What I will do for review
• Discuss a few “exemplar” problems
(typical problems).
• Will discuss in terms of the big ideas and
show you how the procedures and
equations flow from those big ideas
• Goal for you should be to know how to
approach problems conceptually, NOT to
magically “find” the right equation.
Here’s an interesting Q
• Which way does E point at center of insulating sphere
with uniform positive volume charge density ρ, with an
empty hole of radius R/2 situated as shown. Which way
does E point at the center of empty hole?
+ charges in blue region
distributed uniformly
Empty hole
Here’s an interesting Q
•
Which way does E point at center of insulating sphere with uniform
positive volume charge density ρ, and at center of empty hole?
=
+ρ
+
ρ
––
Use superposition and consider the blue positively charged ball and the red
negatively charged ball (with –ρ) separately. The red ball is positioned where the
hole is.
Here’s an interesting Q
•
•
What is net flux for empty hole?
How would you go about finding net flux for blue, charged volume?
Here’s an interesting Q
•
•
What is net flux for empty hole?
How would you go about finding net flux for blue, charged volume?
––
+
Point charges: Three massive charged balls are positioned as shown in the
figure below. A frictionless rod running along the x axis constrains the motion
of Ball #3 to slide along this axis. Ball #1 and Ball #2 are fixed on the y axis…
Could ask:
Fall ‘08
1) Work (or energy needed)
to assemble balls
y
Ball #1: q1 = +1μC
2) Motion of #3
#1 (0,4)
Ball #2: q2 = +2μC
3) What is V or E at some point (e.g.,
Ball #3: q3 = -6μC
origin)
#3
x 4) What is x component of
(8,0)
(0,0)
force on #3
5) What is speed of #3 if released
#2 (0,-4)
when it gets to origin?
Point charges: Three massive charged balls are positioned as shown in the
figure below. A frictionless rod running along the x axis constrains the motion
of Ball #3 to slide along this axis. Ball #1 and Ball #2 are fixed on the y axis…
Could ask:
Fall ‘08
1) Work (or energy needed)
to assemble balls
Ball #1: q1 = +1μC
y
Ball #2: q2 = +2μC
#1 (0,4)
Ball #3: q3 = -6μC
#3
(0,0)
(8,0)
x
What BIG IDEA to apply?
Potential energy stored in system
is the same as work needed
to assemble balls.
#2 (0,-4)
Mathematical procedure: Compute potential energy by assembling the 3 balls one at a
time. Bring ball #1 in first, then ball #2, then ball #3. Work, or energy needed to
bring in ball #1 = 0.
kq1q2
Work needed to bring in #2 in presence of #1: U1, 2 =
r1, 2
Work needed to bring in #3 in presence of #1 & #2: U1,3 + U 2,3 =
U total = U1, 2 + U1,3 + U 2,3
kq1q3 kq2 q3
+
r1,3
r2,3
Point charges: Three massive charged balls are positioned as shown in the
figure below. A frictionless rod running along the x axis constrains the motion
of Ball #3 to slide along this axis. Ball #1 and Ball #2 are fixed on the y axis…
Could ask:
Fall ‘08
2) Motion of #3
Ball #1: q1 = +1μC
y
Ball #2: q2 = +2μC
#1 (0,4)
Ball #3: q3 = -6μC
#3
(0,0)
(8,0)
x
What BIG IDEA to apply?
What is net force on #3 and
how does it change? Recall
#3 constrained to move along
x=-axis
#2 (0,-4)
Reasoning: Y-component of net force irrelevant.
X-component of net force initially points in the –x direction.
This x-component decreases (why) until #3 gets to origin.
When #3 crosses into –x axis, x-component of force reverses and now points in the
+x direction.
What kind of motion does this give rise to?
Point charges: Three massive charged balls are positioned as shown in the
figure below. A frictionless rod running along the x axis constrains the motion
of Ball #3 to slide along this axis. Ball #1 and Ball #2 are fixed on the y axis…
Could ask:
Fall ‘08
3) What is V or E at some point (e.g.,
origin)
Ball #1: q1 = +1μC
y
Ball #2: q2 = +2μC
#1 (0,4)
Ball #3: q3 = -6μC
#3
(0,0)
x
(8,0)
What BIG IDEA to apply?
Coulomb’s law, which defines the
E or V for a point charge
#2 (0,-4)
Mathematical procedure:
If asked for V at origin, write potential for each charge at origin. Potential is scalar
so it’s easy:
kq1 kq2 kq3
V1 + V2 + V3 =
r1
+
r2
+
r3
Electric field is a vector so you have to worry about direction (may need to break up
Into x or y components depending on what is asked). Magnitude of E:
E=
kq
r2
Point charges: Three massive charged balls are positioned as shown in the
figure below. A frictionless rod running along the x axis constrains the motion
of Ball #3 to slide along this axis. Ball #1 and Ball #2 are fixed on the y axis…
Could ask:
Fall ‘08
4) What is x component of
force on #3?
Ball #1: q1 = +1μC
y
Ball #2: q2 = +2μC
#1 (0,4)
Ball #3: q3 = -6μC
#3
(0,0)
#2 (0,-4)
(8,0)
x
What BIG IDEA to apply?
Coulomb’s law. Need to find
X-component of the force whose
magnitude is the following for each
pair of charges:
F =
kqQ
r2
Point charges: Three massive charged balls are positioned as shown in the
figure below. A frictionless rod running along the x axis constrains the motion
of Ball #3 to slide along this axis. Ball #1 and Ball #2 are fixed on the y axis…
Could ask:
Fall ‘08
5) What is speed of #3 if released when
it gets to origin?
Ball #1: q1 = +1μC
y
Ball #2: q2 = +2μC
#1 (0,4)
Ball #3: q3 = -6μC
#3
(0,0)
(8,0)
x
What BIG IDEA to apply?
Conservation of energy. Potential
Energy is converted to kinetic energy.
#2 (0,-4)
Mathematical procedure:
ΔU = ΔKE
1
U f − U i = mv 2
2
ΔU is the change in potential energy of #3 from where it is now (initial) to the origin
(final) in the presence of #1 and #2. Note: Ui and Uf each is made up of 2 terms
E-fields
• By Coulomb’s Law: Break up charge distribution
into little dq, write the E field contribution, dE,
due to each dq, and add up all contributions
(integrate over entire charge distribution).
Remember that E is a vector so need to
integrate x and y pieces separately.
• By Gauss’ Law. Need symmetry (planar,
cylindrical or spherical) to easily compute E,
despite Gauss’ Law being always valid.
An infinite sheet of charge, oriented perpendicular to the x-axis, passes through
x = 0. It has area density σ1 = -5 μC/m2. A thick, infinite conducting slab, also
oriented perpendicular to the x-axis, occupies the region between x = a and x =
b. The conducting slab has a net charge per unit area of σ2 = 2 μC/m2.
(a) Calculate the net x-component of the
electric field at point marked P.
P
(b) Calculate the surface charge densities
on the left-hand, σL, and right-hand, σR,
faces of the conducting slab.
An infinite sheet of charge, oriented perpendicular to the x-axis, passes through
x = 0. It has area density σ1 = -5 μC/m2. A thick, infinite conducting slab, also
oriented perpendicular to the x-axis, occupies the region between x = a and x =
b. The conducting slab has a net charge per unit area of σ2 = 2 μC/m2.
(a) Calculate the net x-component of the
electric field at point marked P.
P
Mathematical procedure:
σ in eqns are positive since
I already determined
direction of E
What BIG IDEA to apply?
Superposition and Gauss’ Law. Compute
E due to the two objects separately, and
add up the two contributions
r
r r
Etot = E1 + E2
r
σ1
(− xˆ )
E1 =
2ε o
r
σ2
(− xˆ )
E2 =
2ε o
r
r r ⎡ σ1 σ 2 ⎤
Etot = E1 + E2 = ⎢
+
⎥ (− xˆ )
⎣ 2ε o 2ε o ⎦
An infinite sheet of charge, oriented perpendicular to the x-axis, passes through
x = 0. It has area density σ1 = -5 μC/m2. A thick, infinite conducting slab, also
oriented perpendicular to the x-axis, occupies the region between x = a and x =
b. The conducting slab has a net charge per unit area of σ2 = 2 μC/m2.
b) Calculate the surface charge densities
on the left-hand, σL, and right-hand, σR,
faces of the conducting slab.
P
What BIG IDEA to apply?
Superposition and Gauss’ Law with fact that
E is 0 in a conductor. You now know E, so
apply Gauss’ Law on left side of conductor.
Mathematical procedure:
σL is + since
E points to left.
How do we get σR?
r
r r ⎡ σ1 σ 2 ⎤
7 μC / m 2
(− xˆ )
+
Etot = E1 + E2 = ⎢
⎥ (− xˆ ) =
2ε o
⎣ 2ε o 2ε o ⎦
7 μC / m 2
σLA
so σ L = ( Etot )ε o =
Etot A =
2
εo
y
Q1 = –2 μC
Q2 = +7 μC
Q3 = –3 μC
a = 3 cm
b = 4 cm
c = 6 cm
d = 7 cm
Q3
Q2
Q1
x
a
d
c
b
insulator
metal
What would you like to know about this?
y
Q1 = –2 μC
Q2 = +7 μC
Q3 = –3 μC
a = 3 cm
b = 4 cm
c = 6 cm
d = 7 cm
Q3
Q2
Q1
a
d
c
b
x
insulato
r
metal
Getting V from E
• Integrate:
r r
V = − ∫ E ⋅dr
f
i
• What if you don’t know E? Find E first (usually by
Gauss’ Law).
• How do you know which point, initial or final, is at
a higher potential? Warning: Lots of – signs
kicking around so likely to mess up. Imagine
releasing a test positive charge between i and f;
which way will it move? It always goes from
higher to lower potential.
Capacitor big ideas
• C=Q/V and depends only on geometry
• Capacitors connected in series have the
same charge
• Capacitors connected in parallel have the
same potential difference between the
plates.
• Inserting a dielectric between plates of a
capacitor increases capacitance
• Combining capacitors in series and parallel
Fall 08
A
C3
6V
C2
C4
C2 = 2 μF
C3 = 3 μF
C4 = 4 μF
1. Find magnitude of the charge Q2 on each
of the plates of capacitor C2.
2. Find total energy stored in capacitors.
3. Find V3.
4. If dielectric with κ= 5.4 inserted into C2,
what is change in energy stored in this
capacitor?
B
1. Easy!! Apply definition of capacitance.
2. Combine capacitors into a single capacitor. Use energy form: (1/2)CtotV2
3. Combine #3 and #4 into one capacitor. Use definition of capacitance
to find Q3,4. This is the charge on each of these two capacitors (they are
in series). Use definition of capacitance for #3 only to get V3.
4. Initial energy before dielectric is (1/2)C1V2. After dielectric is inserted the
capacitance goes up by κ. So,
Eafter - Ebefore= (1/2)κC1V2 - (1/2)C1V2 = (κ-1) (1/2)C1V2
F07
V
C1
C1
C2
Capacitor C1 is initially connected to a battery with voltage V,
as shown on the left. The battery is then disconnected and C1
is connected to capacitor C2, as shown on the right. C2 is uncharged
before it is connected to C1. What is the final voltage across C2?
What BIG IDEAs to apply?
1) Initial charge on C1, Q, is easy to find when connected to battery
2) C1 and C2 are in parallel, so they have the same voltage
3) Q is conserved: shared between C1 and C2 (no gain or loss of initial Q)
4) Definition of capacitance: C=q/V
Mathematical procedures:
•
•
Initial charge on C1: Q=C1V
Voltage same on C1 and C2: V1 = V2,
Q1 Q2
=
C1 C2
3) Charge conserved: Q = C1V = Q1 + Q2
4) Two eqns in Q1 and Q2. Solve for Q1 and use V=Q1/C1 to get V1
5) Should get: V1 = VC1/(C1+C2)