Lecture 7 Circuits Chp. 28
• Cartoon -Kirchoff’s Laws• Opening Demo- transmission lines• Physlet • Topics
– Direct Current Circuits– Kirchoff’s Two Rules– Analysis of Circuits Examples– Ammeter and voltmeter– RC circuits
• Demos– Ohms Law– Power loss in transmission lines– Resistivity of a pencil– Blowing a fuse
• Warm-up problems
Transmission line demo
Direct Current Circuits
1. The sum of the potential charges around a closed loop is zero. This follows from energy conservation and the fact that the electric field is a conservative force.
2. The sum of currents into any junction of a closed circuit must equal the sum of currents out of the junction. This follows from charge conservation.
Example (Single Loop Circuit)
No junction so we don’t need that rule.
How do we apply Kirchoff’s rule?
Must assume the direction of the current – assume clockwise.
Choose a starting point and apply Ohm’s Law as you go around the circuit.
a. Potential across resistors is negative
b. Sign of E for a battery depends on assumed current flow
c. If you guessed wrong on the sign, your answer will be negative
Start in the upper left hand corner.
21321
21
1132221 0
rrRRR
EEi
irEiRirEiRiR
++++−
=
=−+−−−−−
21321
21
rrRRR
EEi
++++−
=
Put in numbers.
VE
VE
rr
RRR
5
10
1
10
2
1
21
321
==
Ω==Ω===Suppose:
32
5
11101010
510=
++++−
=i amp
VE
VE
10
5
2
1
==Suppose:
32
5
32
105 −=
−=i
amp
We get a minus sign. It means our assumed direction of current must be reversed.
Note that we could have simply added all resistors and get the Req. and added the EMFs to get the Eeq. And simply divided.
32
5
)(32
)(5
R .e
.=
Ω==
VEi
q
eqamp
Sign of EMF
Battery 1 current flows from - to + in battery +E1
Battery 2 current flows from + to - in battery -E2
In 1 the electrical potential energy increases
In 2 the electrical potential energy decreases
Example with numbers
Quick solution:
AE
I
R
VVVVE
q
eq
i
i
i
i
16
10
R
16
102412
.e
.
6
1
3
1
==
Ω=
=+−=
∑
∑
=
=
Question: What is the current in the circuit?
Write down Kirchoff’s loop equation.
Loop equation
Assume current flow is clockwise.
Do the batteries first – Then the current. AampsV
i
i
625.0625.016
10
0)311551()2412(
==Ω
=
=+++++−+−+
Example with numbers (continued)
Question: What are the terminal voltages of each battery?
12V:
2V:
4V:
€
V =ε − ir =12V − 0.625A • 1Ω =11.375V
V =ε − ir = 2V − 0.625A • 1Ω =1.375V
V =ε − ir = 4V + 0.625A • 1Ω = 4.625V
Multiloop Circuits
Kirchoff’s Rules1. in any loop
2. at any junction
0=∑i
iV
∑∑ = outin ii
Find i, i1, and i2
Rule 1 – Apply to 2 loops (2 inner loops)
a.
b.
Rule 2
a.
0452
03412
12
1
=+−−=−−ii
ii
21 iii +=
We now have 3 equations with 3 unknowns.
0245
03712
0)(3412
21
21
211
=−+−=−−
=+−−
iiii
iii
061215
061424
21
21
=−+−=−−ii
ii
Ai
Ai
Ai
i
0.2
5.0
5.126
39
02639
2
1
1
==
==
=−
multiply by 2
multiply by 3
subtract them
Find the Joule heating in each resistor P=i2R.
Is the 5V battery being charged?
Method of determinants for solving simultaneous equations
5240
12043
0
21
1
21
=−+−=+−−
=−−
iiiiiii
For example solve for i
Ai 226
52
6128
242048
240
043
111
245
0412
110
==+++−
=
−+−−
−−−+
−−−−
=
You try it for i1 and i2.
See Appendix in your book on how to use Cramer’s Rule.
Another example
Find all the currents including directions.
21
121
1
3580
23380
234480
ii
iii
iiVVV
−−=−−−=
−−−++=
012120
010166
0246
1
12
12
=+−=−+−
=++−
iii
ii 0)1(2426 =++− AiLoop 1 Loop 2
Ai 11 =Ai
Ai
2
12
==
multiply by 2
i = i1+ i2
Loop 1
Loop 2
i i
i
i
i1i2
i2
Rules for solving multiloop circuits
1. Replace series resistors or batteries with their equivalent values.
2. Choose a direction for i in each loop and label diagram.
3. Write the junction rule equation for each junction.
4. Apply the loop rule n times for n interior loops.
5. Solve the equations for the unknowns. Use Cramer’s Rule if necessary.
6. Check your results by evaluating potential differences.
How does a capacitor behave in a circuit with a resistor?
Charge capacitor with 9V battery with switch open, then remove battery.
Now close the switch. What happens?
Discharging a capacitor through a resistor
V(t)
Potential across capacitor = V =
just before you throw switch at time t = 0.
Potential across Resistor = iR
at t > 0.RC
QiRi
C
Q ooo
o=⇒=
C
Qo
What is the current I at time t?
€
i(t) =Q(t)
RC
Time constant = RC
What is the current?
€
Q =Q0e−t
RC
€
i =dQ
dt= −
Q0
RCe
−t
RC = −V0
Re
−t
RCIgnore - sign
RC
How the charge on a capacitor varies with time as it is being charged
Ohmmeter
Ammeter
Voltmeter
Warm up set 7Warm up set 7 Due 8:00 am Tuesday 1. HRW6 28.TB.05. [119859] In the context of the loop and junctions rules for electrical circuits a junction is:
where a wire is connected to a battery where three or more wires are joined where a wire is bent where a wire is connected to a resistor where only two wires are joined
2. HRW6 28.TB.18. [119872] Two wires made of the same material have the same length but different diameter. They are connected in parallel to a battery. The quantity that is NOT the same for the wires is:
the electric field the electron drift velocity the current the current density the end-to-end potential difference
3. HRW6 28.TB.26. [119880] The emf of a battery is equal to its terminal potential difference:
only when there is no current in the battery only when a large current is in the battery under all conditions under no conditions only when the battery is being charged