Lesson 4Circuits and Resistance

Class 9Today we will:• learn about current, voltage, and power in circuits.• learn about resistance of materials and how resistance depends on geometry and temperature.• introduce Ohm’s law.

Current, Voltage, and Power in Simple Circuits

Current• Benjamin Franklin didn’t know if current was caused by positive charges moving or negative charges moving, so he took a guess… and got it wrong.

Current

• Current was defined as the direction positive charge in a wire would travel.

Current• In reality, negative charges are moving in the opposite direction to the current.

Current

• However, we usually ignore that and talk about “positive charge carriers” in a wire that move in the direction of the current.

Current

• Current is the charge that moves past a point in the wire per unit time.

dt

dq

t

Nei

Current

• Units of current are amperes or amps.

•

• Car batteries deliver several hundred amperes. • Most electronic circuits run on a few mA.

sec

coulomb1ampere1

Voltage

•Voltage is electric potential in circuits.

•Voltage is provided by batteries or generators.

•A battery pushes electrons out the negative terminal and sucks electrons into the positive terminal.

Voltage

•If we attach a wire to the positive terminal, a few go into the battery, leaving a positive charge on the surface of the wire.

•The electrons stop moving when the surface charge on the wire pulls the electrons in the wire with the same force as the positive charge on the battery.

+ + + + + + + + + + + + +

Circuits

•If we connect a wire from the positive to the negative terminal of the battery, current will continue to flow through the circuit.

+

+

–

–– – – – –

– – – – + + + +

++ + + + +

Circuits

•Charge remains on the surface of the wire. The surface charge is positive near the positive terninal and negative near the negative terminal.

•The charge density is greatest near the terminals of the battery.

•Current flows uniformly through the entire cross-section of the wire.

+

+

–

–– – – – –

– – – – + + + +

++ + + + +

Circuits

• A 10V battery gives 10eV of energy to each electron that passes through it.

• Collisions with atoms in the wire cause each electron to lose 10eV of energy every time it goes around the circuit.

VqU

+

+

–

–– – – – –

– – – – + + + +

++ + + + +

Ground

•The ground acts like a huge conductor.

•Current can flow into the ground or out of the ground without any limits.

•The two circuits below are equivalent.

+ +

Circuits

Definitions:

•An open circuit is one where there is an open switch or a broken wire so that no current flows.

•A closed circuit is one in which there is a continuous path for current to flow from positive to negative.

•A short circuit is one where there is an unintentional current path to ground. Currents, sometimes large, flow where they should not, leading to shock and fire hazards.

Circuits

What good does flowing charge do?

•Produces heat, light.

•Produces magnetic fields – used in motors, vibrators, etc. to give mechanical power.

•Produces electromagnetic radiation – radio waves for communication.

•Electronics: amplification, logic, light detection, radiation detection, cathode-ray tubes, etc.

Power

Each time an electron goes through a battery, it gains energy. The total energy gained per second is:

iVe

ieV

U

UP

1

electronpercharge

1

second

chargeelectroneachof

second

electronsof#electroneachof

P=iV

Where does the energy go?

•Electrons collide with other electrons in atoms and quickly reach terminal velocity – so they don’t keep gaining kinetic energy.

Where does the energy go?

P=iV• In a wire, it goes to heat.

•In other devices it can go to light, mechanical energy, energy of radiation fields, etc.

Resistance

Resistance in a Wire

Definition:

•In general R is a function of I, and V.

•For many materials R is nearly a constant.

•When R is a constant, we call the material “ohmic.”

I

VIVR ),(

Resistors•Devices to increase the resistance in part of a circuit.

•Made of graphite chunks, wire wound around a core, etc.

•Used to

•Produce heat or light

•Adjust current flow and voltages in circuits.

Resistors

• Even if we don’t want the resistance, we often need to account for resistance in cables, electronic devices, etc.

Ohm’s Law

We assume resistors have constant V.

IRV

Resistance has units of ohms, written as an upper case omega.

A

V

1

11

Typical resistances range from a few ohms to several megohms.

Graphite Resistors•Resistance is color coded. 0

1

2

3

4

5

6

7

8

9

5%

10%

20%

5 6 2 10%

seco

nd d

igit

# of

zer

os

first

dig

it

tole

ranc

e

R=5600 Ω (+/-10%)

What Affects Resistance?

•Material

•Length

•Cross-sectional area

•Temperature

Resistance and Geometry

• One block has V, I, R.

Resistance and Geometry

• Take two blocks with I going through each.

• Voltage is Current is

• Resistance is

VV 2

RI

V

I

VR 2

2

II

Resistance and Geometry

• Take two blocks with I going through each.

• Voltage is Current is

• Resistance is

V

22

R

I

V

I

VR

I2

Resistance and Resistivity

AR

• ρ is the resistivity. It depends on the material from which the resistor is made. The units of resistivity are Ωm.

• σ = 1/ ρ is the conductivity

Resistance and Temperature

bTaR

We assume that resistance varies linearly with temperature.

Resistance and Temperature

00 1 TTRR

If T = T0, then R =R0.

Resistance and Temperature

00 1 TTRR

• α is the temperature coefficient of resistivity (resistance).

• α is usually positive.

• α is negative for graphite.

Class 10Today we will:• learn how to determine if two resistors are in series or parallel.• find out how resistors combine when connected in series and parallel.• work examples of series-parallel reduction to find current, voltage and power in resistance networks.

Resistors in SeriesHave the Same Current

• Take two resistors with I going through each.

• Voltage is Current is

Resistance is

21 VVV

212121 RRI

V

I

V

I

VV

I

VR

II

Resistors in Series

21 VVV

11,, RVI 22 ,, RVI

RVI ,,

21 RRR

Resistors in ParallelHave the Same Voltage

21 III VV

• Take two blocks with I going through each.

• Voltage is Current is

Resistance is

21

1121 111

RRV

I

V

I

V

II

V

I

R

Resistors in ParallelHave the Same Voltage

RVI ,,

11 ,, RVI

22 ,, RVI

21 III

21

111

RRR

A Test for Resistors in Series

Look at the wire connecting the two resistors. Is there any junction between the resistors?

The resistors are connected in series.

The resistors are NOT connected in series.

no yes

A Test for Resistors in Parallel

Look at the wire connecting one end of the first resistor to one end of the second resistor. Is there a circuit element (a junction is OK and usually there are junctions) along this wire?

The resistors are NOT connected in parallel.

The resistors are connected n parallel.

yes no

Look at the wire connecting the other end of the first resistor to the other end of the second resistor. Is there a circuit element along this wire?

The resistors are NOT connected n parallel.

yes no

Series-Parallel QuizAnswer the following six

questions to see if you understand what

series and parallel mean.

Resistors A and B are in

1. series

2. parallel

3. neither

Resistors A and B are in

1. series

2. parallel

3. neither

Resistors A and B are in

1. series

2. parallel

3. neither

Resistors A and B are in

1. series

2. parallel

3. neither

Resistors A and B are in

1. series

2. parallel

3. neither

Resistors A and B are in

1. series

2. parallel

3. neither

Quiz Answers1. series2. neither3. neither4. parallel5. series 6. parallel

Series- Parallel Reduction

• Find a combination in series or parallel.

• Combine resistors into a single equivalent resistor.

•Repeat until there is only one resistor.

•The voltage across the resistor is the same as the voltage across the battery.

Series- Parallel Reduction

• Find V, I, R, P for the last step.

•Bootstrap your way back to the beginning, diagram by diagram.

• What if there are resistors that aren’t in series or parallel?

--- You’ll need to use Kirchhoff’s Laws which we’ll learn later.

Now we’ll work some examples…

Find all the currents, voltages, and powers

2A

24V,2A

2A

2A

2A

2A, 20V

2A

2A, 4V

2A

2A, 20V

4V

4V

2A

2A, 20V

4V, 2/3 A

4V, 4/3 A

2A

4V, 4/3 A

2/3 A 2/3 A

2A2A

2A

4V, 4/3 A

2/3 A, 8/3 V 2/3 A, 4/3 V

2A, 16V2A, 4V

2A

4V, 4/3 A

2/3 A, 8/3 V 2/3 A, 4/3 V

2A, 16V2A, 4V

16/9 W 8/9 W

16/3 W

8 W32 W

48 W

Using Meters

• Ammeters measure current. they must be paced in series with other circuit elements so current flows through them. Ammeters should have very small voltage.

• Voltmeters measure voltage. To measure the voltage between two points, you connect the two leads of the meter to those points. Therefore voltmeters are placed in parallel. Voltmeters should have large voltage.

Real Batteries

• Real batteries have internal resistance. When they are placed in a circuit we can represent them as a resistor in series with an ideal battery.

There’s a voltage drop across the internal resistance.

This means that the full voltage of the ideal battery isn’t available to the circuit.

Class 11Today we will:• discuss Kirchhoff’s loop and node equations.• learn how to determine the number of loop and the number of node equations we will need.• write Kirchhoff’s equations for a sample circuit.

Kirchhoff’s Junction Rule

• Current into a junction equals current out of a junction.

• Comes from conservation of charge.

Kirchhoff’s Junction Rule

• It’s like water in pipes – the water flowing into a junction must flow out again.

Kirchhoff’s Loop Rule• The net change in voltage around a closed loop is zero.

• Comes from Conservation of energy

0V

Kirchhoff’s Loop Rule• It’s like moving a ball around any closed path, the change in gravitational potential energy is zero.

0U

Applying Kirchhoff’s LawsThis is a “turn the crank” approach – but it works well.

1.Mark each junction.

A

B

C

D

2. Label all currents – between each pair of junctions. Choose a direction – it doesn’t have to be right!

Here there are 6 currents.

I2

I1

I3

I4 I5

I6

A

B

C

D

3. Number of junction equations: If there are N junctions, there are N-1 junction equations.

Here there are 4 junctions, so there are 3 junction equations.

I2

I1

I3

I4 I5

I6

A

B

C

D

4. Write the junction equations:

Current in = Current out.

I2

I1

I3

I4 I5

I6

652

241

163

:

:

:

IIIC

IIIB

IIIA

A

B

C

D

I2

I1

I3

I4 I5

I6

652

241

163

:

:

:

IIIC

IIIB

IIIA

5. Number of loop equations: Number of currents – Number of junction equations.

Here: 6-3=3, so we need 3 loops.

A

B

C

D

I2

I1

I3

I4 I5

I6

652

241

163

:

:

:

IIIC

IIIB

IIIA

6. At least one loop must cover every circuit element.

A

B

C

D

12

3

I2

I1

I3

I4 I5

I6

652

241

163

:

:

:

IIIC

IIIB

IIIA

7. Put a plus or minus on every resistor (side current goes in is +) and battery (+ is positive terminal).

A

B

C

D

12

3

+

+

+

+

+

+

+

+

I2

I1

I3

I4 I5

I6

652

241

163

:

:

:

IIIC

IIIB

IIIA

8. Write the loop equations.

A

B

C

D

12

3

+

+

+

+

+

+

+

+

0546:3

076623:2

04126:1

245

6536

413

III

IIII

III

I2

I1

I3

I4 I5

I6

8. Write the loop equations.

A

B

C

D

12

3

+

+

+

+

+

+

+

+

When you go around the loop, ignore the currentarrows! Follow the loop in the direction of the loop arrow!

652

241

163

:

:

:

IIIC

IIIB

IIIA

8. Solve the system of equations using your favorite method. I’ll usually just ask for the equations.

0546:3

076623:2

04126:1

245

6536

413

III

IIII

III