Chapter 20 – 23

ELECTRICITY, ELECTRICITY, CIRCUITS, AND CIRCUITS, AND EVERYTHING ELSE EVERYTHING ELSE IN BETWEENIN BETWEEN

Introduction

The field of electromagnetism is as far-reaching as the field of mechanics. Electromagnetism is manifested in many ways in our daily lives,

both seen and unseen.Observed electric and magnetic phenomena as early

as 700 BC. Found that amber, when rubbed, became electrified and

attracted pieces of straw or feathers. Electric comes from the Greek word elecktron, which is amber.

James Clerk Maxwell (1831 – 1879) is a key figure in the development of the field of electromagnetism. His contributions to the field of electromagnetism is comparable

to Newton’s contribution to the field of mechanics.

Properties of Electric Charges

Two types of charges exist: They are called positive

and negative.Like charges repel. Unlike charges attract

one another.Nature’s basic carrier

of positive charge is the proton. Protons do not move from

one material to another because they are held firmly in the nucleus.

Properties of Charge

Nature’s basic carrier of negative charge is the electron. Gaining or losing electrons is how an object

becomes charged.Electric charge is always conserved.

Charge is not created, only exchanged. Objects become charged because negative charge

is transferred from one object to another. This is just like conservation of energy.

Properties of Charge

Charge is quantized, meaning charge occurs in discrete bundles. All charge is a multiple of a fundamental unit of

charge, symbolized by e. Electrons have a charge of –e. Protons have a charge of +e. e = 1.6 x 10-19 C The SI unit of charge is the Coulomb (C).

Creating A Charge

When two unlike materials are rubbed charge has a tendency to be transferred. A negative charge is transferred in the form of electrons

passing between the materials being rubbed. The triboelectric series gives us an idea of which

materials will be more likely to gain electrons and which is more likely to lose electrons.

This phenomenon occurs to some degree when almost any two non-conducting substances are rubbed.

This phenomenon is what causes static electricity.

Example: Creating A Charge

Rubbing a glass rod on a silk cloth.

Glass is higher up on the triboelectric series and is more likely to be positive and thus gives up its electrons to the silk which is lower down on the series.

Conductors

Conductors are materials in which the electric charges move freely. Copper, aluminum

and silver are good conductors.

When a conductor is charged in a small region, the charge readily distributes itself over the entire surface of the material.

Insulators

Insulators are materials in which electric charges do not move freely. Glass and rubber are

examples of insulators.

When insulators are charged by rubbing, only the rubbed area becomes charged. There is no tendency

for the charge to move into other regions of the material.

Charging by Conduction

Conduction is the process by which charge is transferred from one object to another.

A charged object (the rod) is placed in contact with another object (the sphere).

Some electrons on the rod can move to the sphere.

When the rod is removed, the sphere is left with a charge.

The object being charged is always left with a charge having the same sign as the object doing the charging.

Charging by Induction

Induction is the process by which a charge is created but the objects involved do not touch one another.

A neutral sphere has equal number of electrons and protons.

A negatively charged rubber rod is brought near an uncharged sphere.

A grounded conducting wire is connected to the sphere.

The wire to ground is removed, the sphere is left with an excess of induced positive charge.

Coulomb’s Law

Coulomb shows that an electrical force has the following properties: It is directed along the line

joining the two particles and inversely proportional to the square of the separation distance, r, between them.

It is proportional to the product of the magnitudes of the charges, |q1|and |q2|on the two particles.

It is attractive if the charges are of opposite signs and repulsive if the charges have the same signs. 1736 – 1806

Mathematical Expression of Coulomb’s Law

Mathematically:

ke is called the Coulomb Constant: ke = 8.987 5 x 109 N m2/C2

Typical charges can be in the µC range.Remember that force is a vector quantity.Applies only to point charges.The distance between the charges (r) is

fixed.

2

21e r

qqkF

Characteristics of Particles

Vector Nature of Electric Forces

Two point charges are separated by a distance r.

The like charges produce a repulsive force between them.

The force on q1 is equal in magnitude and opposite in direction to the force on q2.

The unlike charges produce a attractive force between them.

The force on q1 is equal in magnitude and opposite in direction to the force on q2.

Therefore, electrical forces also adhere to Newton’s 3rd law.

Electrical Forces are Field Forces

This is the second example of a field force. Gravity was the first.

Remember, with a field force, the force is exerted by one object on another object even though there is no physical contact between them.

There are some important similarities and differences between electrical and gravitational forces.

Electrical Force Compared to Gravitational Force

Both are inverse square laws.The mathematical form of both laws is the same:

Masses replaced by charges G replaced by ke

Electrical forces can be either attractive or repulsive.

Gravitational forces are always attractive.Electrostatic force is stronger than the gravitational

force.

2

21e r

qqkF 2

21

r

mmGF

Electrical Field

Field forces, like electric and gravitational forces, are capable of acting over a given space. These forces are able to produce an effect even when

there is no physical contact.Faraday developed an approach to discussing

fields. In this approach forces are said to exert a field in a

region of space around an object (gravitational field) or a charge (electrical field).

An electric field is said to exist in the region of space around a charged object. When another charged object enters this electric field,

the field exerts a force on the second charged object.

Electric Field

A charged particle, with charge Q, produces an electric field in the region of space around it.

A small test charge, qo, placed in the field, will experience a force.

Electric Field

Mathematically,

SI units are N / C.Use this for the magnitude of the field.The electric field (E) is a vector quantity.The direction of the field is defined to be the

direction of the electric force that would be exerted on a small positive test charge placed at that point.

2e

o

k Qq r

F

E

Direction of Electric Field: “Negative Field”

The electric field produced by a negative charge is directed toward the charge. A positive test

charge would be attracted to the negative source charge.

Direction of Electric Field: “Positive Field”

The electric field produced by a positive charge is directed away from the charge. A positive test

charge would be repelled from the positive source charge.

Important Final Note About Electric Fields

An electric field at any given point depends only on the charge (q) of the object setting up the field and the distance (r) from that object to a specific point in space.

2e

o

k Qq r

F

E

Sample Problem

An electric field is measured using a positive test charge of 3.0x10-6 C. This test charge experiences a force of 0.12 N at an angle of 15o north of east. What are the magnitude and direction of the electric field strength at the location of the test charge?

Electric Field Lines

A convenient aid for visualizing electric field patterns is to draw lines pointing in the direction of the field vector at any point.

These are called electric field lines and were introduced by Michael Faraday.

The field lines are related to the field in the following manners: The electric field vector is tangent to the electric field

lines at each point. The number of lines per unit area through a surface

perpendicular to the lines is proportional to the strength of the electric field in a given region.

Electric Field Line Patterns

The lines radiate equally in all directions from a positive charge toward a negative charge. Radiates in 3

dimensions.For a positive source

charge, the lines will radiate outward.

For a negative source charge, the lines will point inward.

Electric Field Line Patterns #1

An electric dipole consists of two equal and opposite charges.

The high density of lines between the charges indicates the strong electric field in this region.

Electric Field Line Patterns #2

Two equal but like point charges.

At a great distance from the charges, the field would be approximately that of a single charge of 2q.

The bulging out of the field lines between the charges indicates the repulsion between the charges.

The low field lines between the charges indicates a weak field in this region.

Electric Field Line Patterns #3

Unequal and unlike charges.

Note that two lines leave the +2q charge for each line that terminates on –q.

Lines leaving the greater (+) charge greater than the lines arriving at the (-) charge.

At a great distance from the charges, the field would be approximately that of a single charge of q.

Conductors in Electrostatic Equilibrium

When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium.

An isolated conductor has the following properties: The electric field is zero everywhere inside the

conducting material. Any excess charge on an isolated conductor resides

entirely on its surface. The electric field just outside a charged conductor is

perpendicular to the conductor’s surface. On an irregularly shaped conductor, the charge

accumulates at locations where the radius of curvature of the surface is smallest (that is, at sharp points).

ELECTRIC ENERGYELECTRIC ENERGY

AND CAPACITANCEAND CAPACITANCE

Review of Potential Energy

PE is the energy associated with the position of an object within a system relative to some reference point.

PE is defined for a system rather than an individual object.

PE can also be used to do work or be converted to KE. Conservation of energy W = ∆KE W = -∆PE

Electric Potential Energy

The electrostatic force is a conservative force. Work done by the force is path independent.

It is possible to define an electrical potential energy function with this force. This PE shares the same characteristics as other PE.

Work done by a conservative force is equal to the negative of the change in potential energy. W = -∆PE

Uniform Electric Field

Two metal plates possessing charges equal in size but opposite in charge.

Potential Difference

The potential difference (∆V) between two points A and B is defined as the change in the potential energy (final value minus initial value) of a charge q moved from A to B divided by the size of the charge.

ΔV = VΔV = VBB – V – VAA = ΔPE / q = W/q = ΔPE / q = W/q

Potential Difference

Another way to relate the energy and the potential difference: W= ΔPE = q ΔV

Both electric potential energy and potential difference are scalar quantities.

Units of potential difference: volts V = J/C

A special case occurs when there is a uniform electric field: V = VB – VA= -Ed

Gives more information about units: N/C = V/m

Potential Energy vs. Potential Difference

Potential difference is not the same as potential energy.

Electric potential is characteristic of the field only. Independent of any test charge that may be

placed in the field.Electric potential energy is

characteristic of the charge-field system. Due to an interaction between the field and

the charge placed in the field.

Energy and Charge Movements

A positive charge gains electrical potential energy when it is moved in a direction opposite the electric field.

If a charge is released in the electric field, it experiences a force and accelerates, gaining kinetic energy. As it gains kinetic energy, it loses an equal

amount of electrical potential energy.A negative charge loses electrical

potential energy when it moves in the direction opposite the electric field.

Energy and Charge Movements

When the electric field is directed downward, point B is at a lower potential than point A.

A positive test charge that moves from A to B loses electric potential energy.

It will gain the same amount of kinetic energy as it loses in potential energy.

Summary of Positive Charge Movements and Energy

When a positive charge is placed in an electric field: It moves in the direction of the field. It moves from a point of higher potential

to a point of lower potential. Its electrical potential energy decreases. Its kinetic energy increases.

Summary of Negative Charge Movements and Energy

When a negative charge is placed in an electric field: It moves opposite to the direction of the

field. It moves from a point of lower potential

to a point of higher potential. Its electrical potential energy increases. Its kinetic energy increases. Work has to be done on the charge for it

to move from point A to point B.

Potentials and Charged Conductors

Since W = -q(VB – VA), no work is required to move a charge between two points that are at the same electric potential. W = 0 when VA = VB

All points on the surface of a charged conductor in electrostatic equilibrium are at the same potential.

Therefore, all points on the surface of a charged conductor in electrostatic equilibrium are at the same potential.

Conductors in Equilibrium

The conductor has an excess of positive charge.

All of the charge resides at the surface.

E = 0 inside the conductor.The electric field just outside the

conductor is perpendicular to the surface.

The potential is a constant everywhere on the surface of the conductor.

The potential everywhere inside the conductor is constant and equal to its value at the surface.

Capacitance

A capacitor is a device used in a variety of electric circuits.

The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor (plate) to the magnitude of the potential difference between the conductors (plates).

Capacitance

Units: Farad (F) 1 F = 1 C / V A Farad is very large

Often will see µF or pF

V is the potential difference across a circuit element or device.

Q is the magnitude of the charge on either conductor/plate.

V

QC

Parallel-Plate Capacitor

The capacitance of a device depends on the geometric arrangement of the conductors.

For a parallel-plate capacitor whose plates are separated by air:

d

AC o

Parallel-Plate Capacitor

The capacitor consists of two parallel plates.

Each have area A.They are separated by a distance

d.The plates carry equal and

opposite charges.When connected to the battery,

charge is pulled off one plate and transferred to the other plate.

The transfer stops when Vcap = vbattery.

Sample Problem

A parallel plate capacitor has an area of A = 2.00 cm2 (2.00x104 m2) and a plate separation of d=1.00 mm. Find the capacitance.

Capacitors in Circuits

A circuit is a collection of objects usually containing a source of electrical energy (such as a battery) connected to elements that convert electrical energy to other forms. These objects can be placed in series or in

parallel arrangements. The equivalent capacitance and the equivalent

potential difference can be determined in each case.

A circuit diagram can be used to show the path of the real circuit.

Capacitors in Parallel

The total charge is equal to the sum of the charges on the capacitors. Qtotal = Q1 + Q2

The potential difference across the capacitors is the same. And each is equal to the

voltage of the battery. ∆V = ∆V1 = ∆V2

More About Capacitors in Parallel

The capacitors can be replaced with one capacitor with a capacitance of Ceq. The equivalent capacitor must have

exactly the same external effect on the circuit as the original capacitors.

Ceq = C1 + C2 + …

The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors.

Sample Problem

Determine the capacitance of the single capacitor that is equivalent to the parallel combination of capacitors shown in the figure and the charge on the 12.0 µF capacitor.

Capacitors in Series

When in series, the capacitors are connected end-to-end.

The magnitude of the charge must be the same on all the plates.

Qeq = Q1 = Q2

More About Capacitors in Series

An equivalent capacitor (Ceq)can be found that performs the same function as the series combination.

The potential differences add up to the battery voltage. ∆V = ∆V1 + ∆V2 + …

Sample Problem

Four capacitors are connected in series with a battery, as in the figure. (a) Find the capacitance of the equivalent capacitor. (b) Find the charge on the 12 µF capacitor.

Capacitor Combinations: A Summary

Parallel Capacitors

Series Capacitors

Charge Qtotal = Q1 + Q2

Qeq = Q1 = Q2

Potential Difference

∆V = ∆V1 = ∆V2 ∆V = ∆V1 + ∆V2

CapacitanceCeq = C1 + C2 + …

Sample Problem

Find the equivalent capacitance between a and b for the combination of capacitors shown in the figure. All capacitors are in µF.

CURRENT AND CURRENT AND RESISTANCERESISTANCE

Electric Current

The current is the rate at which the charge flows through a surface. Look at the charges flowing perpendicularly

through a surface of area A:

The SI unit of current is Ampere (A) 1 A = 1 C/s

av

QI

t

Electric Current

The direction of the current is the direction positive charge would flow. This is known as conventional current

direction. In a common conductor, such as copper,

the current is due to the motion of the negatively charged electrons.

It is common to refer to a moving charge as a mobile charge carrier. A charge carrier can be positive or

negative.

Circuits

A circuit is a closed path of some sort around which current circulates.

A circuit diagram can be used to represent the circuit.

Quantities of interest are generally current and potential difference.

Resistance

In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor.

The constant of proportionality is the resistance of the conductor:

VR

I

Resistance

Units of resistance are ohms (Ω): 1 Ω = 1 V / A

Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor:

Ohm’s Law

Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents.

This statement has become known as Ohm’s Law: ΔV = I R A simple rearrangement of the resistance equation.

Ohm’s Law is an empirical relationship that is valid only for certain materials. Materials that obey Ohm’s Law are said to be

ohmic.

Sample Problem

All electric devices are required to have identifying plates that specify their electrical characteristics. The plate on a certain steam iron states that the iron carries a current of 6.4 A when connected to a 120 V source. What is the resistance of the steam iron?

Ohmic vs. Nonohmic

Electrical Energy in a Circuit

In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV. The chemical potential energy of the battery

decreases by the same amount.As the charge moves through a resistor, it

loses this potential energy during collisions with atoms in the resistor. The temperature of the resistor will increase.

Energy Transfer in the Circuit

As the charge moves through the resistor, from C to D, it loses energy in collisions with the atoms of the resistor.

The energy is transferred to internal energy .

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.

Electrical Energy and Power

The rate at which the energy is lost is the power:

From Ohm’s Law, alternate forms of power are:

QV I V

t

22 V

I RR

Electrical Energy and Power

The SI unit of power is Watt (W). I must be in Amperes, R in ohms and V in Volts.

The unit of energy used by electric companies is the kilowatt-hour. This is defined in terms of the unit of power and the

amount of time it is supplied. 1 kWh = 3.60 x 106 J

DIRECT CURRENT DIRECT CURRENT CIRCUITSCIRCUITS

Sources of emf

The source that maintains the current in a closed circuit is called a source of emf. Any devices that

increase the potential energy of charges circulating in circuits are sources of emf.

Examples include batteries and generators.

SI units are Volts The emf is the work

done per unit charge.

emf and Internal Resistance

A real battery has some internal resistance.

Therefore, the terminal voltage is not equal to the emf.

The terminal voltage is ΔV = Vb-Va

ΔV = ε – IrFor the entire circuit, ε = IR + Ir

Resistors in Series

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

The current is the same in all resistors because any charge that flows through one resistor flows through the other. Ieq = I1 = I2 = …

The sum of the potential differences across the resistors is equal to the total potential difference across the combination.

Resistors in Series

Potentials add: ΔV = IR1 + IR2 = I (R1+R2) Consequence of Conservation of

EnergyThe equivalent resistance has

the effect on the circuit as the original combination of resistors. Req = R1 + R2 + R3 + …

The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistors.

Sample Problem

Four resistors are arranged as shown in the figure. Find (a) the equivalent resistance and (b) the current in the circuit if the emf of the battery is 6.0 V.

Resistors in Parallel

The potential difference across each resistor is the same because each is connected directly across the battery terminals. ∆Veq = ∆V1 = ∆V2 The current, I, that enters a

point must be equal to the total current leaving that point.

I = I1 + I2 The currents are generally

not the same. Consequence of

Conservation of Charge.

Resistors in Parallel

Equivalent resistance can replace the original resistances:

The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance. The equivalent is always less than

the smallest resistor in the group.

321eq R

1

R

1

R

1

R

1

Sample Problem

Three resistors are connected in parallel as in the figure. A potential difference of 18 V is maintained between points a and b. (a) Find the current in each resistor. (b) Calculate the power delivered to each resistor and the total power delivered to the three resistors.

Sample Problem

(a) Find the equivalent resistance between points a and b in the figure. (b) Calculate the current in each resistor if a potential difference of 34.0 V is applied between points a and b.

Chapter 20 – 23

ELECTRICITY, ELECTRICITY, CIRCUITS, AND CIRCUITS, AND EVERYTHING ELSE EVERYTHING ELSE IN BETWEENIN BETWEEN THE ENDTHE END