essential university physics · · 2009-08-24•an ideal voltmeter has infinite resistance so it...

Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lecture prepared by Richard Wolfson

Slide 25-1

25 Electric Circuits

Essential University PhysicsRichard Wolfson

2Copyright © 2007 Pearson Education, Inc., publishing as Pearson Addison-Wesley

In this lecture you’ll learn

• To read electric-circuit diagrams

• To analyze simple circuits with series and parallel combinations

• To analyze more complex circuits using loop and node laws

• To use electrical measuring instruments

• About circuits that include capacitors


Circuits and symbols

• Electric circuits are portrayed with diagrams using standard symbols, showing interconnections among their components:

• Most circuits contain a source of “electromotive force,” or emf, a device like a battery that supplies electrical energy.


Resistors in series

• When circuit components are connected in series, current from one component all flows into the next component.

• Therefore the current through series components is the same.

• With two resistors in series, the current I results in voltage drops IR1 and IR2:

• These sum to the battery emf:= IR1 + IR2

• Thus

• Therefore the two resistors behave as a single resistor of resistanceR1 + R2.

• In general, resistors in series add:

Rseries = R1 + R2 + R3 + …

1 2

.IR R

E

E


Resistors in parallel

• When circuit components are connected in parallel,the same voltage appears across each.

• With two resistors in parallel, the battery emf drives currents I1= /R1 and I2= /R2.

• Thus the total current is

• Therefore the two resistors behave as a single resistor whose resistance is R = 1/(1/R1+1/R2).

• In general, resistors in parallel add reciprocally:

1 2

1 1I

R R

E

1

Rparallel

1

R1

1

R2

1

R3

EE E


Clicker question

• The figure shows three circuits, each with three identical resistors. Which two of them are electrically equivalent?

A. Circuits (a) and (c)

B. Circuits (b) and (c)

C. Circuits (b) and (a)


Analyzing circuits with series and parallel

components


Clicker question

• Rank order the voltages across the identical resistors Rat the top of each circuit shown. In (a) the second resistor has the same resistance R, and in (b) the gap is an open circuit (infinite resistance).

A. (c) > (b) > (a)

B. (c) > (a) > (b)

C. (b) > (c) > (a)

D. (a) > (b) > (c)


Multiloop circuits

• Some circuits aren’t amenable to series-parallel analysis.

• Then it’s necessary to use Kirchhoff’s loop and nodelaws:

• The loop law states that the sum of voltage drops around any circuit loop is zero.

• The loop law expresses conservation of energy.

• The node law states that the sum of currents at any circuit node is zero.

• The node law expresses conservation of charge.

Node and loop equations for this circuit:

1 2 3

1 3

2 3

0 (node A)

6 2 0 (loop 1)

9 4 0 (loop 2)

I I I

I I

I I


Clicker question

• The figure shows a circuit with three identical light bulbs and a battery. What happens to each of the other two bulbs if you remove bulb C?

A. Bulb A brightens; bulb B dims.

B. Bulb B brightens; bulb A dims.

C. Bulbs A and B both dim.


Electrical measurements: voltage

• A voltmeter measures the potential difference between its two terminals.

• Connect a voltmeter in parallel with the component whose voltage you’re measuring.

• An ideal voltmeter has infinite resistance so it doesn’t affect the circuit being measured.

• A real voltmeter should have a resistance much greater than resistances in the circuit being measured.

Correct (a) and incorrect (b) ways

to measure the voltage across R2


Clicker question

• An ideal voltmeter is connected between points A and Bin the figure. What value does the voltmeter read?

A.

B.

C.

D.

E

3E

2E

2 3E


Electrical measurements: current

• An ammeter measures the current flowing through itself.

• Connect an ammeter in series with the component whose current you’re measuring.

• An ideal ammeter has zero resistance so it doesn’t affect the circuit being measured.

• A real ammeter should have a resistance much less than resistances in the circuit being measured.

Correct (a) and incorrect (b) ways to

measure the current in the series circuit


Capacitors in circuits

• Capacitors introduce time-dependent behavior to circuits.

• The voltage across a capacitor is proportional to the charge on the capacitor.

• The charge can’t change instantaneously, because that would require an infinite current to move a finite amount of charge onto the capacitor in zero time.

• Therefore, the voltage across a capacitor cannot change instantaneously.


The RC circuit

• The capacitor voltage VC is initially zero.

• Therefore, current flowsthrough the resistor,putting charge on thecapacitor.

• As the capacitor charges,VC increases and thevoltage across theresistor decreases.

• Therefore, the currentdecreases.

• Eventually a steady state isreached, with zero currentand capacitor voltage equal to the battery emf.


Analyzing the RC circuit

• The loop equation for the RCcircuit:

• Differentiating gives

• But dQ/dt is the current, I, so

• This is the differential equation for exponential decay:

• Therefore the voltage is

• The time scale for the changes is determined by the time constant RC.

• In the discharging RC circuit, the capacitor voltage decays exponentially with the same time constant.

0 EQ

IRC

10

dI dQR

dt C dt

dI

dt

I

RC

E t RCI eR

C 1 E E t RCV IR e


Short- and long-term behavior

• For times short or long compared with the time constant RC, a circuit with a capacitor can be analyzed in simple terms:

• For times short compared with RC, an uncharged capacitor acts like a short circuit (e.g., a wire), and a charged capacitor acts like a battery with emf equal to the capacitor voltage.

• For times long compared with RC, a capacitor acts like an open circuit. (a) A circuit with a capacitor

(b) Its short-term equivalent

(c) Its long-term equivalent


Clicker question

• A capacitor is charged to 12 V and then connected to points A and B in the figure, with its positive plate at A. What is the current through the resistor a long time after the capacitor is connected?

A. 6 mA

B. 12 mA

C. 2 mA

D. 3 mA

2-k


Summary• Simple circuits can be analyzed by identifying series and parallel

combinations:

• Resistors in series add: Rseries = R1 + R2 + R3 + …

• Resistors in parallel add reciprocally:

• More complicated circuits requirethe loop and node laws.

• Measuring voltage and current requires

• Connecting voltmeters in parallel

• Connecting ammeters in series

• Capacitors introduce time-dependentbehavior to circuits.

1

Rparallel

1

R1

1

R2

1

R3

essential university physics · · 2009-08-24•an ideal voltmeter has infinite resistance so it...

Documents