Download Electric Current

Electric Current
• Electrons will move between two oppositely-charged
conductors when a conducting wire connects them
– Electric field exerts forces on conduction
+ +
++
electrons, causing them to move
– Flow of electrons stops when the conductors
are at the same potential and electric field + – +
–++––
is zero
– –
––
electrons
• A net flow of charge is an electric current
– Charges of like sign move in one direction in an electric
field E (negative charges move opposite E, positive
charges with E)
• Current = rate of flow of electric
charge past a point
Electric Current
• Definition of current I
– DQ is amount of charge that passes through the
cross-sectional area in time Dt
– SI unit of current is the ampere (A): 1 A = 1 C/s
– Small currents are measured in mA, mA, etc.
I
DQ
Dt
• Direction of current flow is taken as direction positive
charges would flow
– Standard established by Benjamin Franklin
– This is opposite the flow of electrons (the
mobile charge carriers) in a conductor
Moving electrons
– Current flows in direction of the electric field present in the
region
– Negative charge moving in one direction has same
macroscopic effect as positive charge moving in other
direction
CQ 1: Rank the magnitudes of the currents
in the four regions shown below, from
lowest to highest.
A)
B)
C)
D)
E)
dacb
acbd
cabd
dbca
abcd
Circuits
• For currents to flow, a complete circuit is required
(from College Physics,
Giambattista et al.)
• Ammeters measure current while in series with
circuit; voltmeter measures voltage across a device
Microscopic View of Current in a Conductor
• In the absence of an applied electric field, conduction
electrons in a metal are in constant random motion at
(from College Physics,
high speed (~ 106 m/s in copper)
Giambattista et al.)
– Their average velocity is zero due to many
collisions with each other and with atoms
• In a uniform electric field, electrons have a
uniform acceleration between collisions
(opposite to the field)
• Result is a small but non-zero drift speed
(vd) in the direction of the electric force
(~0.1 mm/s for Cu wire)
– Think of air molecules in a gentle breeze
– Current and drift speed related by: I  nqvd A
• n = number of mobile charge carriers / unit volume
e– Drift
• A = cross-sectional area of current flow; q = charge / carrier
Example Problem #17.7
A 200-km-long high-voltage transmission
line 2.0 cm in diameter carries a steady
current of 1000 A. If the conductor is
copper with a free charge density of
8.5  1028 electrons per cubic meter, how
many years does it take one electron to
travel the full length of the cable?
Solution (details given in class):
27 years
Resistance and Ohm’s Law
• For many materials, I is proportional to DV
• The proportionality constant, electrical resistance, is
defined as the ratio of DV to I:
DV
R
– R is measured in ohms (W)
I
– Think of water flow down a river and the resistive effect of
rocks and other obstructions (flow rate analogous to I)
• In ohmic materials, R is constant over a wide range
of values of DV and I
– In that case, there is a linear relationship between DV and I
– Ohm’s Law: DV  IR
• In non-ohmic materials,
R changes with DV or I
– Semiconductor diodes
– Transistors
ohmic
non–ohmic
Resistivity
• Resistance in a circuit arises due to collisions
between electrons and fixed atoms in a conductor
– Collisions inhibit movement of electrons (like friction)
• Resistance depends on the size and shape of the
conductor
– Proportional to its length l and inversely proportional to its
cross-sectional area A
– Think of flow of water through a pipe
• As l increases, there is more friction between water and walls of
pipe and resistance to flow goes up
• As A increases, the pipe can transport more fluid in a given time
interval, so the resistance goes down
• For an ohmic conductor:
Rr
l
A
– r = constant = resistivity of material (units of Wm) – see
Table 17.1 in textbook
Resistors
• A resistor is a circuit element designed to
have a known resistance
• They are commonly found in circuits as little
cylinders with color-coded bands signifying
(from College Physics,
their resistance
Giambattista et al.)
• Current flows through resistor in direction of electric
field, which points from higher to lower potential
– As current moves through a resistor, voltage drops by an
amount DV = IR due to drop in electrical potential energy
– Analogous to water flow: water flows downhill toward lower
potential energy ↔ electric current
flows toward lower potential
• Resistors indicated by zig–zag
pattern in circuit
Example Problem #17.12
Suppose you wish to
fabricate a uniform wire
out of 1.00 g of copper.
If the wire is to have a
resistance R = 0.500 W,
and if all the copper is to
be used, what will be (a)
the length and (b) the
diameter of the wire?
Solution (details given in class):
(a) 1.82 m
(b) 0.280 mm
Temperature Variation of Resistance
• Two primary factors determine the resistivity of a
metal
– Number of conduction electrons per unit volume
– Rate of collisions between an electron and an ion
• The second of these factors is sensitive to
temperature
– At a higher temperature, atoms vibrate with larger
amplitudes
– As a result, electrons collide more frequently with the atoms
and acquire a smaller drift speed (current is smaller for a
given electric field)
– As temperature goes up, resistivity goes up (conductors)
• Resistivity vs. temperature: r  r0 1  a T  T0  (T0 = 20°C)
• Resistance vs. temperature: R  R0 1  a T  T0 
(r0 = r @ 20°C)
(R0 = R @ 20°C)
– a = temperature coefficient of resistivity (see Table 17.1)
Electrical Energy and Power
• Consider the following circuit at right:
– As positive charge q moves from A to B
(through battery), electrical potential
energy of system increases by qDV
– From C to D (through resistor), this charge
loses this energy through collisions with
atoms in the resistor (no net KE gain after PE loss)
– This causes atoms in resistor to vibrate more, increasing
the internal energy of the system (resistor temp. rises)
– The rate at which charges lose energy is DU / Dt
= qDV / Dt = IDV which is rate at which resistor material
gains internal energy
– Power = rate at which energy is delivered to resistor: P  IDV
2
– Alternatively, from DV = IR for a resistor:


D
V
2
PI R
R
CQ 2: What is the energy required to
operate a 60 W light bulb for 1 minute?
A) 1 J
B) 60 J
C) 360 J
D) 3600 J
CQ 3: Interactive Example Problem:
Resistor Power!
Which graphs show the correct
relationship between the indicated
quantities?
A) Graphs A and D
B) Graphs B and C
C) Graphs A, D, and E
D) Graphs B, C, and E
(PHYSLET #9.5.3, copyright Prentice Hall, 2001)
Example Problem #17.30
A toaster rated at 1050 W operates on a 120–
V household circuit and a 4.00-m length of
nichrome wire as its heating element. The
operating temperature of this element is
320°C. What is the cross-sectional area of
the wire?
Solution (details given in class):
4.90  10–7 m2
Example Problem #17.37
The heating element of a coffeemaker
operates at 120 V and carries a current of
2.00 A. Assuming that the water absorbs all
of the energy converted by the resistor
(heating element), calculate how long it
takes to heat 0.500 kg of water from room
temperature (23.0°C) to the boiling point.
Solution (details given in class):
11.2 min
Example Problem #17.57
You are cooking breakfast for yourself and a friend using a
1200-W waffle iron and a 500-W coffeepot. Usually, you
operate these appliances from a 110-V outlet for 0.500 h
each day. (a) At 12 cents per kWh, how much do you spend
to cook breakfast during a 30.0 day period? (b) You find
yourself addicted to waffles and would like to upgrade to a
2400-W waffle iron that will enable you to cook twice as
many waffles during a half-hour period, but you know that
the circuit breaker in your kitchen is a 20-A breaker. Can
you do the upgrade?
Solution (details given in class):
(a) $3.06
(b) No. The circuit needs at least 26.4 A.