Download Q - BYU Physics and Astronomy

Lesson 6
Capacitors and Capacitance
Class 16
Today, we will:
• learn what a capacitor is.
• learn the definition of capacitance.
• find the electric field and voltage inside a
parallel-plate capacitor.
• find the capacitance of the capacitor.
• learn that a dielectric is a material with polar
molecules.
• learn how dielectrics increase capacitance.
• find the energy stored in a capacitor and in the
electric field.
Section 1
Capacitance, Charge, and Voltage
What is a Capacitor?
•Conductors that can hold charge.
•Cables, hands, etc. all have
capacitance.
•For our purposes: two conductors,
one with charge +Q and one with
charge −Q.
What is a Capacitor?
•We “charge” a capacitor by connecting it to a
battery.
+
What is a Capacitor?
•We “charge” a capacitor by connecting it to a
battery.
•When we disconnect
the battery, charge
remains on the
conductors.
What is a Capacitor?
•We “charge” a capacitor by connecting it to a
battery.
•When we disconnect
the battery, charge
remains on the
conductors.
•If we connect the conductors, charge will then flow
from one to the other.
Why Are Capacitors Useful?
•Capacitors can provide uniform electric fields.
We use them to accelerate or deflect charged
beams, etc.
•We can store charge for later use.
•We can charge many capacitors and then
discharge them at one time to produce very large
currents for a short time.
•Capacitors are important in AC (alternating
current = sinusoidal) circuits, but we’ll study that
later.
Charging a Capacitor
When we attach a capacitor to a battery:
•Charge builds up on the conductors.
•The charge on the + conductor is equal and
opposite the charge on the − conductor.
•We call +Q the “charge on the capacitor”
•Voltage builds up on the capacitor until it has the
same voltage as the battery.
•Electric field builds up in the capacitor.
Charging a Capacitor
We find that voltage is proportional to charge.
Q
V
Charging a Capacitor
We find that voltage is proportional to charge.
Q
Q
slope   C
V
V
Capacitance
Q=CV
•If capacitance is large - the capacitor holds a large charge at a small
voltage.
Section 2
Parallel Plate Capacitors
Parallel-plate Capacitors
• made of two plates each of area A (the shape
doesn’t matter)
•plates are separated by a distance d.
Parallel-plate Capacitors
• The electric field is the sum of the electric fields
of a positively charged palate …







Parallel-plate Capacitors
• … and a negatively charged plate.







Parallel-plate Capacitors
• … and a negatively charged plate.














Parallel-plate Capacitors
• The electric fields outside the plates cancel out.














Parallel-plate Capacitors
• The electric fields outside the plates cancel out.
Make the outside fields disappear.














Parallel-plate Capacitors
• The electric fields between the plates add.
Just make the arrows align…














Parallel-plate Capacitors
• The charges move to the inside of the plates.
Move the + and – symbols toward the center.














Parallel-plate Capacitors
• The electric field inside is uniform.
•The electric field outside is small.














Section 3
Electric Field, Voltage, and
Capacitance in a Parallel-Plate
Capacitor
Electric Field of a Capacitor
We can find the electric field in a capacitor
from Coulomb’s law and our knowledge of
field lines!
Electric Field of a Capacitor
The field lines inside a capacitor:
Electric Field of a Capacitor
The field lines inside a capacitor:
Electric Field of a Capacitor
point charge with
a charge Q.
capacitor with a charge Q
and plate area A
+
+
+
+
+
+
+
+
Electric Field of a Capacitor
•Field lines begin on the positive charge in both cases.
•Since the positive charge is the same, the number of field
lines is the same.
+
+
+
+
+
+
+
+
Electric Field of a Capacitor
N
N
E  k , E (r )  k
A
4 r 2
1 Q
E (r ) 
40 r 2
kN 
Q
N
Ek
A
Q
E
0 A
←same N →
N lines
between the
plates!
0
r
+
+
+
+
+
+
+
+
Electric Field of a Capacitor
Q
E
0 A
+
+
+
+
+
+
+
+
Parallel-plate Capacitors
We know how the voltage relates to the electric
field because the electric field is constant.














d
V   Edx  Ed
We always ignore the minus
sign, so V will be positive:
Qd
V  Ed 
0 A
Parallel-plate Capacitors
Now we can find the capacitance:














d
Q 0 A
C 
V
d
Parallel-plate Capacitors
Now we can find the capacitance:














d
Q 0 A
C 
V
d
•If the plate area is large, the capacitor
can hold more charge.
•If the plate separation is small, the
charges on the two plates attract each
other with a stronger force, so the
capacitor can hold more charge.
Parallel-plate Capacitor
Equations
Q  CV
V  Ed
0 A
C
d
Section 4
Dielectrics
Dielectrics
•A dielectric is an insulator with polar molecules
that is placed between the plates of a capacitor.
Dielectrics
•Polar molecules rotate in the electric field of the
capacitor.
Dielectrics
•The net charge inside the dielectric is zero.
Dielectrics
•But there is leftover charge on the surfaces of
the dielectric.
Dielectrics
•This charge produces an electric field that
opposes the electric field of the plates.
E of plates
E of dielectric
Problem Type 1:
Fixed Charge
•A capacitor is charged with a battery to a charge
Q. The battery is removed and a dielectric is
inserted.
Without the dielectric:
Q0  C0V0
With the dielectric:
V  ( E0  E d ) d 
V0

Q  CV  Q0
Q Q0
C 
 C0
V
V0
Problem Type 1:
Fixed Charge
•A capacitor is charged with a battery to a charge
Q. The battery is removed and a dielectric is
inserted.
Q  Q0
V
V0

C  C0
With the dielectric:
V  ( E0  E d ) d 
V0

Q  CV  Q0
Q Q0
C 
 C0
V
V0
Problem Type 1:
Fixed Charge
•A capacitor is charged with a battery to a charge
Q. The battery is removed and a dielectric is
inserted.
 1
Q  Q0
V
V0

C  C0
The electric field of the
dielectric reduces the voltage
across the capacitor, causing
the capacitance to rise.
Problem Type 2:
Fixed Voltage
•A capacitor is connected to a battery with voltage
V and remains connected as a dielectric is
inserted.
Without the dielectric:
Q  C0V0
With the dielectric:
V  V0
Q  CV  C0V0
Problem Type 2:
Fixed Voltage
•A capacitor is connected to a battery with voltage
V and remains connected as a dielectric is
inserted.
With the dielectric:
Q  Q0
V  V0
C  C0
V  V0
Q  CV  C0V0
Problem Type 2:
Fixed Voltage
•A capacitor is connected to a battery with voltage
V and remains connected as a dielectric is
inserted.
Q  Q0
V  V0
C  C0
The charge on the dielectric
pulls additional charge from
the battery to the plates,
causing the capacitance to
rise.
Section 5
Energy in Capacitors and Electric
Fields
Energy in a Capacitor
•Start with two parallel plates with no charge.
•Move one charge from one plate to the other.
•There is no electric field and no force, so it
requires no work.
Energy in a Capacitor
•After the charge is transferred, the capacitor has
a small charge and a small field.
•The field causes a force on the next charge we
move, forcing us to do work.
Energy in a Capacitor
•When the charge on a capacitor is q, the voltage
is q/C and the electric field is V/d=q/Cd.
q
dq
•The force on a small charge dq is F  (dq ) E 
Cd
Energy in a Capacitor
•The work done in moving the charge is
1
dW  Fd  qdq
C
Energy in a Capacitor
•The work done in charging the capacitor to its
final charge Q is:
Q
2
1
Q
1
W   dW   qdq 
 CV 2  U
C0
2C 2
Energy in a Capacitor
1
2
U  CV
2
Energy Density
Energy per unit volume in a an electric field.
In a parallel-plate capacitor of volume v=Ad :
1
1 0 A
1
2
2
Ed    0 E 2 Ad
U  CV 
2
2 d
2
U 1
u   0E2
v 2
Energy Density
The density of the energy stored in any
electric field, not just a capacitor, is:
1
2
u  0E
2
Class 17
Today, we will:
• learn how to combine capacitors in series and
parallel
• find that circuits RC circuits have charges and
currents that depend on exponential functions
• learn the meaning of the exponential time
constant
• find that the exponential time constant for an RC
circuit is τ=RC
Section 6
Capacitors in DC Circuits
Capacitors in Circuits
•In DC circuits, capacitors just charge or
discharge.
•No current flows after a capacitor is fully
charged or discharged.
Capacitors in Circuits
•Describe what happens in this circuit after the
switch is closed.
5Ω
1Ω
12 V

2Ω
20 μF
Capacitors in Circuits
•Initially positive charge on the right plate of the capacitor
pushes charge off the left plate. It is as if the capacitor
were replaced by a wire.
5Ω
20 μF
1Ω
12 V

2Ω
Capacitors in Circuits
•When the capacitor starts charging, it behaves like a
battery that opposes the flow of current.
5Ω
20 μF
─
1Ω
12 V

2Ω
+
Capacitors in Circuits
•Eventually, the capacitor becomes fully charged. No more
current flows. What is the final voltage on the capacitor?
5Ω
20 μF
─
1Ω
12 V

2Ω
+
Capacitors in Circuits
•First, ignore the branch with the capacitor.
•Rtotal=3 Ω. I = 4 A. V across the 1 Ω resistor is IR = 4 V.
5Ω
20 μF
─
1Ω
12 V

2Ω
+
Capacitors in Circuits
•V across the 5 Ω resistor is 0. Why?
•V across the capacitor is 4 V.
•Q on the capacitor CV = 80 μC
5Ω
20 μF
─
1Ω
12 V

2Ω
+
Capacitors in Circuits
•Summary:
In steady state, no current flows through the
capacitor. Just find the voltage across the
capacitor and you can determine the charge.
Section 7
Capacitors in Series and Parallel
Adding Capacitors
Resistors:
Capacitors:
V  IR
1
V Q
C
1
R
C
R and 1/C enter the voltage equations in a similar way.
If you replace R with 1/C in series-parallel equations for
resistors, you get the correct result for capacitors!
Adding Capacitors
Series:
1 1
1
Q  Q1  Q2 , V  V1  V2 ,


C C1 C2
Parallel:
V  V1  V2 , Q  Q1  Q2 , C  C1  C2
Let’s look at the voltage and
charge equations…
•As with resistors, the voltages across two
capacitors in series add to get the total
voltage.
•As with resistors, the voltages across two
capacitors in parallel are the same.
•When we discharge two capacitors in
parallel, the total charge that leaves the
capacitors is the sum of the charges. (Recall
that with resistors the sum of the currents is
the total current in parallel.)
Why are the charges the same on
capacitors in series?
To begin with, there is no charge on either
capacitor.
Why are the charges the same on
capacitors in series?
Before we start charging the two capacitors,
the charge within the dashed box is zero.
Why are the charges the same on
capacitors in series?
q
q
q
q
I
A the capacitors charge, the charge within
the dashed box remains zero.
Why are the charges the same on
capacitors in series?
+Q
–Q
When the left plate of the left capacitor
acquires its final charge +Q, the right plate’s
charge is –Q.
Why are the charges the same on
capacitors in series?
+Q
–Q
+Q
–Q
The charge within the box must remain zero,
so the right capacitor must have the same
charge as the left capacitor.
Section 8
Charging and Discharging
Capacitors and the Time Constant
RC Discharging


I
•Charge a capacitor with a
battery to a voltage V.
•Disconnect the capacitor
and attach it to a resistor.
Q0
•The initial charge is Q=CV.
Q(t)
•The charge decays to zero
– but what is Q(t)?
t
RC Discharging


I


•Look at the voltage around
the circuit. We can use
Kirchoff’s loop rule:
I


Voltage
Q
 IR  0
C
VC 
Q
C
VR  IR
RC Discharging
Voltage


VC 
I

Q
C

VR  IR
Q
 IR  0
C
I  0, Q  0
dQ
I 
dt
The minus sign comes from:
1) I > 0
2) Q is the charge on the capacitor
dQ
0
3) The capacitor is discharging so
dt
RC Discharging
Voltage


VC 
I

Q
C

VR  IR
Q
 IR  0
C
I  0, Q  0
dQ
I 
dt
Q
dQ
R
0
C
dt
dQ
1

Q(t )
dt
RC
RC Discharging
dQ
1

Q(t )
dt
RC
•This is a differential equation, but it is a really easy
one to solve.
•Usually we’ll just give you the solutions to
differential equations.
Qf
dQ
1
dQ
1

dt  

dt  ln

Q
RC
Q
RC
Qi
RC Discharging
dQ
1

dt
Q
RC
Q (t )

Q0
t
dQ
1

dt

Q
RC 0
Q(t )
1
ln

t,
Q0
RC
Q(t )  Q0 e t /
  RC
RC Time Constant
•τ (tau) is called the “RC time constant.” τ = RC.
• τ has units of seconds.
•When τ is big, capacitors charge and
discharge slowly.
•If R is large, not much current flows, so τ is big.
•If C is large, there is a lot of charge that has to
flow, so τ is big.
RC Discharging
Discharging capacitors with three different
time constants.
The time constant is the time it takes the
charge to drop to 1/e of its original value.
1
e
τ=3 s
τ=2 s
τ=1 s
RC Charging
C
V0

R
•A capacitor is initially
uncharged.
•Use a battery with voltage
V0 to charge the capacitor.
The voltage increases to V0.
•The charge increases to
Q=CV0.
RC Charging

I



•We again use Kirchoff’s
Loop rule:

Q
V0  IR   0
C
Voltage
VR  IR
V0
VC 
Q
C
RC Charging

I



•We again use Kirchoff’s
Loop rule:

Voltage
VR  IR
V0
VC 
Q
C
Q
V0  IR   0
C
dQ
I 
0
dt
dQ Q
V0  R
 0
dt C
RC Charging
dQ Q
V0  R
 0
dt C
•This differential equation has the solution:

Q (t )  Q f 1  e
 t /

Q f  CV0
  RC
•Try plugging the solution into the differential
equation and see if it works!
RC Charging
1
1
e
τ=1
s
τ=2
s
τ=3
s
Charging capacitors
with three different
time constants.
The time constant is
the time it takes the
charge to rise to 1-1/e
of its final value.