### Resistivity and Resistance

```Capacitor
EMF
+
Resistor
-
Inductor
EMF
+
-
Resistor
i
What is Current ?
Electric Current:
Current is Conserved
i1 i2 i = i + i
1
2
3
+
-
i3
1A
2A
i
2A
i=?
A. 1
B. 1
C. 2
D. 0
A
A
A
A
1A
i
2A
4A
6A
i=?
A. 1
B. 3
C. 2
D. 4
A
A
A
A
1A
2A
2A
2A
3
A
4
A
i
i=
A. 4 A
C. 8 A
B. 2A
D. 0 A
Current Density:
Current density, J, is the current per unit area through any element of cross
section. It has the same direction as the velocity of the moving charges if
they are positive and the opposite direction if they are negative.
If the current is uniform and
parallel to dA, then
The SI unit for current density is the ampere per square meter (A/m2).
Since charge is conserved during the
transition, the amount of charge and
thus the amount of current cannot
change.
However, the current density
changes—it is greater in the narrower
conductor.
Current Density, Drift Speed:
In a conductor
•Electrons move randomly
•Drift with vd
•vd is opposite applied field
•Assume uniform current density
Resistance and Resistivity:
The resistance R is
The SI unit for resistance is the volt per ampere. This has a special name,
the ohm (symbol W):
In a circuit diagram, we represent a resistor and a resistance with the
symbol
R = r L/A
26.4: Resistance and Resistivity, Variation with Temperature:
The relation between temperature and resistivity for copper—and for
metals in general—is fairly linear over a rather broad temperature range.
For such linear relations we can write an empirical approximation that is
good enough for most engineering purposes:
26.5: Ohm’s Law:
Power in Electric Circuits:
The amount of charge dq that moves between two terminals in time interval
dt is equal to i dt.
This charge dq moves through a decrease in potential of
magnitude V, and thus its electric potential energy decreases in magnitude
by the amount
The power P associated with that transfer is the rate of transfer dU/dt, given
by
The unit of power is the volt-ampere (V A).
Resistances in Series:
Multi-loop Circuits:
For the left-hand loop,
For the right-hand loop,
And for the entire loop,
Resistors in Parallel:
where V is the potential difference between a and b.
From the junction rule,
In which of these circuits are R1 and R2 in parallel
A. a
B. a & c
C. a & b
D. All of them
E. None of them
R1 and R2 are in parallel. Their equivalent
resistance is
A.
B.
C.
D.
Always greater than either R1 or R2
Always less than either R1 or R2
Roughly equal to the mean of R1 and R2
Not enough Info
Consider two identical resistors wired in series (one behind the other). If
there is an electric current through the combination, the current in the
second resistor is
A) equal to
B) half
C) smaller than, but not necessarily half the current through the first
resistor.
Of the four light bulbs in the figure are identical, which circuit puts out more
light? Circuit I or circuit II?
A) I.
B) The two emit the same amount of light.
C) II.

Kirchhoff’s voltage
law (“loop law”)
+

Kirchhoff’s current
law (“junction law”)
+
-
+
n
åV = 0
i
i=1
Conservation of
energy
n
åI
i=1
i
=0
Conservation of
charge
Quick/useful notes with Kirchoff’s rules:
Consider the three identical light bulbs shown in the circuit.
Bulbs B and C are wired in series with each other and are
wired in parallel with bulb A. When the bulbs are connected
to the battery as shown, how does the brightness of each
bulb compare to the others?
a) Bulbs B and C are equally bright,
but bulb A is less bright.
b) Bulbs B and C are equally bright,
but less bright than bulb A.
c) All three bulbs are equally bright.
d) Bulbs A and B are equally bright, but bulb C is less bright.
e) Only bulb A is illuminated.
Consider the circuit shown in the drawing. Two identical light bulbs,
labeled A and B, are connected in series with a battery and are
illuminated equally. There is a switch in the circuit that is initially
open. Which one of the following statements concerning the two
bulbs is true after the switch is closed?
a) Bulbs A and B will be off.
b) Bulbs A and B will be equally
illuminated.
c) Bulb A will be brighter and bulb B
will be off.
d) Bulb A will be off and bulb B will be
brighter.
e) Both bulbs will be dimmer than before the switch was closed.
Two 20-W resistors are connected in parallel. A potential difference of 9 V is
then applied across both resistors. What is the resulting total current
through the two resistors?
a) 0.23 A
b) 0.45 A
c) 0.90 A
d) 2.2 A
e) 4.4 A
ConcepTest 21.11a
Lightbulbs
Two lightbulbs operate at 120 V, but
1) the 25 W bulb
one has a power rating of 25 W while
2) the 100 W bulb
the other has a power rating of 100 W.
3) both have the same
Which one has the greater
4) this has nothing to do
with resistance
resistance?
Since P = V2 / R , the bulb with the lower
power rating has to have the higher
resistance.
Follow-up: Which one carries the greater current?
Two light bulbs, one “50 W” bulb and one “100 W” bulb, are connected in
parallel with a standard 120 volt ac electrical outlet. The brightness of a
light bulb is directly related to the power it dissipates. Therefore, the
100 W bulb appears brighter. How does the brightness of the two bulbs
compare when these same bulbs are connected in series with the same
outlet?
a) Both bulbs will be equally bright.
b) The “100 W” bulb will be brighter.
c) The “50 W” bulb will be brighter.
ConcepTest 21.5a
Parallel Resistors I
1) 10 A
In the circuit below, what is the
2) zero
current through R1?
3) 5 A
4) 2 A
5) 7 A
The voltage is the same (10 V) across each
R2= 2 W
resistor because they are in parallel. Thus,
we can use Ohm’s Law, V1 = I1 R1 to find the
R1= 5 W
current I1 = 2 A.
10 V
Follow-up: What is the total current through the battery?
ConcepTest 21.6a
Current flows through a
Short Circuit
1) all the current continues to flow through
the bulb
connected across the
2) half the current flows through the wire,
the other half continues through the
bulb
bulb, what happens?
3) all the current flows through the wire
lightbulb. If a wire is now
4) none of the above
The current divides based on the
ratio of the resistances. If one of the
resistances is zero, then ALL of the
current will flow through that path.
Follow-up: Doesn’t the wire have SOME resistance?
ConcepTest 21.8a
More Circuits I
What happens to the voltage
1) increase
across the resistor R1 when the
2) decrease
switch is closed? The voltage will:
3) stay the same
R1
With the switch closed, the addition of R2
to R3 decreases the equivalent
S
resistance, so the current from the battery
increases. This will cause an increase in
R3
V
the voltage across R1 .
Follow-up: What happens to the current through R3?
R2
What is the approximate equivalent resistance of the five resistors shown in
the circuit?
a) 21 W
b) 7 W
c) 11 W
d) 14 W
e) 19 W


Voltages are measured
across (in parallel with) the
circuit element(s) in
question. Voltmeters ideally
have infinite resistance.
Currents are measured in
series with the circuit
element(s) in question.
Ammeters ideally have zero
resistance.
ConcepTest 21.9
Even More Circuits
1) R1
Which resistor has the greatest
2) both R1 and R2 equally
current going through it?
Assume that all the resistors
3) R3 and R4
are equal.
4) R5
5) all the same
The same current must flow
through left and right
combinations of resistors. On
the LEFT, the current splits
equally, so I1 = I2. On the
RIGHT, more current will go
through R5 than R3 + R4 since
the branch containing R5 has
less resistance.
V
Follow-up: Which one has the smallest
voltage drop?
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