W10D2
DC Circuits
Today’s Reading Assignment W10D2 DC Circuits &
Kirchhoff’s Loop Rules Course Notes: Sections 7.1-7.5
1
Announcements
PS 8 due Week 11 Tuesday at 9 pm in boxes outside 32-082 or
26-152
Next Reading Assignment W10D3 PS07: PhET: Building a
Circuit 7.1-7.5, 7.10
Exam 3 Thursday April 24
7:30 pm –9:30 pm
Conflict Exam 3 Friday April 25 8-10 am, 10-12 noon
If you have a regularly scheduled MIT activity that conflicts
with Exam 3, contact padour@mit.edu and explain conflict
and request time to take conflict exam.
2
Outline
DC Circuits
Kirchoff’s Laws
Electrical Power
Measuring Voltage and Current
3
Class 12
1
DC Circuits
4
Examples of Circuits
5
Symbols for Circuit Elements
Battery
Resistor
Capacitor
Switch
Equipotential
Junction
Branch
6
Class 12
2
Electromotive Force (EMF)
The work per unit charge around a closed path done
is called electromotive force EMF. Thisis a bad
name because it is not a force. Let fs denote the
force per unit charge, then the EMF is
ε = ∫
 
fs ⋅ d s
closd path
If a conducting closed path is present then
ε = IR
7
Ideal Battery

1)  Inside Battery: chemical force fs , non-zero inside battery
(zero outside), moves
in which
 charges through a region


static electric field Estatic opposes motion Estatic = − fs
2) Ideal battery: net force on charges is zero
3) Potential difference between its terminals
+
+
 


V (+) − V (−) = − ∫ Estatic ⋅ d s = ∫ fs ⋅ d s
−
−


4) Extend path through external circuit where fs = 0
+
 
 
V (+) − V (−) = ∫ fs ⋅ d s = 
∫ fs ⋅ d s = ε
−
8
Kirchhoff’s Loop Rules
9
Class 12
3
Charge Conservation
Sum of currents entering any junction in a circuit
must equal sum of currents leaving that junction.
I1 = I 2 + I 3
10
Sum of Potential Differences
Around a Closed Path
Sum of potential differences across all elements
around any closed circuit loop must be zero.


∑ ΔVi = − ∫ Estatic ⋅ d s = 0
i
Closed
Path
11
DC Circuits Rules for Batteries
and Resistors
12
Class 12
4
Reference System for Circuit
Analysis
1.  Simplify resistors in series/parallel.
2.  Assign current and current direction in each
branch
3.  Assign circulation direction for each loop.
4.  Circulation direction across each element
defines “before” and “after” side of element.
5.  Define electric potential difference across a
circuit element by
ΔV ≡ V (after) − V (before) ≡ V (a) − V (b)
13
Sign Conventions - Battery
Circulating from the negative to positive terminal
of a battery increases your potential
ΔV = +ε
Circulating from the positive to negative terminal
of a battery decreases your potential
ΔV = −ε
14
Sign Conventions - Resistor
Direction of current is the same as direction of electric
field which points to lower electric potential
Circulating across a resistor in the direction of current
decreases the potential
ΔV = − IR
Circulating across a resistor in the opposite direction
of current increases your potential
ΔV = + IR
15
Class 12
5
Series vs. Parallel
Series
Parallel
16
Resistors In Series
The same current I must flow through both resistors
Circulate Clockwise:
ΔV − I R1 + I R2 = 0 ⇒ ΔV = I(R1 + R2 ) = I Req
Req = R1 + R2
17
Resistors In Parallel
Voltage drop across the resistors must be the same
ΔV = ΔV1 = ΔV2 = I1 R1 = I 2 R2 = IReq
I = I1 + I 2 =
ΔV ΔV ΔV
1
1
1
+
=
⇒
= +
R1
R2
Req
Req R1 R2
18
Class 12
6
Concept Q: Resistors In Parallel
Suppose that R1 >> R2 .
Then the equivalent
resistance
1.  Req  R2
2. Req  R1
3. Req  R2 / 2
4. Req  R2 / 2
19
Group Problem: Four Resistors
Four resistors are
connected to a battery
as shown in the figure. The
current in the battery is I,
the battery emf is ε, and
the resistor values are R1 =
R, R2 = 2R, R3 = 4R, R4 =
3R.
Determine the current in
each resistor in terms of I.
20
Internal Resistance
Real batteries have an internal resistance, r, which is
small but non-zero
Circulate clockwise:
Terminal voltage:
ΔV = Vb − Va = ε − I r
(Even if you short the leads you don’t get infinite current)
Class 12
7
Measuring Voltage & Current
22
Measuring Potential Difference
A voltmeter must be hooked in parallel across the
element you want to measure the potential difference
across
1
1
1
= +
Reffective R RVoltmeter

1
R
Voltmeters have a very large resistance, so that
they don’t affect the circuit too much
23
Measuring Current
An ammeter must be hooked in series with the
element you want to measure the current through
Reffective = R + Rammeter
R
Ammeters have a very low resistance, so that
they don’t affect the circuit too much
Class 12
24
8
Concept Question: Measuring
Current
If R1 > R2, compare the
currents measured by the
three ammeters:
1.
2.
3.
4.
5.
6.
7.
I1 > I2 > I3
I2 > I1 > I3
I3 > I1 > I2
I3 > I2 > I1
I3 > I1 = I2
None of the above
Not enough information is given.
P18- 25
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)
26
Concept Question: Bulbs &
Batteries
An ideal battery is hooked to a light
bulb with wires. A second identical
light bulb is connected in parallel to
the first light bulb. After the second
light bulb is connected, the current
from the battery compared to when
only one bulb was connected
1.  is higher.
2.  is lower.
3.  is the same.
P18- 27
Class 12
9
Concept Question: Bulbs &
Batteries
An ideal battery is hooked to a
light bulb with wires. A second
identical light bulb is connected in
series with the first light bulb. After
the second light bulb is connected,
the current from the battery
compared to when only one bulb
was connected
1.  is higher.
2.  is lower.
3.  is the same.
P18- 28
Electrical Power
Power is change in energy per unit time
So power to move current through circuit elements:
P=
d
d
dq
ΔU ) = ( qΔV ) =
ΔV
(
dt
dt
dt
P = I ΔV
29
Power - Battery
Moving from the negative to positive terminal of a
battery increases your potential. If current flows
in that direction the battery supplies power
I
Psupplied = I ΔV = I ε
30
Class 12
10
Power – Resistor (Joule Heating)
Moving across a resistor in the direction of current
decreases your potential. Resistors always
dissipate power
Pdissipated = I ΔV = I 2 R =
ΔV 2
R
31
Concept Question: Power
An ideal battery is hooked to a light bulb
with wires. A second identical light bulb
is connected in parallel to the first light
bulb. After the second light bulb is
connected, the power output from the
battery (compared to when only one bulb
was connected)
1.
2.
3.
4.
5.
Is four times higher
Is twice as high
Is the same
Is half as much
Is ¼ as much
P18- 32
Concept Question: Power
An ideal battery is hooked to a light
bulb with wires. A second identical
light bulb is connected in series with
the first light bulb. After the second
light bulb is connected, the light
(power) from the first bulb (compared
to when only one bulb was connected)
1.
2.
3.
4.
5.
Is four times higher
Is twice as high
Is the same
Is half as much
Is ¼ as much
P18- 33
Class 12
11
Demonstration:
Five Different Types of Resistance
F13
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F 13&show=0
34
Worked Example: Circuit
What is current through each branch of the circuit
shown in the figure below?
35
Worked Example: Simple Circuit
Current conservation at a:
I1 = I 2 + I 3
Lower Loop: Start at a and
circulate clockwise. Then
− I 2 2R1 + I 3 3R1 + ε = 0 ⇒
I 3 = (2 / 3)I 2 − (ε / 3R1 )
Upper Loop: Start at a and circulate counterclockwise. Then
− I 2 2R1 − I1 R1 + 2ε = 0 ⇒ I1 = (2ε / R1 ) − 2I 2
I1 = I 2 + I 3 ⇒ (2ε / R1 ) − I 2 = I 2 + (2 / 3)I 2 − (ε / 3R1 )
Solve for I2:
I 2 = 7ε / 11R1 ⇒ I 3 = ε / 11R1 and I1 = 8ε / 11R1
Class 12
36
12
Demonstration:
Wheatstone Bridge F14
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F 14&show=0
37
Group Problem: Wheatstone Bridge
A circuit consisting of two
resistors with R1=6.0 ohms and
R2=1.5 ohms, a variable resistor,
with resistance Rvar, a resistor of
unknown value Ru, and 9.0 volt
battery, are connected as shown
in the figure. When Rvar is
adjusted to 12 ohms, there is
zero current through the
ammeter. What is the unknown
resistance Ru?
38
Class 12
13