Download Note 11 Circuits

Note 11 Circuits
Electrical Circuit
An electrical circuit has three components.
1. The power source that drives a current.
2. The conductor that guide the current to where they need to go.
3. The device that uses the current.
The following are the quantities that are measured when dealing with a circuit.
Electric Potential
The electric potential is the driver of the circuit. It pushes the charges around. The motion of the
charges is measured using the electric current. The electric potential is measured using the volt,
V = J/C. One volt is the potential that will give one coulomb of charge one joule of energy.
Electric Current
Electrical current is defined as the number of charges flowing through some area per unit time.
I =
Q
Δt
The unit of current is the ampere, A = C/s. When one coulomb of charge flows through an area
every second then the current through that area is one ampere.
Conventional Current
Conventionally, a positive current means positive charges are traveling in the direction of the
current. However, the actual movement is done by the negatively charged electrons. The
electron current flows in the opposite direction.
R
electron current
conventional current
–
+
V
By convention, we think of the current as a flow of positive charges to reduce confusion. This is
called the conventional current.
Direct and Alternating Current
Batteries are direct current sources while wall outlets provide alternating current sources.
Direct currents are constant currents.
direction vary with time.
Alternating current are currents whose magnitude and
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Resistance and Ohm’s Law
Resistivity is a measure of how hard it is for charges to move through a material. Resistance is a
measure of how hard it is for charges to move through an object or device.
Conductivity and conductance are the inverses of these values, respectively. Resistance has the
unit ohm, Ω = V/A. Resistivity has the unit Ω•m.
material
rubber
silicon
iron
copper
resistivity (Ω•m)
1013
1
10-7
10-8
The two are related through the geometry of a device. Here is the resistance, R.
R=ρ
L
A
Resistance is proportional to the resistivity. Resistance goes up with length the charges have to
travel and it goes down with the cross sectional area.
The resistance of certain objects follow Ohm’s law. Ohm’s law states that the current through the
object changes linearly with the applied voltage across it.
ΔV = IR
Resistivity of metals usually goes down as the temperature decreases. At some very low “critical”
temperature, the resistivity goes to zero, this is called superconductivity.
Example
What is the resistance of a copper wire that is 1 m long with a diameter of 1 mm?
R=ρ
L
1m
= 1.68 ×10−8 Ωm
= 0.021 Ω
A
π(0.0005 m)2
Example
What is the resistivity of a material required to make a 100 Ω resistor that is 3 mm in radius and 10
mm long?
ρ=R
A
π(0.0015 m)2
= (100 Ω)
= 0.071 Ωm
L
0.01 m
This must be manufactured using semiconductors like silicon or a carbon-ceramic mixture.
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Power
A battery does work to give charges energy. The charges do work on the circuit elements. The
work done by the flowing charges on a circuit element is
W = −ΔU = −QΔV
The power delivered by the flowing charges to a circuit element is this. This is general.
P=
W
QΔV
Q
=
=
ΔV = I ΔV
Δt
Δt
Δt
The unit for power is the watt. A power of one watt is also defined as one coulomb of charge
passing through a one volt potential difference every second.
If the circuit element is ohmic, then we can apply Ohm’s law. The latter two formula are for an
ohmic object.
ΔV 2
P = I ΔV =
= I 2R
R
Example
A 1.5 V battery delivers 100 mA of current to a lightbulb for 60 s.
lightbulb
0.1 A
1.5 V
How much charge passes through the circuit?
I =
Q
Δt
⇒ Q = I Δt = (0.1 A)(60 s) = 6 C
How much work is done by the battery (or how much energy is used by the lightbulb)?
W = QΔV = (6 C )(1.5 V ) = 9 J
What is the power output of the lightbulb? I won’t assume that the lightbulb is ohmic.
P = I ΔV = (0.1 A)(1.5 V ) = 0.15 W
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Example
A 10 V battery is connected to a 2,000 Ω resistor.
2,000 Ω
9V
How much energy is used by the resistor in one hour? The energy used by the resistor is
W = PΔt =
V2
(9 V )2
Δt =
(3, 600 s) = 146 J
R
(2, 000 Ω)
page 4