Class 12: Outline
Hour 1:
Working with Circuits
Expt. 4. Part I: Measuring V, I, R
Hour 2:
RC Circuits
Expt. 4. Part II: RC Circuits
P12- 1
Last Time:
Resistors & Ohm’s Law
P12- 2
Resistors & Ohm’s Law
R=
ρA
A
∆V = IR
1
Rseries = R1 + R2
Rparallel
1
1
= +
R1 R2
P12- 3
Measuring Voltage & Current
P12- 4
Measuring Potential Difference
A voltmeter must be hooked in parallel across the
element you want to measure the potential difference
across
Voltmeters have a very large resistance, so that
they don’t affect the circuit too much
P12-
Measuring Current
An ammeter must be hooked in series with the
element you want to measure the current through
Ammeters have a very low resistance, so that
they don’t affect the circuit too much
P12-
Measuring Resistance
An ohmmeter must be hooked in parallel across the
element you want to measure the resistance of
Here we are measuring R1
Ohmmeters apply a voltage and measure the
current that flows. They typically won’t work if the
resistor is powered (connected to a battery)
P12-
Experiment 4:
Part 1: Measuring V, I & R
P12- 8
RC Circuits
P12- 9
(Dis)Charging a Capacitor
1. When the direction of current flow is toward
the positive plate of a capacitor, then
Charging
dQ
I =+
dt
2. When the direction of current flow is away from
the positive plate of a capacitor, then
Discharging
dQ
I =−
dt
P12-10
Charging A Capacitor
What happens when we close switch S?
P12-11
Charging A Capacitor
NO CURRENT
FLOWS!
1. Arbitrarily assign direction of current
2. Kirchhoff (walk in direction of current):
Q
∑i ∆Vi = ε − C − IR = 0
P12-12
Charging A Capacitor
dQ
dt
Q dQ
=−
ε− =
R⇒
Q − Cε
RC
C dt
∫
Q
0
t dt
dQ
= −∫
0 RC
Q − Cε
A solution to this differential equation is:
ε (1 − e
Q (t ) = C
− t / RC
)
RC is the time constant, and has units of seconds
P12-13
Charging A Capacitor
Q = Cε 1 − e
(
− t / RC
)
dQ ε − t / RC
= e
I=
dt
R
P12-14
PRS Questions:
Charging a Capacitor
P12-15
Discharging A Capacitor
What happens when we close switch S?
P12-16
Discharging A Capacitor
NO CURRENT
FLOWS!
dq
I =−
dt
q
∑i ∆Vi = C − IR = 0
P12-17
Discharging A Capacitor
Q
t
dq
q
dq
dt
+
=0 ⇒ ∫
= −∫
dt RC
q
RC
0
Q0
Q (t ) = Qo e
− t / RC
P12-18
General Comment: RC
All Quantities Either:
Value(t ) = Value Final (1 − e − t /τ )
Value(t ) = Value0 e − t /τ
τ can be obtained from differential equation
(prefactor on d/dt) e.g. τ = RC
P12-19
Exponential Decay
Very common curve in
physics/nature
(t0,v0)
(t0+τ,v0/e)
Value(t ) = Value0 e − t /τ
How do you measure τ?
1) Fit curve (make sure
you exclude data at
both ends)
2) a) Pick a point
b) Find point with y
value down by e
c) Time difference is τ
P12-20
Demonstrations:
RC Time Constants
P12-21
Experiment 4:
Part II: RC Circuits
P12-22
PRS Question:
Multiloop circuit with Capacitor
in One Loop
P12-23