Lecture 13
Physics II
Chapter 31
RC Circuits
Course website:
http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII
Lecture Capture:
http://echo360.uml.edu/danylov201415/physics2spring.html
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
Steady
current
Time-varying
current
In the preceding sections we dealt with circuits in which the circuits elements
were resistors and in which the currents did not vary with time. Here we
introduce the capacitor as a circuit element, which will lead us to the study of
time-varying currents.
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
RC circuit
(Charging a Capacitor)
Now, we know Kirchhoff’s rules and
let’s apply them
to study an RC circuit
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
Charging a Capacitor
The figure shows an RC circuit, some time after the switch was closed.
,
,
∆
?
We need to analyze it:
Let’s look at the circuit at some arbitrary moment of time t and
apply Kirchhoff’s loop rule:
∆
+∆
+∆
0
0
There are two variables I(t),Q(t), which are dependent:
/
(The resistor current is the rate at which charge is added to the capacitor)
0
0
It is not hard to solve, but we just present the solution
(see the solution at the end of this presentation)
The capacitor charge at time t is:
∆
1
⁄
1
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
⁄
denote   RC (the time constant) and
(full charge of the capacitor)
Resistor Current and Capacitor Voltage
Let’s calculate the resistor current:
⁄
  RC
⁄
1
⁄
⁄
⁄
This current looks like “The Land Run” of 1893 (the Oklahoma Territory)
shown in the movie “Far and Away (https://www.youtube.com/watch?v=jFrVoG-edFc)
No current. Electrons waiting
for a switch to be closed.
Race begins. Electrons are on
the way to their lands.
The first photo of a traveling electron
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
ConcepTest RC circuit 1
In the circuit shown, the capacitor is
originally uncharged. Describe the
behavior of the lightbulb from the instant
switch S is closed until a long time later.
When the switch is first closed, the current is high
and the bulb burns brightly. As the capacitor charges,
The voltage across the capacitor increases causing
the current to be reduced, and the bulb dims.
A) No light.
B) First, it is bright, then dim.
C) First, it is dim, then bright.
D) Steady bright.
RC circuit
(discharging)
We want to analyze the RC circuit:
,
,
∆
?
At t = 0, the switch closes and the charged capacitor
begins to discharge through the resistor.
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
RC circuit (discharging)
The figure shows an RC circuit, some time after the
switch was closed.
Kirchhoff’s loop law applied to this circuit
clockwise is:
Q and I in this equation are the instantaneous values of the capacitor charge and the
resistor current.
0
The resistor current
,
The resistor current is the rate at which charge is removed from the capacitor:
where Q0 is the charge at t = 0
0
denote time constant  as:
ln
The charge on the capacitor of an RC circuit
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
RC circuit (discharging)
Let’s plot it:
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
RC circuit (discharging)
Let’s calculate the resistor current:
∆
∆
∆
I0 is the initial current
The current undergoes the same exponential decay
0.37
2.7
Let’s calculate the voltage of the capacitor:
∆
/
∆
∆
/
the voltage across the capacitor
Now we know everything about the circuit [Q(t), I(t), and ΔV(t)]
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
ConcepTest RC circuit 1
A) Capacitor A.
Which capacitor discharges more
quickly after the switch is closed?
time constant  = RC
  = 12 µs   = 15 µs
   
So the capacitor A discharges faster than B
B) Capacitor B.
C) They discharge at the same rate.
D) Can’t say without
knowing the initial amount of charge.
ConcepTest RC circuit 3
What is the time constant for the discharge of
the capacitor shown in the figure?
A) 5 s
B) 4 s
C) 2 s
D) 1 s
E) The capacitor does not
discharge because the resistors
cancel each other
How about this?
+
=
time constant by definition  = ReqC
 = ReqC =4Ωx1F=4 seconds
 = ReqCeq
ConcepTest RC circuit 2
Figure shows the voltage as a function of time
of a capacitor as it is discharged (separately)
through three different resistors.
Rank in order, from largest to smallest, the
values of the resistances R1, R2, and R3.
∆
time constant by definition  = RC
From the figure we can see that:
   <  
.
∆
.
∆
.
∆
A) R1< R2< R3
B) R1< R3< R2.
C) R2< R3< R1.
D) Not enough information.
My application 
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
Discharging a Capacitor
Charging a Capacitor
I used an RC circuit in my paper.
What you should read
Chapter 31 (Knight)
Sections
 31.9
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Department of Physics and Applied Physics
Derivation
(charging a
capacitor)
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
Thank you
See you on Tuesday
95.144 Danylov Lecture 13
Department of Physics and Applied Physics
ConcepTest
Wheatstone Bridge
An ammeter A is connected
A) l
between points a and b in the
B) l/2
circuit below, in which the four
C) l/3
resistors are identical. The current
D) l/4
through the ammeter is:
E) zero
Since all resistors are identical,
the voltage drops are the same
across the upper branch and the
lower branch.
I
Thus, the potentials at points a
and b are also the same.
Therefore, no current flows.
V