Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Electric Charge and Electric Field
Sections 16.9 - 16.10
Gauss’s Law
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Reading Assignment
Read sections 17.1 - 17.3
Homework Assignment 1
Homework for Chapter 16 (due at the beginning of class on Wednesday, September 1)
Q: 4, 8, 17, 18
P: 12, 14, 28, 46, 64
PE: 23.2
PP: 23.7, 24.2
Life of Galileo
Bertolt Brecht’s Life of Galileo is playing Nov. 17-21 in The Barber Theatre in Cunningham Theatre Center
Ticket prices for students are $4.00
Though you are not required to attend this production, I strongly urge you to
For more details go to http://www3.davidson.edu/cms/x16192.xml
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Coulomb’s Law
The magnitude of the electric force exerted by a point charge q1 on a second point charge q2 is
F12 =
ke q1 q2
r2
=
1
q1 q2
4π0
r2
What are these things?
ke : the Coulomb constant ke = 8.99 × 109 N · m2 /C2
0 = the permittivity of free space 0 = 8.85 × 10−12 C2 /N · m2
q1 and q2 : electric charges of point charge 1 and point charge 2
r: the distance separating the two point charges
Gauss’s Law
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Coulomb’s Law
The magnitude of the electric force exerted by a point charge q1 on a second point charge q2 is
F12 =
ke q1 q2
r2
=
1
q1 q2
4π0
r2
What are these things?
ke : the Coulomb constant ke = 8.99 × 109 N · m2 /C2
0 = the permittivity of free space 0 = 8.85 × 10−12 C2 /N · m2
q1 and q2 : electric charges of point charge 1 and point charge 2
r: the distance separating the two point charges
About Coulomb’s law
It is the governing law of electrostatics (stationary or slow-moving charges)
Gauss’s Law
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Coulomb’s Law
The magnitude of the electric force exerted by a point charge q1 on a second point charge q2 is
F12 =
ke q1 q2
r2
=
1
q1 q2
4π0
r2
What are these things?
ke : the Coulomb constant ke = 8.99 × 109 N · m2 /C2
0 = the permittivity of free space 0 = 8.85 × 10−12 C2 /N · m2
q1 and q2 : electric charges of point charge 1 and point charge 2
r: the distance separating the two point charges
About Coulomb’s law
It is the governing law of electrostatics (stationary or slow-moving charges)
It is not cast in a form that particularly simplifies the work in situations involving symmetry
Gauss’s Law
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Review: electric field lines
The density of field lines is proportional to the
magnitude of the electric field
Imagine that we have a uniform electric field that is
penetrating a plane of area A perpendicular to the field
What can we say about the total number of field lines
penetrating the surface?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Review: electric field lines
The density of field lines is proportional to the
magnitude of the electric field
Imagine that we have a uniform electric field that is
penetrating a plane of area A perpendicular to the field
What can we say about the total number of field lines
penetrating the surface?
The number of field lines penetrating the
surface is proportional to ΦE = EA
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Review: electric field lines
The density of field lines is proportional to the
magnitude of the electric field
Imagine that we have a uniform electric field that is
penetrating a plane of area A perpendicular to the field
What can we say about the total number of field lines
penetrating the surface?
The number of field lines penetrating the
surface is proportional to ΦE = EA
Electric flux
The electric flux ΦE is proportional to the number of
electric field lines penetrating some surface
Electric flux is the product of the magnitude of the
electric field perpendicular to the surface E⊥ and the
surface area A
ΦE = E⊥ A = EA cos θ
Electric flux has units of newton meters squared per
coulomb (N·m2 /C)
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
What if the electric field is not uniform?
Divide the surface into a large number of tiny elements
of area ∆A
The tiny amount of electric flux ∆ΦE passing through
the tiny surface of area ∆A is
∆ΦE = E⊥ ∆A = E ∆A cos θ
We can then sum over all the elements to obtain an
approximation for the total flux through the entire
surface
ΦE ≈
Gauss’s Law
X
E⊥ ∆A =
X
E ∆A cos θ
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
What if the electric field is not uniform?
Divide the surface into a large number of tiny elements
of area ∆A
The tiny amount of electric flux ∆ΦE passing through
the tiny surface of area ∆A is
∆ΦE = E⊥ ∆A = E ∆A cos θ
We can then sum over all the elements to obtain an
approximation for the total flux through the entire
surface
ΦE ≈
X
E⊥ ∆A =
X
E ∆A cos θ
Gaussian surfaces
A Gaussian surface is a hypothetical closed surface
It must be a closed surface, so that a clear distinction
can be made between points that are inside the surface,
on the surface, and outside the surface
It can be any shape you want
Gauss’s Law
Announcements
Gauss’s Law
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Gauss’s law
Gauss’s law states that the net flux through any closed
surface (Gaussian surface) is
ΦE =
Qenc
0
What are these things?
ΦE : the net flux through the Gaussian surface
Qenc : the net charge enclosed by the Gaussian surface
0 = the permittivity of free space
0 = 8.85 × 10−12 C2 /N · m2
Picture of the German scientist and
mathematician Johann Carl Friedrich Gauss
(1777 - 1855).
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
More about Gauss’s Law
Gauss’s law states that the net flux through any closed surface (Gaussian surface) is
ΦE =
Qenc
0
Gauss’s law relates the electric fields at points on a (closed) Gaussian surface and the net charge enclosed
by that surface
Qenc is the algebraic sum of all the enclosed positive and negative charges (therefore, Qenc can be
positive, negative, or zero)
If Qenc is positive, the net flux is outward (positive)
If Qenc is negative, the net flux is inward (negative)
Charge outside the Gaussian surface (no matter how large or how close it may be) is not included in the
term Qenc
The exact form or location of the charges inside the Gaussian surface is not important
→
−
The electric field E , however, is the total electric field resulting from all charges, both inside and outside
the Gaussian surface
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Question
There are five charged lumps of plastic shown above along with an arbitrarily chosen Gaussian surface S. What is
the net electric flux through the surface if q1 = q4 = 3.1 nC, q2 = q5 = −5.9 nC, and q3 = −3.1 nC?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Question
There are five charged lumps of plastic shown above along with an arbitrarily chosen Gaussian surface S. What is
the net electric flux through the surface if q1 = q4 = 3.1 nC, q2 = q5 = −5.9 nC, and q3 = −3.1 nC?
Answer
Use Gauss’s law to find the electric flux through the Gaussian surface
ΦE =
Gauss’s Law
Qenc
0
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Question
There are five charged lumps of plastic shown above along with an arbitrarily chosen Gaussian surface S. What is
the net electric flux through the surface if q1 = q4 = 3.1 nC, q2 = q5 = −5.9 nC, and q3 = −3.1 nC?
Answer
Use Gauss’s law to find the electric flux through the Gaussian surface
ΦE =
Gauss’s Law
Qenc
0
=
q1 + q2 + q3 + q4
0
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Question
There are five charged lumps of plastic shown above along with an arbitrarily chosen Gaussian surface S. What is
the net electric flux through the surface if q1 = q4 = 3.1 nC, q2 = q5 = −5.9 nC, and q3 = −3.1 nC?
Answer
Use Gauss’s law to find the electric flux through the Gaussian surface
ΦE =
=
Qenc
0
=
q1 + q2 + q3 + q4
0
+3.1 × 10−9 C − 5.9 × 10−9 C − 3.1 × 10−9 C + 3.1 × 10−9 C
8.85 × 10−12 C2 /N · m2
2
= −316.4 N · m /C
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Electrostatic equilibrium
A conductor is in electrostatic equilibrium if there is no net motion of charge within it
Gauss’s Law
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Electrostatic equilibrium
A conductor is in electrostatic equilibrium if there is no net motion of charge within it
Properties of a conductor in electrostatic equilibrium
A conductor in electrostatic equilibrium has the following properties:
The electric field inside the conductor is zero everywhere (independent of the shape of the conductor)
The electric field just outside a charged conductor is perpendicular to the surface of the conductor
Any excess charge placed on an isolated conductor will move entirely to the surface (none of the excess
charge will be found within the body of the conductor)
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Claim
The electric field inside the conductor is zero everywhere (independent of the shape of the conductor)
Gauss’s Law
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Claim
The electric field inside the conductor is zero everywhere (independent of the shape of the conductor)
Proof
If the electric field inside the conductor was not zero, it would exert a force on the electrons causing them
to accelerate
If this happened, the conductor would not be in electrostatic equilibrium (which states that there is no net
motion of charge inside the conductor)
A good conductor placed in an external electric field reaches equilibrium almost instantaneously
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Claim
The electric field inside the conductor is zero everywhere (independent of the shape of the conductor)
Proof
If the electric field inside the conductor was not zero, it would exert a force on the electrons causing them
to accelerate
If this happened, the conductor would not be in electrostatic equilibrium (which states that there is no net
motion of charge inside the conductor)
A good conductor placed in an external electric field reaches equilibrium almost instantaneously
Claim
The electric field just outside a charged conductor is perpendicular to the surface of the conductor
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Claim
The electric field inside the conductor is zero everywhere (independent of the shape of the conductor)
Proof
If the electric field inside the conductor was not zero, it would exert a force on the electrons causing them
to accelerate
If this happened, the conductor would not be in electrostatic equilibrium (which states that there is no net
motion of charge inside the conductor)
A good conductor placed in an external electric field reaches equilibrium almost instantaneously
Claim
The electric field just outside a charged conductor is perpendicular to the surface of the conductor
Proof
If the electric field at the conductor’s surface was not perpendicular, it would have a component along the
conductor’s surface that would cause the surface charges to move
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Claim
Any excess charge placed on an isolated conductor will move entirely to the surface. None of the excess charge will
be found within the body of the conductor.
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Claim
Any excess charge placed on an isolated conductor will move entirely to the surface. None of the excess charge will
be found within the body of the conductor.
Proof
Place a Gaussian surface just inside the surface of the conductor
The electric field must be zero for all points on the Gaussian surface (because it is inside the conductor)
Using Gauss’s law, since the net flux through the Gaussian surface is zero, so too must be the net charge
enclosed by the surface
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Scenario
A charged lump of copper with a cavity is in electrostatic equilibrium.
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Announcements
Electric Flux
Gauss’s Law
Scenario
A charged lump of copper with a cavity is in electrostatic equilibrium.
Question #1
What is the net charge on the surface of the cavity?
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Scenario
A charged lump of copper with a cavity is in electrostatic equilibrium.
Question #1
What is the net charge on the surface of the cavity?
Answer
Construct a Gaussian surface just inside the surface of the cavity
Because the copper is in electrostatic equilibrium, the electric flux through the Gaussian surface must be
zero
Therefore, the net charge inside this surface is zero, so we conclude that the net charge on the surface of
the cavity is zero
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Scenario
A charged lump of copper with a cavity is in electrostatic equilibrium.
Question #1
What is the net charge on the surface of the cavity?
Answer
Construct a Gaussian surface just inside the surface of the cavity
Because the copper is in electrostatic equilibrium, the electric flux through the Gaussian surface must be
zero
Therefore, the net charge inside this surface is zero, so we conclude that the net charge on the surface of
the cavity is zero
Question #2
What is the net charge on the outer surface of the copper?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Scenario
A charged lump of copper with a cavity is in electrostatic equilibrium.
Question #1
What is the net charge on the surface of the cavity?
Answer
Construct a Gaussian surface just inside the surface of the cavity
Because the copper is in electrostatic equilibrium, the electric flux through the Gaussian surface must be
zero
Therefore, the net charge inside this surface is zero, so we conclude that the net charge on the surface of
the cavity is zero
Question #2
What is the net charge on the outer surface of the copper?
Answer
Construct a Gaussian surface just inside the outer surface of the copper
Because the copper is in electrostatic equilibrium, the electric flux through the Gaussian surface must be
zero
Therefore, the net charge inside the copper is zero, so we conclude that all of the excess charge resides on
the outer surface of the copper
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Instead, you decide to use a spherical Gaussian surface to surround the charge. What effect will this have
on the net flux?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Instead, you decide to use a spherical Gaussian surface to surround the charge. What effect will this have
on the net flux? The shape of the Gaussian surface does not affect the net flux through the surface
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Instead, you decide to use a spherical Gaussian surface to surround the charge. What effect will this have
on the net flux? The shape of the Gaussian surface does not affect the net flux through the surface
If the net flux through some Gaussian surface is outward, what can you conclude about the charge inside
the surface?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Instead, you decide to use a spherical Gaussian surface to surround the charge. What effect will this have
on the net flux? The shape of the Gaussian surface does not affect the net flux through the surface
If the net flux through some Gaussian surface is outward, what can you conclude about the charge inside
the surface? The net charge inside the surface must be positive
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Instead, you decide to use a spherical Gaussian surface to surround the charge. What effect will this have
on the net flux? The shape of the Gaussian surface does not affect the net flux through the surface
If the net flux through some Gaussian surface is outward, what can you conclude about the charge inside
the surface? The net charge inside the surface must be positive
If the net flux is zero, is there any charge inside the surface?
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Final questions
A cubical Gaussian surface surrounds a point charge Q. If the length of each side of the cube is doubled,
what happens to the total flux through the surface? It is unchanged
Instead, you decide to use a spherical Gaussian surface to surround the charge. What effect will this have
on the net flux? The shape of the Gaussian surface does not affect the net flux through the surface
If the net flux through some Gaussian surface is outward, what can you conclude about the charge inside
the surface? The net charge inside the surface must be positive
If the net flux is zero, is there any charge inside the surface? Perhaps, but the net charge is zero
Gauss’s Law
Announcements
Electric Flux
Gauss’s Law
Electrostatic Equilibrium
Final Questions
Reading Assignment
Read sections 17.1 - 17.3
Homework Assignment 1
Homework for Chapter 16 (due at the beginning of class on Wednesday, September 1)
Q: 4, 8, 17, 18
P: 12, 14, 28, 46, 64
PE: 23.2
PP: 23.7, 24.2
Life of Galileo
Bertolt Brecht’s Life of Galileo is playing Nov. 17-21 in The Barber Theatre in Cunningham Theatre Center
Ticket prices for students are $4.00
Though you are not required to attend this production, I strongly urge you to
For more details go to http://www3.davidson.edu/cms/x16192.xml
Gauss’s Law