Copyright R. Janow - Fall 2015 1 Electricity and Magnetism Lecture 07 - Physics 121 Current,...

1Copyright R. Janow - Fall 2015

Electricity and MagnetismLecture 07 - Physics 121 Current, Resistance, DC Circuits: Y&F Chapter 25 Sect. 1-5Kirchhoff’s Laws: Y&F Chapter 26 Sect. 1

• Currents and Charge• Electric Current i• Current Density J• Drift Speed, Charge Carrier Collisions• Resistance, Resistivity, Conductivity• Ohm’s Law• Power in Electric Circuits• Examples• Circuit Element Definitions• Kirchhoff’s Rules• EMF’s - “Pumping” Charges, Ideal and real EMFs• Work, Energy, and EMF• Simple Single Loop and Multi-Loop Circuits using

Kirchhoff Rules• Equivalent Series and Parallel Resistance formulas

using Kirchhoff Rules.

Copyright R. Janow - Fall 2015

Definition of Current : Net flow rate of charge through some area

Current is the same across each cross-section of a wire

+ -

iCurrent density J may vary [J] = current/area

constant) is (ift

0or it i )dt'i(t'q(t) dt idq

dt

dqi x

Convention: flow is from + to – as if free charges are +Units: 1 Ampere = 1 Coulomb per second

Charge / current is conserved - charge does not pile up or vanish

At any Node (Junction)

outin i ii1

i2

i3 i1+i2=i3

Kirchhoff’s Rules:

(summary)

Current (Node) Rule: currents in = currents out at any node Voltage (Loop) Rule: V’s = 0 for any closed path Voltage sources (EMFs, electro-motive forces) or current sources can supply energy to a circuit Resistances dissipate energy as heat Capacitances and Inductances store energy in E or B fields

Energy orPower

in a circuit:


Junction Rule Example – Current Conservation

7-1: What is the value of the current marked i?

A. 1 A.B. 2 A.C. 5 A.D. 7 A.E. Cannot determine from information given.

5 A

2 A 2 A

3 A1 A

6 A

i

outin i i

3 A

8 A

1 A

= 7 A


Current density J: Current / Unit Area (Vector)

What makes current flow?Microscopic level /1 E J EJ

conductivity

Same current crosses larger or smallerSurfaces, current density J varies+ -

iA

A’

J’ = i / A’(small)

J = i / A(large)

High current densityin this region

Small current densityin this regioni

AdJi area

dAn̂Jdi

units: Amperes/m2 For uniform density: i/A J A J i or

For non-uniform density:

Drift speed: Do electrons in a current keep accelerating?

Yes, for isolated charge in vacuum. No, in a conducting solid, liquid, gas

resistivity


COLLISIONS with ions, impurities, etc. cause resistance Charges move at constant drift speed vD:

• Thermal motions (random motions) have speed • Drift speed is tiny compared with thermal motions.• Drift speed in copper is 108 – 10-4 m/s.

)Tk( m/s10v Boltz2

36th

E field in solid wire

drives current.

APPLIED FIELD = 0Charges in random motion

flow left = flow right

++

--

APPLIED FIELD NOT ZERO Moving charges collide with fixed

ionsand flow with drift velocity vd

+-

Note: Electrons drift rightward but vd and J are still leftward |q| = e = 1.6 x 10-19 C.

For E = 0 in conductor: no current, vD=0, J = 0, i = 0

For E not = 0 (battery voltage not 0):

E J

/volume# :Units carriers charge of density n time unit per area unit crossing carriers charge of# D nv

E vqn J time unit per A area crossing charge netD

A qnv i Dtime tcharge/uni Eqn

v D


EXAMPLE: Calculate the current density Jions for ions in a gas

Assume:

• Doubly charged positive ions• Density n = 2 x 108 ions/cm3

• Ion drift speed vd = 105 m/s

Find Jions – the current density for the ions only (forget Jelectrons) 6

58-19

xD 10 1010 2 10 1.6 2 qnv J coul/ion ions/cm3 m/s cm3/m3

6.4 J A./m2


Increasing the Current

7-2: When you increase the current in a wire, what changes and what is constant?

A. The density of charge carriers stays the same, and the drift speed increases.

B. The drift speed stays the same, and the number of charge carriers increases.

C. The charge carried by each charge carrier increases.D. The current density decreases.E. None of the above

E J EJ

Dqnv J


Definition of Resistance : Amount of current flowing through a conductor per unit of applied potential difference.

evolt/amper 1 1Ohm 1 :units i

VR

R depends on the material & geometry Note: C= Q/V – inverse to R

i iV

E

L

A

Apply voltage to a wire made of a good conductor. - Very large current flows so R is small. Apply voltage to a poorly conducting material like carbon - Tiny current flows so R is very large.

RCircuit

Diagram

V

Resistivity “” : Property of a material itself (as is dielectric constant). Does not depend on dimensions

• The resistance of a device depends on resistivity and also depends on shape• For a given shape, different materials produce different currents for same V• Assume cylindrical resistors

resistor a for L

RA yresistivit

A

L resistance R

m. meters-Ohm :units yresistivit For insulators: infinity

-11 Siemens"" are Units e"conductanc" RG


Example: calculating resistance or resistivity

A

LR

resistanceproportional to length

inversely proportional to cross section area

resistivity

EXAMPLE: Find R for a 10 m long iron wire, 1 mm in diameter

. )/(10

.

A

L R

2m

3-x

m xm.x21

2

1010792

8

Find the potential difference across R if i = 10 A. (Amperes)

V iR V 12

EXAMPLE:

Find resistivity of a wire with R = 50 mdiameter d = 1 mm, length L = 2 m

m. .96x10 /3-10

L

RA 8-

2 x

m

m x x

12

2

101050 3

Use a table to identify material. Not Cu or Al, possibly an alloy


Resistivity depends on temperature:• Resistivity depends on temperature: Higher temperature greater

thermal motion more collisions higher resistance.

Conductivity is the reciprocal of resistivity

-1 E/J EL LJJARiRV :resistor Across

1- .m)( mho"" :units E J

1

i iV

E

L

A Definition:

A

L R

SOME SAMPLERESISTIVITY

VALUES

in m @ 20o C.1.72 x 10-8 copper

9.7 x 10-8 iron2.30 x 10+3 pure silicon

Reference Temperature

Change the temperature from reference T0 to TCoefficient depends

on the materialtcoefficien etemperatur

))TT((

00 1

Simple model of resistivity: = temperature coefficient


Resistivity Tables


Current Through a Resistor

7-3: What is the current through the resistor in the following circuit, if V = 20 V and R = 100 ?

A. 20 mA.B. 5 mA.C. 0.2 A.D. 200 A.E. 5 A.

RV CircuitDiagram

R i V


Current Through a Resistor

7-4: If the current is doubled, which of the following might also have changed?

A. The voltage across the resistor might have doubled.B. The resistance of the resistor might have doubled.C. The resistance and voltage might have both doubled.D. The voltage across the resistor may have dropped by a factor of

2.E. The resistance of the resistor may have dropped by a factor of 2.

RV CircuitDiagram

R i V

Note: there might bemore than 1 right answer


Ohm’s Law and Ohmic materials (a special case) Definitions of

resistance:i/VR (R could depend on applied V)

E/J/ 1 ( could depend on E)

Definition: For OHMIC conductors and devices:• Ratio of voltage drop to current is constant – NO dependance on applied voltage. i.e., current through circuit element equals CONSTANT resistance times applied V • Resistivity does not depend on magnitude or direction of applied voltage

Ohmic Materials e.g., metals, carbon,…

Non-Ohmic Materialse.g., semiconductor diodes

constant slope = 1/R varying slope = 1/R

band gap

OHMICCONDITION dV

di

R

1 is CONSTANT


Resistive Loads in Circuits Dissipate Power

dt Vidq VdU

i

V

+i

-

LOAD

1

2

• Apply voltage drop V across basic circuit element (load)• Current flows through load which dissipates energy • An EMF (e.g., a battery) does work, holding potential V and current i constant by expending potential energy

/R VRi P iR V

Vidt

dq

dq

dU

dt

dU P dissipated Power

only resistors22

load anyfor[Watts]

As charge dq flows from 1 to 2 it loses P.E. = dU

- potential is PE / unit charge- charge = current x time

element absorbs power P=-Vi

1

2iV

+

-

1

2iV

-

+

element supplies power P= +Vi

1

2iV

+

-

1

2iV

-

+

Voltage rise:

+

-V

Active sign convention: Pabsorbed = dot product of Vrise with i

Note: Some EE circuits texts use passive convention: model voltage drops Vdrop as positive.


EXAMPLE:

EXAMPLE:

EXAMPLE:


Basic circuit elements have two terminals

Basic Circuit Elements and Symbols

1 2

Ideal basic circuit elements:

Generic symbol:

Active: voltage sources (EMF’s), current sourcesL

C Cor

R

Passive: resistance, capacitance, inductance

is

Ideal independent voltage source:• zero internal resistance• constant vs for any load

Ideal independent current source: • constant is regardless of load

vs

+

-E

+-vs

DCDC

batteryAC

Dependent voltage or current sources: depend on a control voltage or current generated elsewhere within a circuit


Circuit analysis with resistances and EMFs

ANALYSIS METHOD: Kirchhoff’S LAWS or RULES

Current (Junction) Rule:

outin i i Charge conservation

Loop (Voltage) Rule: loop) (closed 0 V Energy conservation

i

V

slope= 1/R

RESISTANCE:

POWER:

OHM’s LAW:

yresistivit A

LR

i

VR

(resistor)2 RiP iV

dt

dU

dt

dWP

R is independent of V or i

CIRCUIT ELEMENTS ARE CONNECTED TOGETHER AT NODES

i

i1

i2

CIRCUITS CONSIST OF JUNCTIONS (ESSENTIAL NODES) ... and … ESSENTIAL BRANCHES (elements connected in series,one current/branch) i


EMFs “pump” charges to higher energy• EMFs (electromotive force) such as batteries supply energy:

– move charges from low to high potential (potential energy).– maintain constant potential at terminals

– do work dW = Edq on charges (source of the energy in batteries is chemical) – EMFs are “charge pumps”

• Unit: volts (V). Variable name: script E.• Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc.• DC versus AC

+- RE

i

i

CONVENTION:Current flows CW through circuitfrom + to – outside of EMFfrom – to + inside EMF

dq

dW

chargeunit

done work E

dt P dt i dq dW E E

i i Pemf

E E

dt

dW power P

R/V- R i- V i - P 22R

Power is dissipated by resistor:

Power supplied by EMF:


• Zero internal battery resistance• Open switch: EMF = E no current, zero power• Closed switch: EMF E is also applied across load circuit• Current & power not zero

Ideal EMFdevice

R

MultipleEMFs

Assume EB > EA (ideal EMF’s) Which way does current i flow?• Apply kirchhoff Laws to find current• Answer: From EB to EA • EB does work, loses energy• EA is charged up• R converts PE to heat• Load (motor, other) produces motion and/or heat

Real EMFdevice

• Open switch: EMF still = E• r = internal EMF resistance in series, usually small ~ 1 • Closed switch:

• V = E – ir across load, Pckt= iV• Power dissipated in EMF Pemf = i(E-V) = i2r


The Kirchhoff Loop Rule Generates Circuit Equations

• The sum of voltage changes = zero around all closed loops in a multi-loop circuit) • Traversing one closed loop generates one equation. Multiple loops may be needed.• A branch is a series combination of circuit elements between essential nodes.• Assume either current direction in each branch. Minus signs may appear in the result. • Traverse each branch of a loop with or against assumed current direction.• Across resistances, write voltage drop V = - iR if following assumed current direction. Otherwise, write voltage change = +iR. • When crossing EMFs from – to +, write V = +E. Otherwise write V= -E• Dot product i.E determines whether power is actually supplied or dissipated

Follow circuit from a to b to a, same direction as i

Potential around the circuit

EXAMPLE: Single loop (circuit with battery (internal resistance r)

0 iRirERr

Ei

EE

P = iE – i2r

Power in External Ckt

circuitdissipation

batterydrain

batterydissipation

P = iV = i(E – ir)


The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above

Loop Rule: The sum of the potential differences around a closed loop equals zero. Only one loop path exists:

Example: Equivalent resistance for resistors in series

Junction Rule: The current through all of the resistances in series (a single essential branch) is identical. No information from Junction/Current/Node Rule

iiii 321

0321 iRiRiR

eqRi

iR eq 0

321 RRRi

321 RRRReq RR n

iieq

1

CONCLUSION: The equivalent resistance for a series combination is the sum of the individual resistances and is always greater than any one of them.

inverse of series capacitance rule


Example: Equivalent resistance for resistors in parallel

Loop Rule: The potential differences across each of the 4 parallel branches are the same. Four unknown currents. Apply loop rule to 3 paths.

33

22

11 R

i ,R

i ,R

iEEE

321321

111

RRR i i i i E

321

1111

RRRReq

RR

n

i ieq

1

11

011 RiE 022 RiE 033 RiE

Junction Rule: The sum of the currents flowing in equals the sum of the currents flowing out. Combine equations for all the upper junctions at “a” (same at “b”).

eqRi

iR eq 0

The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above.

CONCLUSION: The reciprocal of the equivalent resistance for a parallel combination is the sum of the individual reciprocal resistances and is always smaller than any one of them.

inverse of parallel capacitance rule

i not in these

equations

RR

RRR eq

21

21


7-7: Four identical resistors are connected as shown in the figure. Find the equivalent resistance between points a and c.

A. 4 R.B. 3 R.C. 2.5 R.D. 0.4 R.E.Cannot determine from information given.

Resistors in series and parallel

R

R R

R

c

a

RR

n

i ieq

1

11 RR

n

iieq

1


7-8: Four identical capacitors are connected as shown in figure. Find the equivalent capacitance between points a and c.

A. 4 C.B. 3 C.C. 2.5 C.D. 0.4 C.E. Cannot determine from information given.

Capacitors in series and parallel

C

C C

C

c

a

CC

n

i ieq

1

11 CC

n

iieq

1


EXAMPLE:

EXAMPLE:

+-

R1= 10

R2= 7

R3= 8 E = 7 V

i

+-

E = 9 VR1 R2

i1 i2

i

i

Find i, V1, V2, V3, P1, P2, P3

Find currents and voltage drops

RR

RRR eq

21

21


+-

+-

R1= 10

R2= 15

E1 = 8 V E2 = 3 V

i

i

EXAMPLE: MULTIPLE BATTERIESSINGLE LOOP

A battery (EMF) absorbs power (charges up) when I is opposite to E

i i Pemf

E E


EXAMPLE: Find the average current density J in a copper wire whose diameter is 1 mm carrying current of i = 1 ma.

2amps/m

m3-

x

amps-3

1273 )10 x (.5

10

A

i J

2

Suppose diameter is 2 mm instead. Find J’:

2amps/m 318

4

J

A'

i J' Current i is unchanged

Calculate the drift velocity for the 1 mm wire as above? 28

x3

melectrons/ conduction#CuwheredCu 10 8.49 n ven J

s/mx...

1273

en

J v

xxxCud

82819

10379104981061

About 3 m/year !!

So why do electrical signals on wires seem to travel at the speed of light (300,000 km/s)?

Calculating n for copper: One conduction electron per atom

10 8.49

10 8.96 63.5

10 6.023 x 1 n

3melectrons/

28x

3

/m3

cm6

x 3

gm/cmxgm/mole

atoms/mole23

x atom/electronCu

Copyright R. Janow - Fall 2015 1 Electricity and Magnetism Lecture 07 - Physics 121 Current,...

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