Electrodynamics #1The Electric Current

HW #1 – 4 fits these notes.

I. Electric Current: {Start HW #1}

Introduction:

Our study of electricity so far has concentrated on static systems of charges. In other words, the charges were at rest. This new unit analyzes moving systems of charges.

The first task is to quantify the motion, or flow, of electric charge:

The flow of charge through some point in space is called the current. The concept of electrical current is very similar to the flow of (and currents in) water.

Current is represented by the letter I (or i).Q Q

I ort t

The current is defined as the amount of charge passing by a point in space divided by the elapsed time.

Units:

Current is given its own special unit name, the ampere (A). This is defined as follows:

Q coulombI ampere

t second 1 1

CA

s

Example #1: A wire carries a current of 2.50 mA. How many electrons pass through the wire each second?

32.50 10 A 32.50 10C

s

191.6022 10

each electron

C

161.56 10electrons

s

Driving Force:

For a current to exist, there must be some kind of energy difference between two points in space. Electric charges will move in response to the energy difference. Rather than using potential energy to define the motion of the charges, we use electric potential (V).

V = difference in electric potential between two points in space.

V = energy per unit charge available to push charges into motion, usually provided by a battery or generator.

V

load

connecting wire

The difference in electric potential pushes charges to flow. Direction of flow is based on motion of positive charges (Franklin’s convention).Positive charges flow from high potential (+) to low potential (–).

I

II. Response of Charge Flow: Resistance

When an electric potential is placed across some material, charges will flow through the material.

high V low V

How much charge moves depends on how easily charge can move through the substance. Define the resistance to charge flow as follows:

Vresistance R

I

For a given electric potential, the greater the current flow, the lower the resistance.

V = potential difference moving charges

I = current, flow of charges

R = relative difficulty for charge to flow

Units for Resistance:

Resistance is measured as electric potential per current:

Vresistance R

I

volts V

amperes A

This is given its own special unit name, called an ohm ().

1V

A

Georg Simon Ohm

Example #2: When a wire is connected to a 12.0 V battery, a current of 0.125 A is established in the wire. What is the resistance of the wire?

12.0

0.125

V VR

I A

96.0

Example #3: A certain insulating material has a resistance of 15.0 M. When an electric potential (or “voltage”) of 75.0 kV is placed across this device, how much current exists in the material?

V VR I

I R

3

6

75.0 10

15.0 10

V

35.00 10 5.00I A mA

Ohm’s Law and Resistance:

In general, the resistance of a material is not constant, but varies with temperature and current. A filament light bulb is an example of a device whose resistance changes with temperature and with current carried through the filament.

There are certain materials whose resistance is stable over current load and temperature, and these materials are called “ohmic”. Thus an ohmic material has a constant resistance. The equation for resistance is usually written:

IRV

resembles equation for line

Example #4: When an electric potential difference of 12.0 V is placed across a given load, the load pulls 0.250 A of current. (a) What voltage is required to put 3.25 A of current through the load?

A

V

I

VR

250.0

0.12

0.48

0.4825.3 AIRV V156

(b) What current will the material carry if the electric potential is set to 420 volts?

0.48

420V

R

VI A75.8

Homework: Read and Outline 18.1-18.3 Questions 2-5 Problems 1-11Text book Check Friday.

III. Resistance and Resistivity: {Start HW #2}

In general, resistance is an extrinsic property. Resistance depends on what material is being used to carry a current (for example, copper versus glass). Resistance also depends on the amount and shape of the material carrying the current. For example, five pieces of copper can have very different resistances due to differing amounts and shapes.

Resistivity is an intrinsic property that relates to resistance. Resistivity depends only on what type of material is used, and is not affected by amount or shape.

Five pieces of copper have the same density (intrinsic property) even though they may have different masses (extrinsic property).

The variable used to represent resistivity is the Greek letter “rho”:

resistivity

Resistance is related to resistivity and the shape of the material.

high V low VV

L

A

I

V = voltage difference across the material.

L = length of the material.

A = cross – sectional area of the material.

I = current = flow of charge through the material

R L1

RA

LR

A

The resistivity depends on what material is used, and the value is given on a table on the homework sheet.

Units of Resistivity:

The units of resistivity are given as:

L RAR

A L

2munit m

m

Example #5: (a) What is the resistance of a copper wire that is 0.100 mm in diameter and 12.0 m long?

81.69 10Cu m

2

L LR

A r

8

23

1.69 10 12.0

0.100 102

m m

m

25.8R

(b) What voltage difference must be put across this wire to establish a current of 0.340 mA?

30.340 10 25.8V IR A

38.78 10 8.78V V mV

Example #6: A parallel plate capacitor is made with two conducting parallel plates that have a diameter of 2.00 cm. The parallel plates are separated by a 0.0150 mm, and the space between the plates is filled with fused quartz (an insulator). Determine the resistance of the fused quartz.

mquartz 161075

22

316

1000.1

100150.01075

m

mm

A

LR

161058.3R

Example #7: A wire has a given cross sectional area Ao, length Lo, and resistance Ro. If this wire is cut into two identical lengths, and the two pieces are connected side by side, what will be the new resistance of this combination? Lo

Aocut

L= ½Lo

A=2Ao

LR

A

2 1

2 4

o

o

o o

LL

RA A

1

4 oR R

Electrodynamics #1The Electric Current

HW #1 – 4 fits these notes.

{day #2}

Example #8: A 1.00 gram lump of copper is pulled into a wire that has a resistance of 0.100 . Determine the length and diameter of this wire.

81.69 10Cu m 3

38.99 10 kgCu m

density D

LR

A

DLAVolDm

2LDA

LDLAmR

D

mRL

mkg

kgL

83

3

1069.11099.8

100.01000.1m811.0

Now solve for diameter:L

RA

R

LAr

2

100.0

811.01069.1 8

mm

R

Lr

mr 41009.2

mmrdiameter 418.02

IV. Temperature Dependence of Resistance and Resistivity:

{Start HW #3}

For most materials, heating the material causes the resistivity (and resistance) of the material to increase. This effect can be modeled with the following equation:

oo TT 1 oo TTRR 1

(or R) is the resistivity (or resistance) at some new temperature T

o (or Ro) is the resistivity (or resistance) at some starting temperature To. Usually the starting temperature is 20.0 °C, and the values given in the date table are calibrated to this starting temperature.

is the temperature coefficient of resistivity (or resistance)

Example #9: A copper wire has a resistance of 1.45 at 20.0 °C. What would be the resistance of this wire if it were heated to 165 °C?

CCu 13109.3

oo TTRR 1

CCR C 0.20165109.3145.1 13

27.2R

Example #10: A platinum wire has a resistance of 2.000 at 20.0 °C. What would be the temperature of the wire if the resistance was measured only to be 0.145 ?

CPt 131092.3 oo TTRR 1

oo

TTR

R 1

1o

o R

RTT

1

1

oo R

RTT

1

1

oo R

RTT

1000.2

145.0

1092.3

10.20

13C

CT

CT 217

he resistivity of platinum keeps a linear temperature response over a wide range of temperatures, making it ideal for thermometer work.

V. Power and Resistance: {Start HW #4}

When an electric current passes through an object, work must be done to move the charged particles through the material.

As the charges move through the material, they collide with the atoms of the material.

Each collision transfers energy from the charges to the atoms of the material.

This transfer of energy heats the resistive materials.

The transfer of heat from the charges to the resistive material is given in terms of power, or the rate at which energy is removed from the circuit.

V

The battery feeds power into the circuit, driving the charges around the circuit. This current passes through the resistive load.

R

The resistor saps energy from the current. Overall, energy is conserved. Energy in from the battery is taken out by the resistor.

The amount of power put in or taken out by an element of the circuit is given as: VIP

If Ohm’s law is substituted in, there are two more versions of the equation for power:

RIP 2

R

VP

2

Example #11: Show that the power equation does give units of power.

VIP

voltagecurrentpower voltampere

C

J

s

Cpower

s

J Wwatt

Example #12: An incandescent light bulb is rated at 100 W when run from a 120 V source. (a) What is the current through the light bulb?

VIP

V

PI

V

W

120

100 A833.0

(b) What is the resistance of the bulb? R

VP

2

P

VR

2

W

V

100

120 2

144

Example #13: An electric hot water heater is designed to pull 25.0 A from a 220 V source. (a) What is the power output of this water heater?

VIP VA 2200.25 kWW 50.5500,5

(b) How much time does it take to heat 200 kg of water from 20.0°C to 90.0°C?

t

TmC

time

energyP

Ckg

JheatspecificC

4186

t

TmCP

P

TmCt

W

CCkg

Jkg

t5500

0.704186200

W

Jt

5500

1086.5 7 s41007.1

hours96.2

(c) Energy is usually sold in units of kilowatt hours. Show that this is a unit of energy.

hourkilowattkWh 11

swattskWh 360010001

JkWh 6106.31

(d) If electrical energy costs $0.500 per kilowatt hour, how much does it cost to heat this volume of water?

From part (b), the energy needed is:

Jenergy 71086.5

cost = J71086.5

J

kWh6106.3

1

kWh1

500.0$14.8$

alternate = kW50.5 hours96.2

kWh1

500.0$14.8$

18.8 Microscopic View of Electric Current

18.9 Superconductivity

18.10 Electrical Conduction in the Human Nervous System

18.8 Microscopic View of Electric Current

Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

This drift speed is related to the current in the wire, and also to the number of electrons per unit volume.

(18-10)

Example 18-14. Electron Speeds on a Wire.A copper wire , 3.2 mm in diameter, carries a 5.0 Amp current. Determine the drift speed of the electrons. Assume that one electron per Cu atom is free to move.

“n” is the density of free electrons. This is equal to the density of Cu atoms. A is the cross sectional area. Vd is the drift velocity. See text book for complete calculation.

Calculation shows that the drift velocity of electrons is 4.7 X 10-5 m/sIf the distance bewtween the switch and the lights is 5 meters, how long will it take the lights to come on?

Time = Distance/ Velocity = 29 hours! What is wrong with our model??

29 hours is how long it would take for the electron at the switch to get to the light. The electric field is established at the speed of light.

“The garden hose is full, so the water at the spigot is not the first water out the hose.”

18.9 Superconductivity

In general, resistivity decreases as temperature decreases. Some materials, however, have resistivity that falls abruptly to zero at a very low temperature, called the critical temperature, TC.

18.9 Superconductivity

Experiments have shown that currents, once started, can flow through these materials for years without decreasing even without a potential difference.

Critical temperatures are low; for many years no material was found to be superconducting above 23 K.

More recently, novel materials have been found to be superconducting below 90 K, and work on higher temperature superconductors is continuing.

18.10 Electrical Conduction in the Human Nervous System

The human nervous system depends on the flow of electric charge.

The basic elements of the nervous system are cells called neurons.

Neurons have a main cell body, small attachments called dendrites, and a long tail called the axon.

Signals are received by the dendrites, propagated along the axon, and transmitted through a connection called a synapse.

This process depends on there being a dipole layer of charge on the cell membrane, and different concentrations of ions inside and outside the cell.

This applies to most cells in the body. Neurons can respond to a stimulus and conduct an electrical signal. This signal is in the form of an action potential.

The action potential propagates along the axon membrane as a wave.

Therapies for treating phantom sensations.

Problem Set: 49 – 53. Consult Book.49.Assume one cubic meter of copper.

Assume one electron per atom of copper. N is a mole. A is cross sectional area. Express the density in kg/cubic meter. Look up the atomic mass of copper.

50.Same assumption as 49 for part c.51.Use the drift velocity equation and sketch

the motion of both particles relative to the electric field.

52. Δ V = E Δx53. Use the Definition of velocity.