chapter 20 electric charge, force, and field. properties of electric charges two types of charges...
TRANSCRIPT
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Chapter 20
Electric Charge, Force,and Field
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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
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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
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Particle Summary
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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
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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
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Superposition Principle
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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?
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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Particle Summary
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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
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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
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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
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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)
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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.)
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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
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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
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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
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Answers to Even Numbered Problems
Chapter 20:
Problem 14
1.6 × 1020
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Answers to Even Numbered Problems
Chapter 20:
Problem 26
5.2 × 1011 N/C
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Answers to Even Numbered Problems
Chapter 20:
Problem 42
(1.6 iˆ - 0.33 jˆ) Nor 1.7 N at an angle of − 11°