2/1/07 184 Lecture 15 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 15

Title: Direct Current

2/1/07 184 Lecture 15 2

AnnouncementsAnnouncementsAnnouncementsAnnouncements

Homework Set 4 is active and is due next Tuesday morning at 8:00 am

Today: Quick review of the material of the past 4 weeks and we will start with “Direct Currents”

Midterm 1 will take place in class next Thursday• Bring a calculator• Bring a no. 2 pencil• Bring your ID

2/1/07 184 Lecture 15 3

OutlineOutlineOutlineOutline

Review of Chapters 16 – 19 Introduction to Chapter 20 = Direct Current

2/1/07 184 Lecture 15 4

Review: Electrostatics (1)Review: Electrostatics (1)Review: Electrostatics (1)Review: Electrostatics (1)

Electric charge can be either positive or negative; like charges repel and unlike charges attract each other. An object with equal amounts of positive and negative charge is electrically neutral. The total charge of an isolated system is always conserved.

The electric force F between two charges, q1 and q2, separated by a distance r is given by Coulomb’s Law:

The constant k is called Coulomb’s constant and is given by

F kq1q2

r2F k

q1q2

r2

Opposite charges: F is attractive (-)Like charges: F is repulsive (+)

k 8.99 109 N m2

C2k 8.99 109

N m2

C2k

1

40

k 1

400 8.85 10 12

C2

N m20 8.85 10 12

C2

N m2

2/1/07 184 Lecture 15 5

Review: Electrostatics (2)Review: Electrostatics (2)Review: Electrostatics (2)Review: Electrostatics (2)

We may define the unit of charge in terms of the charge of one electron• An electron is an elementary particle with charge q =

-e where: e = 1.60210-19 C• A proton has the charge q = +e

2/1/07 184 Lecture 15 6

Clicker: ElectrostaticsClicker: ElectrostaticsClicker: ElectrostaticsClicker: Electrostatics

Calculate the magnitude of the force (in N) between a gold nucleus and an electron on an orbit with radius 4.88×10-12 m around the nucleus. The gold nucleus has a charge of +79e.

A) 7.7 10-4 N B) -1.56 10-3 N C) 8.9 10-5 N

k 8.99 109 N m2

C2k 8.99 109

N m2

C2

e = 1.60210-19 C

2/1/07 184 Lecture 15 7

Clicker: ElectrostaticsClicker: ElectrostaticsClicker: ElectrostaticsClicker: Electrostatics

Calculate the magnitude of the force (in N) between a gold nucleus and an electron on an orbit with radius 4.88×10-12 m around the nucleus. The gold nucleus has a charge of +79e.

A) 7.7 10-4 N

N107.7 142

21 r

qkqF

2/1/07 184 Lecture 15 8

Review: Electric Field (1)Review: Electric Field (1)Review: Electric Field (1)Review: Electric Field (1) The electric force on a charge q due to an

electric field E is given by

The electric field at any point is equivalent to the sum of all the sources of electric field at that point:

Electric field from a point charge:

The electric field points radially away from a positive charge and radially toward negative charges.

A system of two oppositely charged point particles is called an electric dipole. • p is the magnitude of the dipole moment• q is the magnitude of one of the opposite charges• d is the distance between the charges• p points from the negative to the positive charge

EqF

dqp

2/1/07 184 Lecture 15 9

Review: Electric Field (2)Review: Electric Field (2)Review: Electric Field (2)Review: Electric Field (2) The electric flux through a surface A is defined as

Gauss’ Law:

Gauss’ Law says that the electric flux through a closed surface is proportional to the net charge enclosed by this surface.

0 q0 q

E

20r

2kr

E

20

E

20

E 0

E 0

conducting wire Infinite non-conductingcharged sheet

Infinite conducting plane

The electric field inside a closed conductor is 0

qAdE

0

2/1/07 184 Lecture 15 10

Review: Electric Field (3)Review: Electric Field (3)Review: Electric Field (3)Review: Electric Field (3) The electric field inside a spherical shell of charge q is zero

The electric field outside a spherical shell of charge q is the same as the field from a point charge q

Electric field from charge distributed uniformly throughout a sphere of radius r

• r2 > r

• r1 < r

E 1

40

q

r2E

1

40

q

r2

E r1 qtr140r

3

kqtr1r3

E r1 qtr140r

3

kqtr1r3

2/1/07 184 Lecture 15 11

Review - Potential Energy Review - Potential Energy Review - Potential Energy Review - Potential Energy

When an electrostatic force acts on charged particles, assign an electric potential energy, U

If the system is changed from initial state i to the final state f, the electrostatic force does work, W

The change in electric potential energy is U is equal to the charge q times the change in electric potential V, U=qV

Equipotential surfaces (or lines) represent adjacent points in space that have the same potential.

Calculate the change in the electric potential from the electric field by integrating the field in a particular direction,

2/1/07 184 Lecture 15 12

Review - Electric PotentialReview - Electric PotentialReview - Electric PotentialReview - Electric Potential Taking the convention that the electric potential is zero at infinity

we can express the electric potential in terms of the electric field as

Calculate the electric field from gradients of the electric potential in each component direction

The electric potential from a point charge q at a distance r is given by:

The electric potential can be expressed as an algebraic sum of all sources of electric potential

V Vii1

n

kqi

rii1

n

Ex V

x; Ey

V

y; Ez

V

z

V kq

rV

kq

r

In particular for a system of point charges:

2/1/07 184 Lecture 15 13

Clicker - Electric Field Clicker - Electric Field Clicker - Electric Field Clicker - Electric Field

F kq1q2

r2F k

q1q2

r2 Like charges: F is repulsive (+)

BC

A

Use:=qt/(Volume of sphere)

Volume=4/3R3

r1<R

2/1/07 184 Lecture 15 14

Clicker - Electric Field Clicker - Electric Field Clicker - Electric Field Clicker - Electric Field

F kq1q2

r2F k

q1q2

r2 Like charges: F is repulsive (+)

BC

A

Use:=qt/(Volume of sphere)

Volume=4/3R3

X

r1<R

2/1/07 184 Lecture 15 15

Review: Capacitance (1)Review: Capacitance (1)Review: Capacitance (1)Review: Capacitance (1)

The definition of capacitance is

The capacitance of a parallel plate capacitor is given by• A is the area of each plate• d is the distance between the plates

The capacitance of a spherical capacitor is

• r1 is the radius of the inner sphere• r2 is the radius of the outer sphere

C q

VC

q

V

C 0A

dC

0A

d

C 40

r1r2

r2 r1C 40

r1r2

r2 r1

2/1/07 184 Lecture 15 16

Review: Capacitance (2)Review: Capacitance (2)Review: Capacitance (2)Review: Capacitance (2)

The capacitance of an isolated spherical conductor is

The capacitance of a cylindrical capacitor is

Placing a dielectric between the plates of a capacitor increase the capacitance by

The electric potential energy stored in a capacitor is given by

C q

V

L

20

ln r2 / r1

20L

ln r2 / r1 C

q

V

L

20

ln r2 / r1

20L

ln r2 / r1

C 40RC 40R

C CairC Cair

U 1

2CV 2U

1

2CV 2

2/1/07 184 Lecture 15 17

Review: Capacitance (3)Review: Capacitance (3)Review: Capacitance (3)Review: Capacitance (3)

The equivalent capacitance for n capacitors in parallel is

The equivalent capacitance for n capacitors in series is

Ceq Cii1

n

Ceq Cii1

n

1

Ceq

1

Cii1

n

1

Ceq

1

Cii1

n

2/1/07 184 Lecture 15 18

Electric Current

Nature is simple – we can understand it and in some ways even control it.

That is the origin of technology.

The human race has developed a remarkable technology of electric current..

2/1/07 184 Lecture 15 19

Direct CurrentDirect CurrentDirect CurrentDirect Current

We will study charges in motion. Electric charge moving coherently from one

region to another is called electric current. Current is flowing through light bulbs, iPods,

and lightning strikes. Current usually consists of mobile electrons

traveling in conducting materials. Direct current is defined as a current that flows

only in one direction in the conductor.

Most of our electric technology is based on Alternating Current – Chapter 24

2/1/07 184 Lecture 15 20

Electric CurrentElectric CurrentElectric CurrentElectric Current

We define the electric current i as the net charge passing a given point in a given time.

Random motion of electrons in conductors, or the flows of electrically neutral atoms, are not electric currents in spite of the fact that large amounts of charge are moving past a given point.

If net charge dq passes a point in time dt we define the current i to be

i dq

dti

dq

dt

2/1/07 184 Lecture 15 21

Electric Current (2)Electric Current (2)Electric Current (2)Electric Current (2)

The amount of charge q passing a given point in time t is the integral of the current with respect to time given by

We will use charge conservation, implying that charge flowing in a conductor is never lost.

Therefore the same amount of charge must flow through one end of the conductor as the charge that exits from the other end of the conductor.

q dq idt0

t

q dq idt0

t

2/1/07 184 Lecture 15 22

The AmpereThe AmpereThe AmpereThe Ampere

The unit of current is coulombs per second, which has been given the unit ampere, named after French physicist André-Marie Ampère,  (1775-1836)

The ampere is abbreviated as A and is defined by

Some typical currents are• Flashlight - 1 A• The starter motor in a car - 200 A• iPod - 50 mA• In a lightning strike (for a short time) - 100000 A

1 A 1 C

1 s1 A

1 C

1 s

2/1/07 184 Lecture 15 23

BatteriesBatteriesBatteriesBatteries

We use of batteries as devices that provide direct currents in circuits.

If you examine a battery, you will find its voltage written on it.

This voltage is the potential difference it can provide to a circuit.

You will also find their ratings in units of mAh. This rating provides information on the total charge

that they can deliver when fully charged. The quantity mAh is another unit of charge:

1 mAh (10 3 A)(3600 s) 3.6 As 3.6 C

2/1/07 184 Lecture 15 24

2/1/07 184 Lecture 15 25

The flow of electrons is always from anode—to--cathode outside of the cell (i.e., in the circuit) and from cathode—to--anode inside the cell. Inside a chemical cell, ions are carrying the electrons from cathode—to--anode inside the cell.

Anode (negative terminal): Zinc powderCathode (positive terminal): Manganese dioxide (MnO2) powderElectrolyte: Potassium hydroxide (KOH)

The half-reactions are:At the cathode…2 MnO2 + H2O + 2 e- —>Mn2O3 + 2 OH-At the anode…Zn + 2 OH- —> ZnO + H2O + 2 e- The overall reaction is:Zn + 2MnO2 —> ZnO + Mn2O3 + [E=1.5 V]

2/1/07 184 Lecture 15 26

Alkaline battery Al Kaline batter

2/1/07 184 Lecture 15 27

CurrentCurrentCurrentCurrent

Current is a scalar. Current has a sign but not a direction. We will represent the direction of the current flowing in

a conductor using an arrow. This arrow represents whether the net current is

positive or negative in a conductor at a given point but does not represent a direction in three dimensions.

Physically, the charge carriers in a conductor are electrons that are negatively charged.

However, as is conventionally done, we define positive current as the net flow of positive charge carriers past a given point per unit time.

2/1/07 184 Lecture 15 28

Circuits

In this circuit, electrons flow around the circuit counterclockwise. (The conventionally defined current is clockwise; remember, electrons are negative charges.) The electrons can’t disappear so the current requires a whole loop!

Chemical action pumps electrons from the positive terminal (+) to the negative terminal () in the battery.The emf (electromotive force, or electric field) pushes electrons around the wire from () to (+).

Ohm’s Law V = I R

+

2/1/07 184 Lecture 15 29

Current DensityCurrent DensityCurrent DensityCurrent Density Let’s consider current flowing in a conductor. Taking a plane through the conductor, we can

define the current per unit area flowing through the conductor at that point as the current density J

We take the direction of J as the direction of the velocity of the charges crossing the plane.

If the cross sectional area is small, the magnitude of J will be large.

If the cross section area is large, the magnitude of J will be small.

J

2/1/07 184 Lecture 15 30

Current Density (2)Current Density (2)Current Density (2)Current Density (2)

The current flowing through the surface is

… where dA is the differential area elementperpendicular to the surface.

If the current is constant andperpendicular to the surface, thenand we can write an expression forthe magnitude of the current density

J i

AJ

i

A

AdJi