuit.

When the capacitor is fully charged, the voltage across the plates will be equal to the voltage provided by the source of emf and the total charge qtot on the capacitor will be qtot = CVemf.

2/13/07 184 Lecture 20 18

Capacitor ChargingCapacitor ChargingCapacitor ChargingCapacitor Charging Going around the circuit in a counterclockwise direction we

can write

We can rewrite this equationremembering that i = dq/dt

The solution of this differential equation is

… where q0 = CVemf and = RC

Vemf VR VC Vemf iR q

C0

Rdq

dtq

CVemf

dq

dtq

RCVemfR

q(t) q0 1 et

The term Vc is negative since the top plate of the capacitor is connected to the positive - higher potential - terminal of the battery. Thus analyzing counter-clockwise leads to a drop in voltage across the capacitor!

2/13/07 184 Lecture 20 19

Capacitor Charging (2)Capacitor Charging (2)Capacitor Charging (2)Capacitor Charging (2)

We can get the current flowing in the circuit by differentiating the charge with respect to time

The charge and current as a function of time are shown here ( = RC)

i dq

dtVemfR

e

t

RC

q(t) q0 1 e

t

Math Reminder:

2/13/07 184 Lecture 20 20

Now let’s take a resistor R and a fully charged capacitor C with charge q0 and connect them together by moving the switch from position 1 to position 2

In this case current will flow in the circuit until the capacitor is completely discharged.

While the capacitor is discharging we can apply the Loop Rule around the circuit and obtain

Capacitor DischargingCapacitor DischargingCapacitor DischargingCapacitor Discharging

iR VC iR q

C0 R

dq

dtq

C0

2/13/07 184 Lecture 20 21

Capacitor Discharging (2)Capacitor Discharging (2)Capacitor Discharging (2)Capacitor Discharging (2) The solution of this differential equation for the charge

is

Differentiating charge we get the current

The equations describing the time dependence of the charging and discharging of capacitors all involve the exponential factor e-t/RC

The product of the resistance times the capacitance is defined as the time constant of a RC circuit.

We can characterize an RC circuit by specifying the time constant of the circuit.

q q0et

RC

i dq

dt

q0

RC

e

t

RC

2/13/07 184 Lecture 20 22

Example: Time to Charge a CapacitorExample: Time to Charge a CapacitorExample: Time to Charge a CapacitorExample: Time to Charge a Capacitor Consider a circuit consisting of a 12.0 V battery,

a 50.0 resistor, and a 100.0 F capacitor wired in series.

The capacitor is initially uncharged. Question:

• How long will it take to charge the capacitor in this circuit to 90% of its maximum charge?

Answer:• The charge on the capacitor as a function of time is

q t q0 1 et

RC

2/13/07 184 Lecture 20 23

Example: Time to Charge a Capacitor (2)Example: Time to Charge a Capacitor (2)Example: Time to Charge a Capacitor (2)Example: Time to Charge a Capacitor (2)

We need to know the time corresponding to

We can rearrange the equation for the charge on the capacitor as a function of time to get

q t q0 1 et

RC

q t / q0 0.90

0.10 et

RC Math Reminder: ln(ex)=x

ms 5.11)10.0ln( RCt

2/13/07 184 Lecture 20 24

Example: More RC Circuits Example: More RC Circuits Example: More RC Circuits Example: More RC Circuits

A 15.0 k resistor and a capacitor are connected in series and a 12V battery is suddenly applied. The potential difference across the capacitor rises to 5V in 1.3 s. What is the time constant of the circuit?

Answer: 2.41 s

2/13/07 184 Lecture 20 25


A 15.0 k resistor and a capacitor are connected in series and a 12V battery is suddenly applied. The potential difference across the capacitor rises to 5V in 1.3 s. What is the capacitance C of the capacitor?

A) 161 pF B) 6.5 pF C) 0 D) 49 pF

2/13/07184 lecture 201 phy 184 spring 2007 lecture 20 title:

Documents

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