Exam 2 Lectures

Circuits

Charges in Motion

• Now we will start to talk about charges in motion instead of static charges

• Consider the three cases below– A pipe of flowing water – A wire without a potential difference– A wire with a potential difference between the

two ends

Definitions• Electric Current—charges in motion, or a

stream of moving charges

• Steady state—constant flow in time: for every electron entering, an electron must leave.

• Ampere—unit of current 1 Amp = 1C/s

• Current density—vector quantity which has the direction of E through a surface and magnitude of current per unit area.

• Drift speed—speed electrons drift through a conductor with a current in it

Current

• The flow of conduction electrons through a metal wire (conductor)

• In steady state

• Current I is a scalar not a vector

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i 0

itq

• By convention the arrow is drawn in the direction (+) charges would move

• Current can be from the movement of electrons, positive ions, or both

Current Density• Current density

• Current density is a vector and we must use vector math

• The same direction as the E field in the wire

• Relationship between J and E

AdJiareacurrent

J

EJJE

&

Drift Speed• Electrons move in the direction opposite

the E field with a drift speed vd

• vd tiny compared to the random motion speed of 106 m/s from Brownian motion

• Know how to find n (carrier density)

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vd510

d

d

nevJneJ

nAei

v

Batteries• Almost any 2 different solid conductors immersed

in an active solution (electrolyte) functions as a battery

• The chemical energy stored in the interatomic bonds is converted to electrical potential energy as the solution and the conductors become involved in the chemical reaction

• The electrolyte is a solution which dissolves the ions formed by the leaving electrons allowing the ions to move in the solution

• One of the conductors becomes the cathode (gains electrons) and the other becomes the anode (loses electrons)

• A salt bridge is necessary for letting the ions flow

Batteries cont

• emf—potential difference that can be used to supply energy and sustain a current. Also voltage measured across the terminals of the battery when no current is being drawn from or delivered to it

• If the batteries are connected oppositely: + terminal to – terminal then the voltages subtract

• For big i & low V – put battery cells in parallel.• For small i & big V – put battery cells in series. • For big i & big V – put rows of parallel battery cells

in series

Definitions• Conductivity—the ability of a material to

conduct electricity. is not necessarily a constant, it could be a tensor or it could be a function of E.

• Resistivity—the inverse of conductivity

• Resistance—the ability of a material to resist the flow of electric charge

• Ohm—the unit of resistance. 1 = 1 V/A

Definitions cont• Resistors—devices in a circuit to control the

current level in various parts of the circuit. Isotropic materials—materials whose electrical properties are the same in all directions (conductivity and resistivity)

• Ohm’s Law—usually stated V = iR or J = E. Not all devices follow this law, some are not directly proportional to V (R a constant), for some R is a function of V (R = f(V)). (isotropic materials)

• Resistivity is a property of the material, and resistance is a property of the object

• Resistance depends on the geometry of the conductor (resistor)

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AL

R

• Resistivity depends on the properties of the material and temperature

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Ohm’s Law

• Ohm’s law is true for many substances, but there are many materials and devices that are nonohmic

• A device obeys Ohm’s law when its R is independent of the magnitude and polarity of V

• A material obeys Ohm’s law when is independent of the magnitude and direction of E

V iR

• Most modern electronic devices are nonohmic and their usefulness or proper operation depends on how they violate Ohm’s Law

b) An ohmic device – a resistorc) A nonohmic device – a pn junction diode

– For a resistor, resistance is a constant of proportionality between current and the voltage difference and is independent of V and i

– For a resistor, resistance does not depend on either i or V, but on the properties of the material making up the resistor

No tolerance band – ±20%Silver band – ±10%Gold band – ±5%

Microscopic View of Ohm’s Law• Look at the motion of free conduction

electrons

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mE

meEt

neJ

v

meEt

atv

meE

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22

Different Types of Conductors• Conductors – materials that allow the flow of

charge• Insulators – materials that don’t allow the

flow of charge• Semiconductors—materials that are

intermediate between conductors and insulators

• Doping—adding minute amounts of impurities to semiconductors to change their resistivity.

• Superconductors—materials with no resistance to the movement of electric charge through them

Types of Conductors cont

Materials Conductors Semiconductors Insulators

Resistivity < 10-5 m 10-5 < < 105 m > 105 m

Examples Ag and Cu Si and Ge Glass,

rubber

Energ

y

Conductor Semiconductor Insulator

Conduction Band

Valance Band

Comparison Conductors to Semiconductors

• Semiconductors have smaller n• Semiconductors have a much higher • Semiconductors temperature coefficient of

resistivity is large and negative • In conductors n is large but nearly constant. As

T increases, v increases and t decreases > o

• In semiconductors t still decreases but n starts out small and increases fast with temperature. < o as n increases

2ne

m

Power

• The power or the joule heating of the resistor is how fast a resistor heats up

• The rate of energy transfer from battery to some other device

• This energy could be a conversion of electrical potential energy to some other form of energy such as mechanical work, thermal energy, stored chemical energy, light or etc

RiP 2

iVP

Circuit Devices

• Resistor – device in a circuit to control the current level in various parts of the circuit.

• Capacitor – device in a circuit which store energy in an electric field

• Battery – device in a circuit which produces a potential difference

• Conductor – material through which current flows

Capacitor

Definitions• Ideal emf device—has no internal resistance

and = potential difference between the terminals

• Real emf device—does have internal resistance and > potential difference between the terminals (some energy lost probably as heat)

Emf Devices• An emf device does work on (transfers

energy to) charge carriers

• Energy comes from:1. In batteries or fuel cells—chemical energy

2. In electric generator—mechanical forces

3. In thermopile—temperature differences

4. In solar cell—sun or solar energy

• 2 ways to calculate the current i in a simple single loop circuit

1. Energy method

2. Potential method

Energy Method• Using conservation of energy

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RiiR

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2

Potential Method• Using the potential differences

• A battery from low to high potential

from high to low potential

• A resistor from low to high potential

• from high to low potential

VV

iRV iRV

aa ViRV

RiiRiR

0

• Using the sign of the first terminal

• Battery – from low to high V – from high to low V

• Resistor – from low to high V– from high to low V

V E

V E

V iR

V E

V iR

a aV iR V 0iR iR

iR

Internal Resistance of Battery

rRiiRir

This internal resistance comes from the resistance of the internal components of the battery and is irremovable

Resistors in Series and Parallel• Resistors in Series

• Resistors in Parallel

321 RRRRequivalent

321

1111RRRRtotal

Circuit Facts

Parallel Series

321 VVVVtotal 321 VVVVtotal

321 qqqqtotal

321 iiiitotal 321 qqqqtotal

321 iiiitotal

321 RRRRtotal

321 CCCCtotal 321

1111CCCCtotal

321

1111RRRRtotal

Circuit Facts & Kirchhoff’s Laws• If you have two batteries in a circuit, the

battery with the larger emf determines the direction of the current

• Loop Rule: the algebraic sum of the changes in potential encountered in a complete transversal of any loop of a circuit must be 0 (conservation of energy)

• Junction Rule: the sum of the currents entering any junction must be equal to the sum of the currents leaving that junction (conservation of charge)

Meters• Ammeter—an instrument used to measure

currents. In series & low resistance

• Voltmeter—an instrument used to measure potential differences. In parallel & high resistance

• Ohmmeter—an instrument used to measure resistance of an element.

• Multimeter—a single meter which can measure all of the above

RC Circuits• RC circuit – circuit in which the current

varies with time

Charging Discharging

C

t

eRdt

dqi

C

t

eCq 1

C

t

eCq

V 1

C

t

o eqq

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to eRCq

dtdq

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RCC