RC Circuits!
Lecture 15
Announcements
•
Exam on Thursday.
•
Check the website
•
Practice exam & solutions
•
Seating chart
•
Study hard!
RC Demo
Charging: Overview of the action.
I
R
Q=0
Vb
+
–
C
Q=0
Q = CVb
+
+
–
–
++ ++
–– ––
Charging: Overview of the action.
I
R
Q=0
Vb
+
–
C
Q=0
Q = CVb
+
+
–
–
My capacitor heuristic:
•
Small t (large ν) →
C is a short circuit
•
Large t (small ν) →
C is an open circuit
++ ++
–– ––
Charging: Setting up the equations
I
R
Q
Vb
+
–
++ ++
C
–– ––
Kirchhoff Voltage Law:
Vb + IR + Q/C = 0
Aside: Does current really
flow through a capacitor?
•
Current through 3
surface
•
The current through
the gap is always 0!
•
“Current through the
capacitor” = change in Q on
plates
++++++
I(t) = 0
– – – – – –
Aside: Does current really
flow through a capacitor?
•
++++++
I(t) = 0
– – – – – –
“Current through the
capacitor” = change in Q on
plates
dQ
I=
dt
Charging: Setting up the equations
I
R
Q
Vb
+
–
++ ++
C
–– ––
Kirchhoff Voltage Law:
dQ
Vb + R
+ Q/C = 0
dt
Initial State:
Q=0
Charging: Solving the equations
Overhead.
Charging: Solution
RC time constant:
h
Q(t) = CVb 1 e
h
VC (t) = Vb 1 e
⌧ ⌘ RC
i
t/⌧
t/⌧
i
I
R
Q
Vb
+
++ ++
C
–
Vb
I(t) =
e
R
–– ––
t/⌧
Quick check of the units….
•
RC time constant units:
[⌧ ] = [R][C] = [V ][I]
1
[Q][V ]
1
= [dQ/dt]
1
[Q] = s
Discharging: Overview of the action.
I
R
Q = CVb
Vb
+
–
++ ++
C
–– ––
+
–
++ ++
–– ––
Q=0
+
–
Discharging: Setting up the equations
I
Kirchhoff Voltage Law:
R
Q
C
++ ++
–– ––
RI + Q/C = 0
Initial State:
Q(0) = CVb
Discharging: Solution
I
R
Q
RC time constant:
Q(t) = CVb e
C
⌧ ⌘ RC
t/⌧
++ ++
–– ––
I(t) =
Vb
e
R
t/⌧
Energy flow in an RC circuit.
•
Power:
P = IV
+
•
Energy stored: U =
2
1
CV
2
–
++ ++
–– ––