Electric Currents. Resistors

(Chapters 27-28)

• Electric current I

• Resistance R and resistors

• Relation between current and resistance: Ohm’s Law

• Resistivity ρ

• Energy dissipated by current. Electric power

• Electromotive force: emf, ε

• Simple resistive circuits:

• Series and parallel circuits. Circuits reducible to simple combinations

• Circuits non-reducible to series and parallel: Kirchhoff Rules

• Resistors and capacitors combined: dc RC circuit

Electric Current – Charge Carrier Motion in a Conductor

• In an electrically insulated wire, the electrons undergo thermal random motion

• When an potential difference is set up between the ends of a wire, the surface charge

gets nonuniformly distributed, producing an electric field inside the wire, such that

electrons tend to drift against the average field producing an electric current, I

• The nature of the charges carrying the current depends on the nature of the material

Ex: The zigzag lines represent the drift against

the electric field of an electron in a conductor.

The drift is analog with the parabolic

trajectories of a ball drifting due to gravity

down a wall with pegs.

Ex: electrons in metals, electrons and ions in plasma, holes in semiconductors, etc.

–-e

+ + +

+ + +

+ + +E g

m

ion cores pegs

• When a circuit is completed, the electric field sets in the wires with the speed of

light (of the order of 108 m/s), and all free electrons throughout the material move

very fast; however, their motion is impeded by collisions in the material, so their

drift speed, vd, is much smaller, (of the order of 10 –4 m/s)

more positive

V+

more negative

V –

+

+

+

–

–

–

a b cEa

+++++

+

+

Enet

EbEc

Enet

Eb

dv

• So, whenever electric charges of like signs drift, an electric current is said to exist

• In order to define it quantitatively, consider at the charge flowing through a wire

perpendicular to a cross-sectional surface of area A

SI

1CAmpere (A)

1sI

Comments:

• Conventional current direction: the direction of the current is the direction of the

drift of positive charge: that is, in the direction of the average field, or from high

potential to low potential

• Albeit its directionality, the current is not a vector

• The current across a potential difference is the same through any cross-section, that

is, the carriers drift identically as long as the conductive substance is the same

• The colloquial short for “Amperes” is “Amps”

• Order of magnitude: flashlight bulb ~1A, sensitive electronics ~μA, high-power

devices (such as large electro-magnets) ~kA

Electric Current – Definition

Def: The rate at which the electric charge flows through this

surface, that is, the amount of charge dQ flowing per unit of

time dt through the surface, is called an electric current

dQI

dt

Q

A

• To find the relationship between the macroscopic

electric current and the microscopic details of the

electric-carrier drift, consider elementary charge

carriers of charge e drifting with constant drift speed

vd through a current carrying conductor of cross-

sectional area A, with the charge concentration given

by:

• Therefore, combining with the definition of current, we find that the current is related

to the drift speed vd as following:

Q en Q neA x

A x

d

Q xI neA neA

t tI v

Electric Current – Relation to drift speed. Current density

• Then, as shown on the figure,

• So, if the drift speed is small (many collisions), the

current is small and vice versa.

• If the carriers are electrons (charge –e), the current

opposes the drift velocity.

Number of carriersn

Volume

dvI

J nqA

Def: The current per unit cross-sectional area of a conductor

is a vector called current density:

Quiz 1: We learned that the electron drift speed is relatively small: of the order of 10 –4 m/s.

How come, when the circuit in the figure is closed, the bulb light up almost instantaneously?

a) Actually it doesn’t: in my house we wait minutes

until bulbs light up…

b) Since the energy of the electric field is delivered to

the bulb before electrons from the battery reach to it

c) Because the “sea of electrons” fill all the circuit.

When the circuit is closed, the field is set with the

speed of light and electrons start to drift through

each cross-section of the circuit including the bulb.

Quiz 2: Both segments of the shown wire are made of the same metal. Current I1 flows into

segment 1 from the left. How does current I1 in segment 1 compare to current I2 in segment 2?

a) I1 > I2

b) I1 = I2

c) I1 < I2

d) There’s not enough information to compare them

Ohm’s Law – Statement• Experiments show that in most metals the current density J depends on:

• the electric field E

• the tabulated properties of the material: resistivity ρ or conductivity σ = ρ–1

• The dependency is given by

Comments:

• Ohm’s Law does not state the proportionality E ~ J, but the constancy of ρ

Ohm’s Law: In an ohmic (or linear) material at constant temperature, the

resistivity and conductivity remain constant for any electric field and the current

density is proportional to the field:

constant

J E E J

0 01 T T

ρ is the resistivity at temperature T

ρ0 is the resistivity at a reference

temperature T0 (usually taken to be 20° C)

is the temperature coefficient of resistivity

ρ0

ρ

T0 T

Slope = αρ0

ρ

T

conductor seminconductor

ρ

Tc T

superconductor

Ex: Only some metals follow this behavior

through the whole range of temperatures. The

resistivity of superconductors drops sharply to

zero under a critical T, while the resistivity of

semiconductors decreases with increasing T

• For most metals, resistivity increases

approximately linearly with

temperature T, over limited T-ranges:

Resistance and Resistors – A new element of circuit• Ohm’s law can be reformulated in terms of the potential difference Vab responsible

for the electric field driving a current through a wire:

abV IR

SI

1VOhm (Ω)

1AR

abab

V IE J V I

L A

L

A Va

I

JE

L

AVb

• The quantity R is termed the resistance of the conductor:

length parallel with current

area perpendicular on current

ab a b a bV V V IR V V

+ -

R

I

a b

Analogy:

circuit of

pipes with

water in

laminar

flow

Pump: pressure difference ↔ potential

Constricted pipe ↔ resistance

Flow rate ↔ current

High Low

• The elements of circuit with resistance are called resistors. Symbol:

• In a resistor in circuit with a battery, the electric potential decreases in the direction

of the current – the resistor determines a voltage drop – while the potential increases

across a battery – it determines a voltage raise – since a battery does work on the

carriers

I

II

Problems:

1. Temperature dependence of current: A resistor with uniform cross-section and

temperature coefficient α = 410-3 C-1 is heated from T0 = 20oC to T = 170oC. All this time a

constant potential difference is applied across the device. If the current through the resistor at

T0 is I0 = 2.0 A what is the current at temperature T?

2. Resistance and directionality: A metallic solid parallelepiped with resistivity ρ has length

L, width a and height b. Find the resistance of the object if

a) a potential difference is applied on the ab cross-sections

b) the potential difference is applied perpendicular on L

Quiz 3: An ohmic material is probed for drawn

current I by applying an increasing potential

difference V across the sample. Which of the

shown I vs. V graphs (called IV-characteristics)

represents the likely dependency of I on V? V

I

V

I

a) b)

Quiz 4: What geometrical aspect of the IV-characteristics is a measure of the resistance?

a) The intercept of V axis divided by the respective current

b) The slope of the graph

c) The inverse of the slope of the graph

Meters in a Circuit – Measuring current and voltage

A

V

• It must be mounted in line with the element along which the current is measured: all

the charge passing through the element must also pass through the meter

• In order to measure current without modifying it, the ammeter must have a very

small resistance, so no potential difference across it

• A voltmeter is used to measure potential difference. Symbol:

• An ammeter is used to measure current. Symbol:

• It must be connected to the two ends of the element

• In order to measure the voltage without affecting it, the voltmeter must carry no

current so its resistance is very large

Energy Dissipated in a Resistor – Qualitative approach

• We’ve seen that the capacitor is an element of circuit

that stores electric energy. What about a resistor? What

does a resistor do from an energy point of view?

• Consider an element of positive charge, dq, moving

around a closed circuit from point a back to point a

• As the charge moves through the battery from a to b, the

potential energy of the charge increases by Vdq (on behalf

of the chemical energy of the battery)

• As the charge moves through the resistor R, from c to d,

it looses energy in collisions with the atoms of the resistor:

the energy is transferred to the internal energy

ground: taken as

having zero potential

• When the charge returns to a, the net result is that some chemical energy of the

battery has been delivered to the resistor and caused its temperature to rise: we say

that the energy was dissipated across the resistor

• Notice that as long the charge flowing around the circuit (that is, the current) looses

energy in the resistor, the battery must resupply it in order to maintain the current

• Resistance acts like friction in a mechanics: without resistance (friction), the current

would flow (charges would move) around the circuit without the need of a battery

• Say that a circuit carries a charge dq through a difference of potential V across a

resistor R that dissipates energy dU = Vdq

• The rate at which the energy U is dissipated is the electric power of the resistor:

• Using Ohm’s law, we can find two useful alternative forms:

dU dq

P V P IVdt dt

2

2 VP I R

R

• The unit of energy used by electric companies is the kilowatt-

hour: defined in terms of the unit of power and the amount of

time it is supplied: 1 kWh = 3.60106 J

Energy Dissipated in a Resistor – Electric power

Quiz:

5. Which of the resistors on the right will draw a larger current?

6. Which of the two resistors will get hotter?

a) A b) B c) Both draw the same current

a) A b) B c) Both will get equally hot d) They don’t get hot

• Direct-Current or dc circuits are traveled by currents in only one direction: the

magnitude of the currents along the circuit branches may vary, but not their direction

• Alternating-Current circuits or ac circuits are traveled by currents in directions

alternating with a certain periodicity

• Different elements of circuit behave differently in dc and ac circuits

Types of Circuits

Electromotive Force – Sources and internal resistance

SI

Volts V

Def: An device – such as a battery or generator – that maintains the current in a

closed circuit is said to provide an electromotive force or emf, ε

• A real battery has some internal resistance r, so some

of its energy is dissipated internally

• Therefore, the voltage Vab across a real battery when

it drives a current is not equal to the emf

• The voltage Vab is applied to the external circuit of

resistance R (called load), so

ab rV V Ir

IR Ir

The internal resistance can

be represented as a resistor

in series with the battery

• Notice that the emf ε is equal to the terminal

voltage when the current is zero (also called the

open-circuit voltage)

• If the emf source is mounted across a load, a

current I is driven and the difference of

potential across the battery decreases by Ir

• The current provided by the battery depends

both on the load R and the internal resistance r

• When R r, the source is considered as ideal

• Ordinary batteries have a small internal

resistance, such that the voltage delivered is less

than the nominal voltage which is the emf

• The battery power is distributed both to the

load and internal resistance:

• So, when R r, most of the power delivered

by the battery is transferred to the load

I R r

IR Ir

2 2I I R I r

Ex: Voltage diagram

The voltage V raised by the battery

by ε drops both across the internal

resistance r and the load R

r Rε

Electromotive Force – Comments

• In general, for n resistors in series:

• The series equivalent resistance is always greater than any of the individual resistors

ab bcV V V

1 2 sIR IR IR

1 2 sR R R

1 2 3 ...s nR R R R R

Electric Circuits – Resistors in series

• The current is the same in all resistors because

any charge that flows through one resistor flows

through the other:

• As a consequence of energy conservation around

the circuit, the sum of the potential differences

across the resistors is equal to the total potential

difference across the combination

1 2I I I

• When two or more resistors are connected end-to-end, they are said to be in series

Vab Vbc

Rs

• The potential difference across each resistor is

the same

• As a consequence of charge conservation, the

current, that enters into a junction must be equal

to the total current leaving that junction, so,

using for instance junction a,

1 2 I I I

1 2p

V V V

R R R

1 2

1 2 1 2

1 1 1p

p

R RR

R R R R R

1 2

1 1 1 1..

p nR R R R

2 abV V

1 abV V

Electric Circuits – Resistors in parallel

• When two or more resistors are connected to the same two points in the circuit, they

are said to be in parallel

1 2V V V

• In general, for n resistors in parallel:

• So, the parallel equivalent resistance is always less than the smallest resistor

Rp

Problems:

3. Current with and without internal resistance: A battery with emf ε and internal

resistance r = 2.0 Ω delivers a current I0 = 100 mA when connected to a load R = 70 Ω.

Calculate the current I through the circuit if the internal resistance of the battery were zero.

4. Analysis of combined circuit: An electric circuit contains an ideal battery with ε and four

given resistors, R1, R2, R3, R4, arranged as in the figure.

a) Calculate the equivalent resistance of the circuit.

b) Calculate the current through resistor R4 and the potential difference Vbc across it

c) Calculate the potential difference Vab across points a and b on the circuit.

d) Calculate the currents I1 and I2 through R1 and R2,3 respectively.

e) Say that point c is “grounded”. What are the potentials in points a, b, and d with respect to

this ground? What changes if the ground changes?

ε

4R

1R

3R

2R

b

a

c

d

Kirchhoff’s Rules – Statements

• Resistors can be connected so that the circuits formed cannot be reduced to a single

equivalent resistor as a combination of parallel and series arrangements

• These generic circuits can be analyzed – that is, can be described in terms of

currents carried along various branches, and potential differences between different

points in the circuit – using Kirchhoff’s Rules:

1. Junction Rule

A statement of Charge Conservation

The sum of the currents entering any junction must equal the sum of the currents

leaving that junction

2. Loop Rule

A statement of Energy Conservation

The sum of the potential differences across all the elements around any closed

circuit loop must be zero

0I

0V

How to use the rule:

• Assign symbols and directions to the currents in all

branches of the circuit

• The directions can be arbitrary: if a direction is

chosen incorrectly, the current resulting after solving

the equations will be negative, with a correct

magnitude

• Write the junction rule for as many junctions as

needed as needed, as long as each time you write an

equation you include in it a current that has not been

used in a previous junction rule equation

• In general, the number of times the junction rule

can be used is one fewer than the number of junction

points in the circuit

Kirchhoff’s Rules – Junction rule

Ex: Hydrodynamic analog:

the current entering into a

junction splits into partial

currents (a) as water flowing

into bifurcating pipes (b)

in outI I

around a loop

0iV Kirchhoff’s Rules – Loop rule

How to use the rule:

• Circuits contain loops of electric elements such as resistors and batteries

• Choose an arbitrary direction to travel around a circuit loop Ex: clockwise

Resistors:

• If the resistor is traveled in the direction of the current: V = –IR (voltage drop)

• If the resistor is traveled opposite to the current: V = +IR (voltage raise)

Batteries:

• If the source of emf is traveled in the direction of the emf: V = +ε (voltage raise)

• If the source of emf is traveled opposite to the emf: V = –ε (voltage drop)

loop

6 . Applying Kirchhoff’s Rules – two batteries:

An electric circuit contains 4 resistors and two ideal

batteries, as in the figure. Write out Kirchhoff rules

for this circuit.

Problems:

5. Applying Kirchhoff’s Rules – one battery: An

electric circuit contains an ideal battery with ε = 6 V and

four given resistors, R1 = 10 Ω, R2 = 12 Ω, R3 = 2 Ω,

arranged as in the figure. Use Kirchhoff Rules to calculate

the currents I, I1, I2 through the branches of the circuit.

ε

a

b

ε12R

b

a

ε2

dc

3R

4R

1R

2R1R

1. Draw the circuit diagram and assign labels and symbols to all quantities

2. Assign directions to the currents – no need for the directions to be all correct; a current with

an incorrectly chosen direction will eventually come out negative

3. Apply the junction rule to any junction in the circuit

4. Apply the loop rule to as many loops as are needed to solve for the unknowns

5. Solve the equations simultaneously for the unknown quantities. Check your answers

Kirchhoff’s Rules – Problem solving strategy

RC Circuit – Functionality

• A more complex behavior is expected when resistors are connected in the same

circuit with capacitors forming an RC dc-circuit: in these circuits, currents will be

unidirectional (dc), but their magnitudes will vary with time

ready for charging

ready for discharging

• When the RC circuit is completed is series with a battery

ε, the capacitor C starts to charge and the circuit is called

in charging regime

• Due to the presence of resistance R, the flow of charge is

slowed down, such that the capacitor will take time to

approach its maximum charge Q = Cε

• The capacitor builds a potential opposing the battery, so

the current i – which is initially ε/R when there is no

charge on the capacitor – decreases until the capacitor is

fully charged and the current in the circuit tends to zero

• Subsequently, if the battery is removed, the capacitor

will enter in a discharging regime gradually releasing its

charge through the resistor like a finite-charge reservoir

• Initially the capacitor drives the current ε/R opposite to

the current in the charging regime, but the current

decreases since the charge on the capacitor depletes

• Using Kirchhoff rules around the circuit, one ca obtain an

equation for how the charge on the capacitor increases with

time exponentially, tending to Qfinal= Cε:

• Consequently, the current through resistor decreases with

time from I0 = ε/R to zero:

1 t RC

finalq Q e

0

t RCi I e

• The constant = RC is called the time constant and represents the time required

for the charge to increase from zero to 63.2% of its maximum

• In a circuit with a large time constant, the capacitor charges and discharges slowly

charging…

RC Circuit – Charging regime

time constant

τ = RC

• When the battery is removed, the capacitor discharges and

it can be shown that the charge decreases exponentially with

time, from Q0 asymptotically to zero:

0

tq Q e

discharging…

0

ti I e 0 00

V QI

R RC

RC Circuit – Discharging regime

time

constant τ

maximum voltage across the capacitor

• The current decreases exponentially from

I0 (but in opposite direction than when

charging)

Problems:

7. Charging an RC circuit: Demonstrate that the time dependency of the charge on the plates

of a capacitor in an RC-circuit charged by a battery with emf ε is, indeed, given by

8. RC circuit: An RC circuit is connected to a battery with emf ε = 10 V. The capacitance is C

= 0.50 μF and the resistance is R = 4.0 MΩ.

a) Calculate the current and the amount of charge accumulated

in the capacitor at t = 0.20 s in the charging regime

b) How long will it take until the charge reaches half of its

maximum value

c) If the battery is removed (discharging regime), how long will

it take until the charge in the capacitor is half the maximum

charge

d) What is the current through the resistor at that moment?

1 1t RC t RC

fq C e Q e