Chapter 20

Electric Charge, Force,and Field

Properties of Electric Charges

• Two types of charges exist (named by Benjamin Franklin): positive and negative

• Like charges repel and unlike charges attract one another

• Nature’s most basic positive charges are the protons (held firmly in the nucleus and do not move from one material to another)

• Nature’s most basic negative charges charge are the electrons – an object becomes charged by gaining or losing electrons

Benjamin Franklin 1706 – 1790

Properties of Electric Charges

• Electric charge is always conserved (not created, only exchanged) in an isolated system

• Objects become charged because negative charge (electrons) is transferred from one object to another

• Charge is quantized (a multiple of a fundamental unit of charge, e): electrons have a charge of – e and protons have a charge of + e

• The SI unit of charge is the Coulomb (C)

e = 1.6 x 10-19 C Charles Coulomb 1736 – 1806

Particle Summary

Coulomb’s Law

Electric force is:

• Along the line joining the two point charges

• Inversely proportional to the square of the separation distance, r, between the particles

• Proportional to the product of the magnitudes of the charges, |q1| and |q2| on the two particles

• k = 8.9875 x 109 N m2/C2 : Coulomb constant

• Attractive if the charges are of opposite signs and repulsive if the charges have the same signs

1 212 2

ˆkq q

F rr

Electric Forces

• Electric forces are vector quantities

• Electric force on q1 is equal in magnitude and opposite in direction to the force on q2

• Electric force is exerted by one object on another object without physical contact between them – field force

• The superposition principle applies: resultant force on any one charge equals the vector sum of the forces exerted by the other individual charges that are present

Superposition Principle

Chapter 20Problem 40

A charge 3q is at the origin, and a charge -2q is on the positive x-axis at x = a. Where would you place a third charge so it would experience no net electric force?

Electric Field

• Electric field exists in the region of space around a charged object (source charge)

• When another charged object q0 (test charge) enters this electric field, the field exerts an electric force F on the test charge

• Electric field:

• SI units: N / C FE

q

Electric Field

• The field is produced by some charge or charge distribution, separate from the test charge

• The existence of an electric field is a property of the source charge; the presence of the test charge is not necessary for the field to exist

• The test charge serves as a detector of the field

FE

q

Direction of Electric Field

• The direction of the vector of electric field is defined as the direction of the electric force that would be exerted on a small positive test charge placed at that point

• The electric field produced by a negative charge is directed toward the charge (attraction)

• The electric field produced by a positive charge is directed away from the charge (repulsion)

Relationship Between F and E

• If q is positive, the force and the field are in the same direction; if q is negative, the force and the field are in opposite directions

• Coulomb’s law, between the source and test point charges, can be expressed as

• Then

2ˆo

e e

qqkr

F r

2ˆe

eo

qk

q r

FE r

.F qE

Superposition of Electric Fields

• At any point P, the total electric field due to a group of source charges equals the vector sum of the electric fields of all the charges

2ˆii i

i

kqE E r

r

Continuous Charge Distribution

• Charge ultimately resides on individual particles, so that the distances between charges in a group of charges may be much smaller than the distance between the group and a point of interest

• In this situation, the system of charges can be modeled as continuously distributed along some line, over some surface, or throughout some volume

Continuous Charge Distribution• Divide the charge distribution into

small elements, each of which contains Δq

• Calculate the electric field due to one of these elements at point P

2ˆ

e

qkr

E r

Continuous Charge Distribution• Evaluate the total field by summing

the contributions of all the charge elements

2ˆ

e

qkr

E r

2 20ˆ ˆlim

i

ie i eq

i i

q dqk k

r r

E r r

Charge Densities

• Volume charge density: when a charge is distributed throughout a volume:

dq = ρ dV; [ρ] = [Q ] / [V] with units C/m3

• Surface charge density: when a charge is distributed over a surface area:

dq = σ dA; [ σ ] = [ Q ] / [ A ] with units C/m2

• Linear charge density: when a charge is distributed along a line:

dq = λ dℓ; [ λ ] = [ Q ] / [ ℓ ] with units C/m

Charge Densities

• Linear charge density: when a charge is distributed along a line:

dq = λ dℓ; [ λ ] = [ Q ] / [ ℓ ] with units C/m

2kE

y

Problem-Solving Strategy

• Categorize (individual charge? group of individual charges? continuous distribution of charges?) and take advantage of any symmetry to simplify calculations

• For a group of individual charges: use the superposition principle, find the fields due to the individual charges at the point of interest and then add them as vectors to find the resultant field

• For a continuous charge distribution: a) the vector sums for evaluating the total electric field at some point must be replaced with vector integrals; b) divide the charge distribution into infinitesimal pieces, calculate the vector sum by integrating over the entire charge distribution

Chapter 20Problem 46

A 1.0-µC charge and a charge 2.0-µC are 10 cm apart. Find a point where the electric field is zero.

Electric Field of a Uniform Ring of Charge (Example 20.6)

cos2r

dqkdE ex cos

22 ax

dqke

r

x

ax

dqke 22

2222ax

x

ax

dqke

2/322 ax

xdqkE ex

dqax

xke2/322

2/322 ax

xQke

Electric Field of a Uniformly Charged Disk (Problem 71)

• The ring has a radius R and a uniform charge density σ

• Choose dq as a ring of radius r

• The ring has a surface area 2πr dr

Electric Field of a Uniformly Charged Disk (Problem 71)

2/322 rx

xdqkdE e

x

R

ex

rx

xrdrkE

02/322

2

dAdq )2( rdr

2/322

2

rx

xrdrke

R

erx

rdrxk0

2/322

2

R

erx

rdxk0

2/322

2

R

erx

xk0

22

2

22

12Rx

xke

Motion of Charged Particles• When a charged particle is placed in an electric field,

it experiences an electrical force

• If this is the only force on the particle, it must be the net force

• The net force will cause the particle to accelerate according to Newton’s second law

• If the field is uniform, then the acceleration is constant

• If the particle has a positive (negative) charge, its acceleration is in the direction of (opposite) the field

e q m F E a

Particle Summary

Electric Dipole• An electric dipole consists of two

charges of equal magnitude and opposite signs separated by 2a

• The electric dipole moment p is directed along the line joining the charges from –q to +q and has a magnitude of p ≡ 2aq

• Assume the dipole is placed in a uniform field, external to the dipole (it is not the field produced by the dipole) and makes an angle θ with the field

• Each charge has a force of F = Eq acting on it

Electric Dipole• The net force on the dipole is zero

• The forces produce a net torque on the dipole:

2Eqa sin θpE sin θ

• The torque can also be expressed as the cross product of the moment and the field:

p E

Electric Dipole

ir

a

r

qki

r

a

r

qkE ee ˆˆ

22

22 ayr

iay

qakE e ˆ2

2/322

ayiy

qakE e ,ˆ

23

Classification of Substances vs. Their Ability to Conduct Electric Charge

• Conductors are materials in which the electric charges move freely in response to an electric force (e.g., copper, aluminum, silver, etc.)

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

• Insulators (dielectrics) are materials in which electric charges do not move freely (e.g., glass, rubber, etc.)

• When insulators are charged (by rubbing), only the rubbed area becomes charged (no tendency for the charge to move into other regions of the material)

Classification of Substances vs. Their Ability to Conduct Electric Charge

• Semiconductors – their characteristics are between those of insulators and conductors (e.g., silicon, germanium , etc.)

An Atomic Description of Dielectrics• Molecules are said to be polarized when a

separation exists between the average position of the negative charges and the average position of the positive charges

• Polar molecules are those in which this condition is always present

• Molecules without a permanent polarization are called nonpolar molecules

• The average positions of the positive and negative charges act as point charges, thus polar molecules can be modeled as electric dipoles

An Atomic Description of Dielectrics• A linear symmetric molecule has no permanent

polarization (a)

• Polarization can be induced by placing the molecule in an electric field (b)

• Induced polarization is the effect that predominates in most materials used as dielectrics in capacitors

An Atomic Description of Dielectrics

• In the absence of an electric field the molecules that make up the dielectric (modeled as dipoles) are randomly oriented

• An external electric field produces a torque on the molecules partially aligning them with the electric field; alignment of dipoles reduces the electric field

Answers to Even Numbered Problems

Chapter 20:

Problem 14

1.6 × 1020

Answers to Even Numbered Problems

Chapter 20:

Problem 26

5.2 × 1011 N/C

Answers to Even Numbered Problems

Chapter 20:

Problem 42

(1.6 iˆ - 0.33 jˆ) Nor 1.7 N at an angle of − 11°