 # Resistivity and Resistance

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Resistivity and Resistance

+ - Capacitor EMF Resistor Inductor

+ - EMF Resistor

i What is Current ?

Electric Current:

Current is Conserved i2 i3 i1 i1 = i2+ i3 + -

1 A 2 A 2 A i = ? 1 A 2 A 0 A i

1 A i 2 A 6 A i = ? 1 A 3 A 2 A 4 A 4 A

1 A 2 A 2 A 2 A 3A 4A i i = 4 A B. 2A C. 8 A 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. Always greater than either R1 or R2 B. Always less than either R1 or R2 C. Roughly equal to the mean of R1 and R2 D. 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? I. B) The two emit the same amount of light. C) II.

Kirchhoff’s Laws Kirchhoff’s voltage law (“loop law”)
Kirchhoff’s current law (“junction law”) + - Keep track of the conventions Conservation of energy Conservation of charge

Quick/useful notes with Kirchoff’s rules:

Consider the three identical light bulbs shown in the circuit
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
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- resistors are connected in parallel
Two 20- 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) A b) A c) A d) 2.2 A e) 4.4 A

ConcepTest 21.11a Lightbulbs
Two lightbulbs operate at 120 V, but one has a power rating of 25 W while the other has a power rating of 100 W. Which one has the greater resistance? 1) the 25 W bulb 2) the 100 W bulb 3) both have the same 4) this has nothing to do with 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
2) zero 3) 5 A 4) 2 A 5) 7 A In the circuit below, what is the current through R1? 10 V R1= 5 W R2= 2 W The voltage is the same (10 V) across each resistor because they are in parallel. Thus, we can use Ohm’s Law, V1 = I1 R1 to find the current I1 = 2 A. Follow-up: What is the total current through the battery?

ConcepTest 21.6a Short Circuit
1) all the current continues to flow through the bulb 2) half the current flows through the wire, the other half continues through the bulb 3) all the current flows through the wire 4) none of the above Current flows through a lightbulb. If a wire is now connected across the bulb, what happens? 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 across the resistor R1 when the switch is closed? The voltage will: 1) increase 2) decrease 3) stay the same With the switch closed, the addition of R2 to R3 decreases the equivalent resistance, so the current from the battery increases. This will cause an increase in the voltage across R1 . V R1 R3 R2 S Follow-up: What happens to the current through R3?

What is the approximate equivalent resistance of the five resistors shown in the circuit?
b) 7  c) 11  d) 14  e) 19 

Electrical Measurements
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
2) both R1 and R2 equally 3) R3 and R4 4) R5 5) all the same Which resistor has the greatest current going through it? Assume that all the resistors are equal. 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 R1 R2 R3 R5 R4 Follow-up: Which one has the smallest voltage drop?