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FOR SCIENTISTS AND ENGINEERS

A STRATEGIC APPROACH

4/E

PHYSICS

RANDALL D. KNIGHT

Chapter 23 Lecture

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Chapter 23 The Electric Field

IN THIS CHAPTER, you will learn how to calculate and use the electric field.

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Chapter 23 Preview

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Chapter 23 Preview

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Chapter 23 Preview

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Chapter 23 Preview

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Chapter 23 Preview

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Chapter 23 Reading Questions

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What device provides a practical way to produce a uniform electric field?

A long thin resistor
A Faraday cage
A parallel-plate capacitor
A toroidal inductor
An electric field uniformizer

Reading Question 23.1

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What device provides a practical way to produce a uniform electric field?

A long thin resistor
A Faraday cage
A parallel-plate capacitor
A toroidal inductor
An electric field uniformizer

Reading Question 23.1

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For charged particles, what is the quantity q/m called?

Linear charge density
Charge-to-mass ratio
Charged mass density
Massive electric dipole
Quadrupole moment

Reading Question 23.2

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For charged particles, what is the quantity q/m called?

Linear charge density
Charge-to-mass ratio
Charged mass density
Massive electric dipole
Quadrupole moment

Reading Question 23.2

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Which of these charge distributions did not have its electric field determined in Chapter 23?

A line of charge
A parallel-plate capacitor
A ring of charge
A plane of charge
They were all determined.

Reading Question 23.3

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Which of these charge distributions did not have its electric field determined in Chapter 23?

A line of charge
A parallel-plate capacitor
A ring of charge
A plane of charge
They were all determined.

Reading Question 23.3

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The worked examples of charged-particle motion are relevant to

A transistor.
A cathode ray tube.
Magnetic resonance imaging.
Cosmic rays.
Lasers.

Reading Question 23.4

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The worked examples of charged-particle motion are relevant to

A transistor.
A cathode ray tube.
Magnetic resonance imaging.
Cosmic rays.
Lasers.

Reading Question 23.4

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Chapter 23 Content, Examples, and QuickCheck Questions

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Four Key Electric Fields: Slide 1 of 2

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Four Key Electric Fields: Slide 2 of 2

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Electric Field of a Point Charge

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The Electric Field

The electric field was defined as

where _on_q is the electric force on test charge q.

The SI units of electric field are therefore Newtons per Coulomb (N/C).

= _on_q / q

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The Electric Field of Multiple Point Charges

Suppose the source of an electric field is a group of point charges q₁, q₂, …
The net electric field E_net is the vector sum of the electric fields due to each charge.
In other words, electric fields obey the principle of superposition.

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What is the direction of the electric field at the dot?

QuickCheck 23.1

E. None of these.

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What is the direction of the electric field at the dot?

QuickCheck 23.1

E. None of these.

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Problem-Solving Strategy: The Electric Field of Multiple Point Charges

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Problem-Solving Strategy: The Electric Field of Multiple Point Charges

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What is the direction of the electric field at the dot?

QuickCheck 23.2

E. The field is zero.

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What is the direction of the electric field at the dot?

QuickCheck 23.2

E. The field is zero.

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When r >> d, the electric field strength at the dot is

QuickCheck 23.3

A.

B.

C.

D.

E.

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When r >> d, the electric field strength at the dot is

QuickCheck 23.3

A.

B.

C.

D.

E.

Looks like a point charge 4Q at the origin.

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Electric Dipoles

Two equal but opposite charges separated by a small distance form an electric dipole.
The figure shows two examples.

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The Dipole Moment

It is useful to define the dipole moment p, shown in the figure, as the vector:

The SI units of the dipole moment are C m.

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The Dipole Electric Field at Two Points

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The Electric Field of a Dipole

The electric field at a point on the axis of a dipole is

where r is the distance measured from the center of the dipole.

The electric field in the plane that bisects and is perpendicular to the dipole is

This field is opposite to the dipole direction, and it is only half the strength of the on-axis field at the same distance.

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Example 23.2 The Electric Field of a Water Molecule

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Electric Field Lines

Electric field lines are continuous curves tangent to the electric field vectors.
Closely spaced field lines indicate a greater field strength.
Electric field lines start on positive charges and end on negative charges.
Electric field lines never cross.

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Electric Field Lines of a Point Charge

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The Electric Field of a Dipole

This figure represents the electric field of a dipole using electric field lines.

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Two protons, A and B, are �in an electric field. Which proton has the larger acceleration?

QuickCheck 23.4

Proton A
Proton B
Both have the same acceleration.

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Two protons, A and B, are in an electric field. Which proton has the larger acceleration?

QuickCheck 23.4

Proton A
Proton B
Both have the same acceleration.

Stronger field where field lines are closer together.

Weaker field where field lines are farther apart.

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QuickCheck 23.5

An electron is in the plane that bisects a dipole. What is the direction of the electric force on the electron?

E. The force is zero.

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QuickCheck 23.5

An electron is in the plane that bisects a dipole. What �is the direction of the electric force on the electron?

E. The force is zero.

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Continuous Charge Distributions

Linear charge density, which has units of C/m, is the amount of charge per meter of length.

The linear charge density of an object of length L and charge Q is defined as

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If 8 nC of charge are placed on the square loop of wire, the linear charge density will be

QuickCheck 23.6

800 nC/m
400 nC/m
200 nC/m
8 nC/m
2 nC/m

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If 8 nC of charge are placed on the square loop of wire, the linear charge density will be

QuickCheck 23.6

800 nC/m
400 nC/m
200 nC/m
8 nC/m
2 nC/m

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Continuous Charge Distributions

The surface charge density of a two-dimensional distribution of charge across a surface of area A is defined as

Surface charge density, with units C/m², is the amount of charge per square meter.

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A flat circular ring is made from a very thin sheet of metal. Charge Q is uniformly distributed over the ring. Assuming w << R, the surface charge density η is

QuickCheck 23.7

Q/2πRw
Q/4πRw
Q/πR²
Q/2πR²
Q/πRw

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QuickCheck 23.7

The ring has two sides, each of area �2πRw.

A flat circular ring is made from a very thin sheet of metal. Charge Q is uniformly distributed over the ring. Assuming w << R, the surface charge density η is

Q/2πRw
Q/4πRw
Q/πR²
Q/2πR²
Q/πRw

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Problem-Solving Strategy: The Electric Field of a Continuous Distribution of Charge

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Problem-Solving Strategy: The Electric Field of a Continuous Distribution of Charge

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The Electric Field of a Finite Line of Charge

Example 23.3 in the text uses integration to find the electric field strength at a radial distance r in the plane that bisects a rod of length L with total charge Q:

The Electric Field of a Line of Charge

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At the dot, the y-component of the electric field due to the shaded region of charge is

QuickCheck 23.8

A.

B.

C.

D.

E.

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At the dot, the y-component of the electric field due to the shaded region of charge is

QuickCheck 23.8

A.

B.

C.

D.

E.

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An Infinite Line of Charge

The electric field of a thin, uniformly charged rod may be written

If we now let L → ∞, the last term becomes simply 1 and we’re left with

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A Ring of Charge

Consider the on-axis electric field of a positively charged ring of radius R.
Define the z-axis to be the axis of the ring.
The electric field on the �z-axis points away from the center of the ring, increasing in strength until reaching a maximum when |z| ≈ R, then decreasing:

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A Disk of Charge

Consider the on-axis electric field of a positively charged disk of radius R.
Define the z-axis to be the axis of the disk.
The electric field on the �z-axis points away from the center of the disk, with magnitude:

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Example 23.5 The Electric Field of a Charged Disk

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Example 23.5 The Electric Field of a Charged Disk

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A Plane of Charge

The electric field of a plane of charge is found from the on-axis field of a charged disk by letting the radius �R → ∞.
The electric field of an infinite plane of charge with surface charge density η is

For a positively charged plane, with η > 0, the electric field points away from the plane on both sides of the plane.
For a negatively charged plane, with η < 0, the electric field points toward the plane on both sides of the plane.

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A Plane of Charge

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Two protons, A and B, are next to an infinite plane of positive charge. Proton B is twice as far from the plane as proton A. Which proton has the larger acceleration?

QuickCheck 23.9

Proton A
Proton B
Both have the same acceleration.

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Two protons, A and B, are next to an infinite plane of positive charge. Proton B is twice as far from the plane as proton A. Which proton has the larger acceleration?

QuickCheck 23.9

Proton A
Proton B
Both have the same acceleration.

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A Sphere of Charge

A sphere of charge Q and radius R, be it a uniformly charged sphere or just a spherical shell, has an electric field outside the sphere that is exactly the same as that of a point charge Q located at the center of the sphere:

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The Parallel-Plate Capacitor

The figure shows two electrodes, one with charge +Q and the other with –Q placed face-to-face a distance d apart.
This arrangement of two electrodes, charged equally but oppositely, is called a parallel-plate capacitor.
Capacitors play important roles in many electric circuits.

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The figure shows two capacitor plates, seen from the side.
Because opposite charges attract, all of the charge is on the inner surfaces of the two plates.
Inside the capacitor, the net field points toward the negative plate.
Outside the capacitor, the net field is zero.

The Parallel-Plate Capacitor

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The electric field of a capacitor is

where A is the surface area of each electrode.

Outside the capacitor plates, where E₊ and E_– have equal magnitudes but opposite directions, the electric field is zero.

The Parallel-Plate Capacitor

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Three points inside a �parallel-plate capacitor are marked. Which is true?

QuickCheck 23.10

E₁ > E₂ > E₃
E₁ < E₂ < E₃
E₁ = E₂ = E₃
E₁ = E₃ > E₂

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Three points inside a �parallel-plate capacitor are marked. Which is true?

QuickCheck 23.10

E₁ > E₂ > E₃
E₁ < E₂ < E₃
E₁ = E₂ = E₃
E₁ = E₃ > E₂

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The Ideal Capacitor

The figure shows the electric field of an ideal parallel-plate capacitor constructed from two infinite charged planes.
The ideal capacitor is a good approximation as long as the electrode separation d is much smaller than the electrodes’ size.

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A Real Capacitor

Outside a real capacitor and near its edges, the electric field is affected by a complicated but weak fringe field.
We will keep things simple by always assuming the plates are very close together and using E = η/ ₀ for the magnitude of the field inside a parallel-plate capacitor.

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Example 23.6 The Electric Field Inside a Capacitor

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Example 23.6 The Electric Field Inside a Capacitor

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Example 23.6 The Electric Field Inside a Capacitor

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Uniform Electric Fields

The figure shows an electric field that is the same—in strength and direction—at every point in a region of space.
This is called a uniform electric field.
The easiest way to produce a uniform electric field is with a parallel-plate capacitor.

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Motion of a Charged Particle in an Electric Field

Consider a particle of charge q and mass m at a point where an electric field E has been produced by other charges, the source charges.
The electric field exerts a force F_on_q = qE.

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Motion of a Charged Particle in an Electric Field

The electric field exerts a force F_on_q = qE on a charged particle.
If this is the only force acting on q, it causes the charged particle to accelerate with

In a uniform field, the acceleration is constant:

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“DNA fingerprints” are measured with the technique of gel electrophoresis.
A solution of negatively charged DNA fragments migrate through the gel when placed in a uniform electric field.
Because the gel exerts a drag force, the fragments move at a terminal speed inversely proportional to their size.

Motion of a Charged Particle in an Electric Field

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A proton is moving to the right in a vertical electric field. A very short time later, the proton’s velocity is

QuickCheck 23.11

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A proton is moving to the right in a vertical electric field. A very short time later, the proton’s velocity is

QuickCheck 23.11

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Which electric field is responsible for the proton’s trajectory?

QuickCheck 23.12

A.

B.

C.

D.

E.

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Which electric field is responsible for the proton’s trajectory?

QuickCheck 23.12

A.

B.

C.

D.

E.

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Dipoles in a Uniform Electric Field

The figure shows an electric dipole placed in a uniform external electric field.
The net force on the dipole is zero.
The electric field exerts a torque on the dipole that causes it to rotate.

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The figure shows an electric dipole placed in a uniform external electric field.
The torque causes the dipole to rotate until it is aligned with the electric field, as shown.
Notice that the positive end of the dipole is in the direction in which E points.

Dipoles in a Uniform Electric Field

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Which dipole experiences no net force in the electric field?

QuickCheck 23.13

A.

B.

C.

Dipole A
Dipole B
Dipole C
Both dipoles A and C
All three dipoles

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Which dipole experiences no net force in the electric field?

QuickCheck 23.13

A.

B.

C.

Dipole A
Dipole B
Dipole C
Both dipoles A and C
All three dipoles

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Which dipole experiences no net torque in the electric field?

QuickCheck 23.14

Dipole A
Dipole B
Dipole C
Both dipoles A and C
All three dipoles

A.

B.

C.

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Which dipole experiences no net torque in the electric field?

QuickCheck 23.14

Dipole A
Dipole B
Dipole C
Both dipoles A and C
All three dipoles

A.

B.

C.

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Dipoles in a Uniform Electric Field

The figure shows a sample of permanent dipoles, such as water molecules, in an external electric field.
All the dipoles rotate until they are aligned with the electric field.
This is the mechanism by which the sample becomes polarized.

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The Torque on a Dipole

The torque on a dipole placed in a uniform external electric field is

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