Download PHYS 202 Notes, Week 4

PHYS 202 Notes, Week 4
Greg Christian
February 9 & 11, 2016
Last updated: 02/09/2016 at 10:16:38
This week we multiloop and RC circuits, and review for Thursday’s
exam.
Multiloop Circuits
So far, you’ve learned to solve for currents and potential drops in relatively simple circuits, making combinations of resistors in series and in
parallel. However, in some cases the circuits may be too complicated
for this to be practical. For example, there can be more than one emf
source in a circuit, which complicates things.
To understand how to solve more complicated “multiloop” circuits,
we can make use of some equations called Kirchhoff’s Rules. But first
we need to define a couple of new concepts:
1. Junction: A junction is any point on a circuit where three or more
conductors meet.
2. Loop: A loop is any closed conducting path in a circuit.
Figure 1 shows examples of what are and aren’t junctions and loops
in a multiloop circuit.
Kirchhoff’s rules take advantage of the concepts of loop and junction to define two rules that help with solving circuits:
1. Kirchhoff’s Junction Rule: The algebraic sum of all currents going into
and out of a junction is zero:
I = 0.
• Definitions
– Junction: A circuit point where
three or more conductors meet.
– Loop: Any closed conducting path.
• Kirchhoff’s Rules
– The algebraic sum of currents into
and out of a junction is zero.
– The algebraic sum of all potential
differences in any loop is zero.
Important equations
• Kirchhoff’s Junction Rule
∑
• Kirchhoff’s Loop Rule
∑V=0
loop
Not a Junction
R1
junction
V = 0.
(2)
Note that this applies to both potential increases (e.g. due to an
emf source) and to potential drops (e.g. due to a resistor), with the
following rules applying:
• Currents flowing into a junction are considered positive while
currents flowing out of a junction are negative.
• The current flowing into a circuit element equals that flowing out
of it.
ℰ
+ -
∑
closed loop
Junction
ℰ
(1)
2. Kirchhoff’s Loop Rule: The algebraic sum of all potential differences
in any closed loop is zero:
I=0
junction
Not a Junction
+ -
∑
Important points
R3
R2
Loop 1
Not a Junction
Junction
Loop 2
Not a Junction
Figure 1: Illustration of junctions and
loops in a circuit.
phys 202 notes, week 4
2
• Crossing a battery from the − to + gives a positive potential of
+E . Crossing from the + to − terminal gives −E .
• Crossing a resistor in the direction of current flow gives a negative
potential − IR. Crossing a resistor against the current direction
gives a positive potential + IR.
Measurement Instruments
There are two types of basic measurement instruments that can be
used to probe circuits:
1. Ammeters measure the current flowing through a circuit branch.
Ammeters are a “series device,” places in-line with the circuit branch
in which you want to measure current. An ideal ammeter would
have zero resistance to avoid altering the circuit it’s measuring. In
reality this number is very low but not quite zero.
2. Voltmeters measure the potential difference, or voltage, between two
points. Voltmeters are a “parallel device,” placed in parallel to the
points you want to measure the potential drop across. An ideal
voltmeter would have infinite resistance so that no current flows
through it and it doesn’t alter the circuit.
Important points
• Ammeters measure current (ideal:
zero resistance).
• Voltmeters measure potential (ideal:
infinite resistance).
Figure 2 shows example placements of a voltmeter and an ammeter in
a circuit.
Figure 2: Example placements of a voltmeter (labeled V) and an ammeter (labeled A) in a circuit.
phys 202 notes, week 4
3
RC Circuits
Until now, we’ve just dealt with circuits that are steady state, i.e. a constant current flowing through resistors due to an emf source. However,
you can make combinations of resistors, emf sources, and capacitors
where the current becomes time dependent.
Figure 3 shows an example of such an RC Circuit. There’s a switch
in the circuit, initially open, that is closed at some time t0. Here’s what
happens:
• Initially, there is an open switch in the circuit, which means no
current flows at all. The potential drop across the resistor is zero
and the capacitor is uncharged.
• When you first close the switch, the charge on the capacitor is still
zero. But current immediately begins flowing. At this instant it
looks just like a resistor + emf, with current
I0 =
E
.
R
(3)
• Next the capacitor begins charging up. During this time, the charge
on the capacitor is increasing, decreasing the current flowing through
the resistor. The equation describing this is:
E = iR +
q
.
C
(4)
This equation can be solved to get the current i and charge q as a
function of time:
i = I0 e−t/RC
h
i
q = Q f 1 − e−t/( RC) .
Important points
• In an RC circuit, the current and
charge on the capacitor are functions
of time.
• If the capacitor is initially uncharged, current decreases exponentially and charge increases exponentially (“charging up”).
• If the capacitor is initially charged,
both current and charge decrease exponentially (“discharging”).
Important equations
• Capacitor initially uncharged and
charged up by a emf source:
E = iR + q C
i = I0 e− t/( RC)
h
i
q = Q f 1 − e− t/( RC)
I0 = E /R
Qf = EC
• Capacitor initially charged and then
discharging:
i = I0 e−t/( RC)
q = Q0 e−t/( RC) .
(5)
(6)
What’s happening is that the current is decreasing exponentially,
while the charge is increasing exponentially. Basically you’re trading current for charge on the capacitor.
• Eventually, the capacitor becomes fully charged (at t = ∞). From
the equations above you can figure out what happens (remember:
e−∞ = 0). The current drops to zero and the capacitor takes its
full charge value (as if it were the only thing connected to the emf
source):
Q f = E C.
(7)
Figure 4 shows plots of the exponential decay of current and exponential increase of charge in such an RC circuit.
In Eqns. (5) and (6), the quantity RC is called the time constant, denoted by the symbol τ:
τ = RC.
(8)
Figure 3: RC circuit.
phys 202 notes, week 4
RC Circuit
This is the characteristic time which describes how fast the current
decreases or charge increases:
1
RC = 1
I0 = 1
Qf = 1
0.8
Current [A]
• Small τ: the capacitor charges quickly.
• Large τ: the capacitor charges slowly.
4
So far we’ve considered the case of charging up a capacitor that’s
initially uncharged. However, there can also be a situation where the
capacitor is initially charged to some value Q0 and then discharges. Here
the equations describing current and charge are both exponential decays:
0.6
0.4
I0 / e
0.2
0
0
0.5
1
RC
1.5
2
2.5
3
2
2.5
3
Time [s]
0.9
(9)
q = Q0 e−t/( RC) .
(10)
This means that the current is initially at its maximum value I0 , at
the moment the discharge and exponentially decays down to zero.
Likewise, the charge on the capacitor also exponentially decays to zero
from its maximum value of Q0 .
In the case where the circuit was charged by some battery with emf
E , the initial charge is
Q0 = E C,
(11)
and the initial current is
I0 =
E
.
R
(12)
Qf / e
0.8
0.7
Charge [C]
i = I0 e−t/( RC)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
RC
Time [s]
Figure 4: RC circuit with the capacitor
initially uncharged.