Chapter 27
Circuits
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
“Invention is the most important
product of man’s creative brain.
The ultimate purpose is the
complete mastery of mind over
the material world, the
harnessing of human nature to
human needs.”
- Nikola Tesla
David J. Starling
Penn State Hazleton
PHYS 212
Circuits
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Objectives
(a) Apply Kirchhoff’s loop and junction rules for voltages
and currents to electric circuits and relate these rules to
first principles.
(b) Explain the role of internal resistance in real electric
circuits and describe how including internal resistance
affects the behavior of a circuit.
(c) Determine equivalent resistances of a set of resistors in
series, parallel, and more complex arrangements.
(d) Describe the behavior of an RC circuit and determine
the characteristic time constant for the system.
Multi-Loop Circuits
RC Circuits
Chapter 27
Circuits
Work, Energy and EMF
A battery does work on charge in a circuit.
I
How much work does it do, per charge?
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This is called the electromotive force, or emf.
I
Definition:
E=
I
dW
dq
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
(1)
Examples: batteries, generators, solar cells, fuel cells
(H), electric eels, human nervous system
Work, Energy and EMF
Chapter 27
Circuits
Work, Energy and EMF
Let’s look at the energy in this circuit:
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
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The battery does work: dW = Edq
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The resistor dissipate energy: Pdt = i2 Rdt
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Recall the definition of current: dq = idt
Work, Energy and EMF
Let’s look at the energy in this circuit:
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
I
Putting this together:
dW = Pdt (conservation of energy)
E dq = i2 R dt
Ei = i2 R
E
= iR
The emf E of an emf device is the potential V that it
produces.
Work, Energy and EMF
Question 1:
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Consider a circuit that contains an ideal battery and a
Multi-Loop Circuits
RC Circuits
resistor to form a complete circuit. Which one of the
following statements concerning the work done by the
battery is true?
(a) No work is done by the battery in such a circuit.
(b) The work done is equal to the thermal energy
dissipated by the resistor.
(c) The work done is equal to the work needed to move a
single charge from one side of the battery to the other.
(d) The work done is equal to the emf of the battery.
(e) The work done is equal to the product iR.
Single-Loop Circuits
A few rules to remember:
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
I
Resistance Rule: the potential drops by iR across a
Multi-Loop Circuits
resistor in the direction of the current; the potential
RC Circuits
increases by iR in the opposite direction.
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Emf Rule: the potential increases by E from the
negative to the positive terminal, regardless of the
direction of current.
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Loop Rule: the sum of the changes in potential
encountered in a complete traversal of any loop of a
circuit must be zero.
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Junction Rule: the sum of the currents entering a
junction equals the sum of the currents leaving that
junction.
Single-Loop Circuits
Resistance Rule
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
I
The potential drops by iR across a resistor in the
Multi-Loop Circuits
direction of the current; the potential increases by iR in
RC Circuits
the opposite direction.
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Vf − Vi = −iR
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∆V = −iR
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Or, Vi − Vf = +iR (against the current)
Single-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Emf Rule
Single-Loop Circuits
Multi-Loop Circuits
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The potential increases by E from the negative to the
positive terminal, regardless of the direction of current.
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The roles of Vi and Vf have switched.
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Vi − Vf = E
RC Circuits
Single-Loop Circuits
Loop Rule
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
I
The sum of the changes in potential encountered in a
complete traversal of any loop of a circuit must be zero.
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Start at Vi and end at Vi :
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Vi − iR + E = Vi
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Opposite direction: Vi − E + iR = Vi .
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Both equations: E − iR = 0
Multi-Loop Circuits
RC Circuits
Single-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Junction Rule
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The sum of the currents entering a junction equals the
sum of the currents leaving that junction.
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For junction b: i2 = i1 + i3
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For junction d: i1 + i3 = i2
Multi-Loop Circuits
RC Circuits
Single-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Let’s look at a “real” battery:
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
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Most batteries have some internal resistance r.
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This resistance makes the battery voltage seem smaller
than the emf E.
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The larger the current, the worse the effect.
Chapter 27
Circuits
Single-Loop Circuits
Work, Energy and EMF
Let’s look at a “real” battery:
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
I
Let’s sum up the changes:
E − ir − iR = 0
i =
E
r+R
Chapter 27
Circuits
Single-Loop Circuits
Another example:
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
I
Let’s sum up the changes:
E − iR1 − iR2 − iR3 = 0
i =
E
R1 + R2 + R3
This is the same if we replace all three series resistors with
Req = R1 + R2 + R3 .
Chapter 27
Circuits
Single-Loop Circuits
For resistors in series:
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
We replace them with an equivalent resistance:
Req = R1 + R2 + R3 =
3
X
i=1
Ri
(2)
Single-Loop Circuits
Question 2
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Which resistors, if any, are connected in series?
Multi-Loop Circuits
RC Circuits
(a) R1 and R2
(b) R1 and R3
(c) R2 and R3
(d) R1 , R2 and R3
(e) None
Single-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Example 1
Multi-Loop Circuits
RC Circuits
Find the potential Vba across the battery in the figure below:
Single-Loop Circuits
Example 2
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
We can throw in a wire to “ground” the circuit:
∆V doesn’t change.
(a) Since Va = 0, Vb = 8 V.
(b) Since Vb = 0, Va = −8 V.
RC Circuits
Single-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Example 3
Single-Loop Circuits
Multi-Loop Circuits
Consider the following circuit:
E1 = 4.4 V, E2 = 2.1 V
r1 = 2.3 Ω, r2 =1.8 Ω, R = 5.5 Ω
(a) What is the current i?
(b) What is the potential difference across battery 1?
RC Circuits
Single-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Example 3
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
A plot of the change in potential:
Multi-Loop Circuits
Three parallel resistors:
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
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Resistors in parallel all have the same voltage V
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But they may have different currents
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In this example, i = i1 + i2 + i3
Chapter 27
Circuits
Multi-Loop Circuits
Three parallel resistors:
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
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Each current is given by i1 = V/R1
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This gives:
V V V
i = i1 +i2 +i3 =
+ +
=V
R1 R2 R3
1
1
1
+
+
R1 R2 R3
Multi-Loop Circuits
For three parallel resistors, we can use an equivalent
resistance:
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
I
such that V = iReq , with
1
1
1
1
=
+
+
Req
R1 R2 R3
Multi-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
In summary:
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
Multi-Loop Circuits
Chapter 27
Circuits
Work, Energy and EMF
Example 4
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
Find the current through each resistor:
R1 = R2 = 20 Ω, R3 = 30 Ω, R4 = 8.0 Ω and E = 12 V.
Multi-Loop Circuits
Example 5
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
Find the direction and magnitude of the current in each
branch:
R1 = 2 Ω, R2 = 4 Ω, E1 = 3 V, E2 = 6 V.
Chapter 27
Circuits
RC Circuits
Consider a circuit with a resistor and a capacitor
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
With the battery connected, we find:
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E − iR − q/C = 0 (loop rule)
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i = dq/dt
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This gives:
R
dq
q
+ =E
dt
C
Chapter 27
Circuits
RC Circuits
Work, Energy and EMF
Single-Loop Circuits
dq
q
+ =E
dt
C
This is a differential equation
R
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The solution, with q(0) = 0:
q(t) = CE(1 − e−t/RC )
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Don’t believe me? Let’s check it!
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Note: V = q/C = E(1 − e−t/RC )
Multi-Loop Circuits
RC Circuits
Chapter 27
Circuits
RC Circuits
The combination τ = RC is the capacitive time constant.
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Single-Loop Circuits
If t = τ , then
Multi-Loop Circuits
−RC/RC
q(τ ) = CE(1 − e
I
) = 0.63CE
τ is the time it takes for the capacitor to charge about
63%
Work, Energy and EMF
RC Circuits
Chapter 27
Circuits
RC Circuits
Once the capacitor is charged, let’s remove the battery:
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
With the battery bypassed, we find:
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−iR − q/C = 0 (loop rule)
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i = dq/dt
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This gives:
R
dq
q
+ =0
dt
C
Chapter 27
Circuits
RC Circuits
dq
q
+ =0
dt
C
The solution, with q(0) = CE:
R
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q(t) = CEe−t/RC
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The battery discharges with the same time constant!
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
RC Circuits
Example 6 (E = 9 V, R = 10 Ω, C = 5 µF)
Chapter 27
Circuits
Work, Energy and EMF
Single-Loop Circuits
Multi-Loop Circuits
RC Circuits
The switch is left in position a for a long time. At t = 0, the
switch is thrown to position b.
What is the charge on C and the current through R at
(a) At t = 0
(b) At t = 2 s.
(c) As t → ∞