Download Lecture 8 - The Local Group

Chapter 25
Current & Resistance
Electric Current
●
●
Electric current is the rate of flow of charge
through some region of space (or a wire).
The SI unit of current is the ampere (A)
●
●
1A=1C/s
The symbol for electric current is I
dQ
I=
dt
Direction of Current
●
●
●
The charges passing
through the area could be
positive or negative or both
It is conventional to assign
to the current the same
direction as the flow of
positive charges
Convention: direction of
current flow is opposite
the direction of the flow
of electrons
#thanksbenfranklin
Charge Motion in a Conductor
●
Electric field forces cause the electrons to move in the wire and
create a current
• Most electron motion in
a conductor is random,
caused by collisions.
• The net “drift velocity”
is very slow.
Current and Drift Speed
̣
̣
̣
̣
Wire of cross-section area A
n = charge carriers per Volume
nAΔx = number of charges that
move through A in time Δt
Total charge is ΔQ = (nAΔx)e
Drift Velocity
dQ
I=
= n q Avd
dt
Electron drift is SLOW
Current:
I ' 1 A = 1 C/s
Density of electrons in wire:
Electron charge:
Area of wire:
A ⇠ 10
Definition
of current:
v=
q ⇠ 10
6
n ⇠ 1029 /m3
19
C
m2
Drift Velocity
dQ
I=
= nqAv
dt
I
1C/s
= 29
nqA
10 · 10 19 · 10
6 C/m
= 10
4
m/s
wikipedia
slower than a snail
faster than continental drift
Why is the current flow nearly instantaneous?
If drift velocities are ~10-4 m/s, why do the lights turn on instantly?
Answer: a conductor has mobile charges all along its length.
Turning on a switch establishes an electric field through the conductor almost
instantaneously, which means all mobile charges start drifting immediately.
•A charge doesn’t need to travel from the switch to the light-bulb for the light-bulb
to turn on – there are already charges inside the light-bulb, and the light-bulb turns
on as soon as they start drifting, which happens almost instantly.
Current and Work
We just showed that the kinetic energy of the charges does not increase appreciably.
So Where does the energy go?
•Due to the frequent collisions of the moving charges, the energy is transferred to the (stationary)
ions of the material, increasing their vibrational energy, and thus their temperature.
Bulk of the work done by the electric field goes into heating the conductor.
(this is ‘wasted’ energy, lots of room to increase efficiency)
Quiz: current
These four wires are made of the same metal. Rank in order, from largest to
smallest, the electron currents ia to id:
A.id>ia>ib>ic
B.ib=id>ia=ic
C.ic>ib>ia>id
D.ic>ia=ib>id
E.ib=ic>ia=id
dQ
I=
= n q Avd
dt
Quiz: current
These four wires are made of the same metal. Rank in order, from largest to
smallest, the electron currents ia to id:
A.id>ia>ib>ic
dQ
I=
= n q Avd
dt
B.ib=id>ia=ic
C.ic>ib>ia>id
D.ic>ia=ib>id
E.ib=ic>ia=id
(πr )v
2
(πr )2v
2
(π 4r )v
2
⎛ 1 2⎞
⎜⎝ π r ⎟⎠ 2v
4
Quiz: current and drift speed
Two copper wires of different diameter are joined end to end, and a current flows in
the wire combination.
When electrons move from the larger-diameter wire into the smaller-diameter wire,
A.their drift speed increases.
B.their drift speed decreases.
C.their drift speed stays the same.
D.not enough information given to decide
Quiz: current and drift speed
Two copper wires of different diameter are joined end to end, and a current flows in
the wire combination.
When electrons move from the larger-diameter wire into the smaller-diameter wire,
A.their drift speed increases.
B.their drift speed decreases.
C.their drift speed stays the same.
D.not enough information given to decide
Analogy: water is a pipe (when radius of pipe decreases, the speed increases).
The current I is constant for the whole wire.
If A decreases, v has to decrease to compensate.
dQ
I=
= nqAv
dt
Current density
Current:
dQ
I=
= n q Avd
dt
Take out the area factor => current density: J =
I
= n q vd ⎡ A 2 ⎤
⎣ m⎦
A
In some cases, it is useful to treat the current density as a vector:
• q > 0, vd same direction as E
• q < 0, vd opposite direction as E
Total current through a surface:
J describes how charges flow at a certain point
• Direction of J changes around the circuit
• Magnitude of J = I/A can change along the circuit (e.g. when A changes)
Lecture 8
Conductivity, Resistance,
& Circuits
Announcements
— Ch 24 HW due Thursday @ 8am
— Ch 25 HW (yet to be assigned) will be due next week
— Quiz tomorrow will cover Ch 23 and Ch 24
— Keep an eye out for a future email (from me) regarding
a survey worth extra credit…
Conductivity
●
Currents flow in response to electric fields.
●
For some materials (e.g. metals), the current density
is [nearly] directly proportional to the electric field
●
The constant of proportionality, σ, is called the
conductivity of the conductor
J=σE
“Ohm’s Law”
current density:
Ohm’s Law
●
●
●
●
I
= n q vd
A
Ohm’s law : for many materials, the
current density is proportional to the
electric field: J = σ E
Most metals obey Ohm’s law - Materials
that obey Ohm’s law are called ohmic
Not all materials follow Ohm’s law
●
●
J=σE
J=
Materials that do not obey Ohm’s law
are said to be nonohmic
Ohm’s law is not a fundamental law
of nature
Ohm’s law is an empirical
relationship valid only for certain
materials
●
Specifically when conduction properties
are determined by collisions of
electrons with atoms
●
●
●
●
Ohm (1789 -1854)
German physicist
Formulated idea of
resistance
Discovered Ohm’s Law
Conductivity & Resistivity
J=σE
Conductivity (σ) has units of [A/Vm] =
•Good conductors have high conductivity
•Insulators have low conductivity.
ρ =1/σ
[Ωm]-1
• V/A = Ω “ohm”
• V = “volt”
• A = “amp”
ρ = E/J
Resistivity (ρ) of a material is the ratio of the magnitudes of J and E
•The greater the resistivity, the larger the electric field required to produce a
given current density.
When a material obeys Ohm’s Law it
means that the resistivity (and/or
conductivity) of a material is a constant
(independent of E at a given temperature)
ρ = E/J
Resistivity
Enormous differences in resistivity of conductors and insulators
•Good conductors have low resistivity
•Insulators have high resistivity
•Bigger resistivity => larger E-field needed to produce a given current density
Resistivity & Temperature
Resistivity of materials depends on temperature.
•Metallic materials: When you heat something up, the atoms jiggle
around more – harder for electrons to get through.
In some cases, resistivity follows a linear relation with
temperature:
ρ (T ) = ρ0 ⎡⎣1 + α (T − T0 )⎤⎦
• ρ0: resistivity at reference T0 (commonly 0oC or 20oC)
• α: temperature coefficient of resistivity depends on the
material
•The resistivity of a metal typically increases with temperature
•For semiconductor, resistivity decreases with temperature, in a
non-linear way.
•For special materials called superconductors, the resistivity goes
to zero below a certain “critical temperature” (Tc).
Superconductivity
=> resistivity becomes zero below a
certain temperature Tc.
1911: discovered in mercury (Tc = 4.2 K)
1986: the highest Tc superconductor was ~20 K => need liquid Helium
(expensive) or liquid Hydrogen (explosive) to maintain.
1987, an alloy with Tc well above the boiling point of liquid Nitrogen (cheap
and safe) was discovered. The current record for highest T at atmospheric
pressure is about 138 K.
- “Room temperature” superconductivity is a highly sought after goal.
➡ Would revolutionize power-distribution, electronics and transportation.
Resistance
!"
!"
Ohm’s law tells us E & J related by resistivity ρ: E = ρJ
There is a similar relation between V (~qE) and I (~J x area )
V = IR
R = Resistance is related to resistivity ρ via:
ρ <= set by material (e.g. copper)
ρL
R=
Ω]
[
A
●
SI units of resistance are ohms (Ω)
●
1Ω=1V/A
●
Resistance in a circuit arises due to
collisions between the electrons
carrying the current with the fixed
atoms inside the conductor
L & A <= set by shape (e.g. wire)
Resistivity ρ is a property of the material
Resistance R depends on material AND shape of resistor.
V = IR
This formula is more useful than the formal Ohm’s law
because V and I are easier to measure than E and J
Resistance
●
●
●
The voltage applied across the
ends of a conductor is
proportional to the current through
the conductor.
The constant of proportionality is
called the resistance of the
conductor R = ΔV /I
SI units of resistance are ohms
(Ω)
●
●
1Ω=1V/A
€
Resistance
in a circuit arises due
to collisions between the electrons
carrying the current with the fixed
atoms inside the conductor
V = IR
R
V
I
Example
15 cm
Block of steel 1.2cm×1.2cm×15cm.
ρsteel = 20×10-8 Ωm
1.2cm
1.2cm
a)Resistance between square ends:
ρL (20 ×10 [Ωm])(0.15[m])
R=
=
= 2.1×10 [Ω ]
A
(1.2 ×10 ) ⎡⎣m ⎤⎦
−8
−4
−2 2
2
b)Resistance between the two rectangular faces:
(
)(
)
−8
−2
ρL 20 ×10 [Ωm] 1.2 ×10 [m]
−6
R=
=
=
1.3
×10
Ω]
[
−2
−2
2
A
1.2 ×10 15
. ×10 ⎡⎣ m ⎤⎦
(
)(
)
R=
ρL
Ω]
[
A
current–voltage relationships
V = IR
I = (1/R) V
current–voltage relationships
V = IR
I = (1/R) V
V ≠ IR
Diodes operate like valves! Current flows only
in one direction. Important for logic functions
circuits.
Electric Circuits
Current can only flow at a steady rate in a
conductor that is part of a closed loop or
“circuit”.
In an isolated conductor, an electric field could
produce a current momentarily, but charge would
accumulate at the two ends.
•This accumulation of charge would create a second
electric field opposite to the one causing the current.
•The electric field due to the build-up of charge at the
ends would very quickly cancel the original electric
field, and the current would stop.
Circuits and Electric Potential Energy
In moving through a conductor with
resistance, the electric field does work on a
charge and the charge’s potential energy
always decreases:
ΔV = IR
ΔU = qΔV
To keep current going, there must be some part in the
circuit where the potential energy increases (e.g. a battery).
•This part of the circuit behaves like the pump in a water fountain.
•Without it, the water would fall to the bottom of the fountain and
stay there.
•The pump sends water back to the top of the fountain, so it can fall
to the bottom again.
Quiz: Resistor and PE
Electrons in an electric circuit pass through a resistor. The wire has the same diameter on
each side of the resistor.
Compared to the potential energy of an electron before entering the resistor, the potential
energy of an electron after leaving the resistor is
A.greater.
B.less.
C.the same.
D.not enough information given to decide
Hint: Potential Energy
U = q0V
Quiz: Resistor and PE
Electrons in an electric circuit pass through a resistor. The wire has the same diameter on
each side of the resistor.
Compared to the potential energy of an electron before entering the resistor, the potential
energy of an electron after leaving the resistor is
A.greater.
B.less.
C.the same.
D.not enough information given to decide
Hint: Potential Energy
U = q0V
In general, energy is dissipated in the resistor, so the charge carriers (electrons) must loose
energy.
Current flows from high-potential to lower-potential, thus the potential is lower at the end of
the resistor.
However, the electrons move opposite to the current, so they move towards higher electric
potential V.
The potential energy of an electron is U = −eV , thus it decreases when V increases.
Charge is not “used up”
It is a common mistake to think that charge is
somehow consumed or “used up” as it passes
through an electric circuit.
The current is the same at every point in a simple loop circuit even if the wire
thickness is different at various places.
Charge can’t be created or destroyed, and it can’t accumulate anywhere in this
simple type of circuit.
Just like an ornamental water fountain, the rate of water flowing out of the top is
the same as the rate of water flowing into the bottom. Water isn’t “used up”
anywhere.
emf
A 1.5 V flashlight battery is an
example of a source of emf
- Non-electrostatic forces in the
battery do 1.5 J of work on
every Coulomb of charge that
passes through it
An emf converts chemical (or some other form) of energy into electrical energy. It
acts like a “pump” in an electric circuit: moves charge from a lower potential to a
higher potential. (We avoid the historical, misleading term “electromotive force”)
- emf is not a force, but rather something like an electric potential.
- emf has same unit as electric potential: Volts
- Written as the symbol ε if we want to distinguish it from an electric potential V
sources of emf
In an ideal source of emf, which is isolated (not connected to a circuit),
there is a balance between
• the electrostatic force (Fe) on a charge, which tends to move charges to lower electric potential, in the direction of
the electric field and
• the non- electrostatic forces, Fn, which tend to move charges to higher electric potential, opposite the direction of the electric field
If a charge is moved from b (lower potential) to a (higher potential) inside the source, the electric field does negative work, and the charge’s electric potential energy increases by: ΔU = qV
e
ab
The work done by the non-electrostatic force must then equal the change
in the electric potential energy:
Wn = qε = ΔUe = qVab
Thus, in an ideal source:
ε = Vab
sources of emf and circuits
When a source of emf is connected to a circuit
the potential difference between the poles of
the source creates an electric field in the wire,
which results in a current flowing from a to b:
Vab = IR
When a + charge flows around the circuit:
• Potential rises by ε as it passes through
the EMF source
• Potential drops by Vab=IR as it passes
through the rest of the circuit
Internal resistance of emf sources
In a real (as opposed to ideal) emf source, charges moving
from the negative pole to the positive pole inside the source
encounter resistance.
- use the variable r = “internal resistance” of the source.
If this resistance obeys Ohm’s Law, it is constant, and
independent of the current.
=> Flowing charges experience a drop in potential equal to Ir
I=
ε
R+ r
Vab = ε − Ir
The potential difference between the poles of a source
with internal resistance is then:
Vab = IR = ε − Ir
•Vab is called the “terminal voltage”, and is less than the emf
The current in the circuit outside the source is
determined by Vab (not ε directly)
This depends on I
and thus changes
depending on the
circuit the battery
is hooked to.
emf source
Vab = ε − Ir
This is given by the battery
An emf source is a source of constant emf
- NOT a source of constant current
- NOT a source of constant voltage
•The current through a source of emf will depend on the resistance of
the circuit it is attached to.
•The terminal voltage will decrease as the current through the source
increases.
Vab = ε − Ir
Current through external circuit connected to emf source is still given
by Vab = IR. This means:
Vab = E
Ir = IR
E
I=
R+r
FYI: Circuit diagram Symbols
Example: What is the ammeter reading for I?
Vab = E
Ir = IR
E
I=
R+r
I=
12V
= 2A
4⌦ + 2⌦
Quiz: What is the voltmeter reading for Vab?
Vab = E
Ir = IR
E
I=
R+r
I=
a)12V
b)10V
c)8 V
d)6V
12V
= 2A
4⌦ + 2⌦
Quiz: What is the voltmeter reading for Vab?
Vab = E
Ir = IR
E
I=
R+r
I=
a)12V
b)10V
c)8 V
d)6V
12V
= 2A
4⌦ + 2⌦
Va0 b0 = IR = (4⌦)(2A) = 8V
OR
Va0 b0 = Vab = E
IR = 12V
2A(2⌦) = 8V
Electrical Power
1Joule
=1Watt
1sec
Charge goes a to b: PE of
the system increases: QΔV
- chemical energy in the
battery must decrease
by this same amount
●
●
Charge goes c to d:
same energy is lost
(transformed into heat
from the resistor).
The rate at which the system loses potential energy as the charge
passes through the resistor is equal to the rate at which the system
gains internal energy in the resistor
The power is the rate at which the energy is delivered to the resistor
Power = dU/dt
Power Current
U = qV
!
dU dq
= V
dt
dt
!
P = IV
Useful Equations for Power
P = IV
Batteries & Resistors
2
P= I R
2
V
P=
R
Power Delivered to a Resistor
(V=IR)
Announcements
— Ch 24 HW due Thursday @ 8am
— Ch 25 HW (yet to be assigned) will be due next week
— Quiz tomorrow will cover Ch 23 and Ch 24
— Keep an eye out for a future email (from me) regarding
a survey worth extra credit…