Download - DC Circuits
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DC Circuits
Currents. Resistors. Batteries.
Kirchhoff’s Loop Rules. Power
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Examples of Circuits
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t
QI
av
Current: flow of charge
Average current: Iav= Charge Q
flowing across area A in time t
t
QI
d
d
Instantaneous current: differential limit of Iav
Units of current: Coulomb/second=Ampere (A)
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Direction of the current
Direction of current is in direction of flow of positive charge
or, opposite direction of flow of negative charge
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Current density J (1)J: current/unit area
A
IJ
points in the direction of the current
Units A/m2
J
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Current density J (2)If the ara is not perpendicular to the current
An area or surface perpendicular to the direction of the current
)cos(n JAJAI
: angle between normal to A and current
nA
IJ
)cos(n AA
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Why does current flow?If an electric field is set up in a conductor, charge will move (making a current in the direction of E)
Note that when current is flowing, the conductor is not an equipotential surface (and Einside0)!
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Microscopic picture (1)
Drift speed is velocity forced by applied electric field in the presence of collisions, it is typically 4x10-5m/s, or 0.04 mm/s!
To go one meter at this speed takes about 10 hours
Thermal velocity is around 103 km/s !
How can this be?
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Microscopic picture (2)
vd : drift velocity
n : number of carriers per unit volume
q : charge of each carrier (normally e)
A : suface perpendicular to vd
AqnvI d
dqnvJ
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Conductivity and resistiviy
Ability of current to flow depends on density of charges & rate of scattering.
Two quantities summarize this:
: conductivity
: resistivity 1
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Microscopic Ohm’s Law
And depend only on the microscopic properties of the material,
not on its shape
1
or JEEJ
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The voltage drops in a resistorThe electric field brings about a voltage drop in a resistor
ElVVV ab
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What is the relationship between V and current?
ElVVV ab
Ohm’s Law (1)
A
lIl
A
IJlElV
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R has units of ohms ()=Volt/Amp
Ohm’s Law (2)
A
lR IRV
Then, units of : m and : -1m-1
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Examples of Circuits
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Symbols for circuit elements
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Electromotive force -Battery Moving from the negative to positive terminal of a battery increases your potential
Think:
Ski Lift
ab VVV
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Internal resistanceReal batteries have an internal resistance, r, which is small but non-zero
IrVVV ab Terminal voltage:
Real battery=ideal battery in series with a resistance
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Sign conventions - ResistorMoving across a resistor in the direction of current decreases your potential
ab VV Think:
Ski Slope
Voltage drop:
Voltage decreases in the direction of the current
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Sign conventions - CapacitorMoving across a capacitor from the negatively to positively charged plate increases your potential
ab VVV
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Resistors in seriesThe same current I must flow through both resistors
eq2121 )( IRRRIIRIRVVV ca
21eq RRR
N
iiRR
1eq
or
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Resistors in parallelVoltage drop across the resistors must be the same
eq221121 IRRIRIVVV
N
i iRR 1eq
11
eq212121
11
R
V
RRV
R
V
R
VIII
21eq
111
RRR or
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Measuring V, I, R
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Measuring Potential Difference A voltmeter must be hooked in parallel across the element you want to measure the potential difference across
Voltmeters have a very large resistance, so that they don’t affect the circuit too much
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Measuring CurrentAn ammeter must be hooked in series with the element you want to measure the current through
Ammeters have a very low resistance, so that they don’t affect the circuit too much
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Measuring ResistanceAn 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)