Download Current, Resistance and Circuits

Current, Resistance and Circuits
Key Contents
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Electric current
Current density and drift velocity
Resistance and resistivity
Ohm’s law
Power in electric circuits
The emf device
Single-loop circuits
Multi-loop circuits
RC circuits
Electric Current
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An electric current is a stream of moving charges.
However, not all moving charges constitute an electric current.
To have that, here must be a net flow of charge through a
surface.
Consider a flow of water through a garden hose.
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The flow of water through a garden hose represents the directed flow of
positive charge (the proton in the water molecules) at a rate of perhaps
several million coulombs per second. There is however no net transport
of charge because there is a parallel flow of negative charge (the
electrons in the water molecules) of exactly the same amount moving in
exactly the same direction!
Electric Current: the most common example
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The free electrons (conduction
electrons) in an isolated length of
copper wire are in random motion
at speeds of the order of 106 m/s. If
you pass a hypothetical plane
through such a wire, conduction
electrons pass through it in both
directions at the rate of many
billions per second—but there is no
net transport of charge and thus no
current through the wire.
However, if you connect the ends
of the wire to a battery, you slightly
bias the flow in one direction, with
the result that there now is a net
transport of charge and thus an
electric current through the wire.
Electric Current
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The figure shows a section of a conductor, part of a conducting loop in which current
has been established. If charge dq passes through a hypothetical plane (such as aa’)
in time dt, then the current i through that plane is defined as:
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The charge that passes through the plane in a time interval extending from 0 to t is:
Electric Current
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Under steady-state conditions, the current is the same for planes aa’, bb’,
and cc’ and for all planes that pass completely through the conductor, no
matter what their location or orientation.
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The SI unit for current is the coulomb per second, or the ampere (A):
Electric Current: Conservation of Charge, and Direction of Current
Current Density
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The magnitude of current density, J, is equal to the current per unit area through
any element of cross section. It has the same direction as the current. This is a local
property.
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If the current is uniform across the surface and parallel to dA, then J is also uniform
and parallel to dA, and then
Here, A is the total area of the surface.
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The SI unit for current density is the ampere per square meter (A/m2).
Current Density
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Figure 26-4 shows how current
density can be represented with a
similar set of lines, which we can
call streamlines.
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The current, which is toward the
right, makes a transition from the
wider conductor at the left to the
narrower conductor at the right.
Since charge is conserved during
the transition, the amount of
charge and thus the amount of
current cannot change.
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However, the current
changes—it is greater
narrower conductor.
density
in the
Current Density, Drift Speed
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What really happens with an electric current flowing in a
conducting wire?
Current Density, Drift Speed
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When a conductor has a current passing through it, the electrons move randomly, but
they tend to drift with a drift speed vd in the direction opposite that of the applied
electric field that causes the current. The drift speed is tiny compared with the speeds
in the random motion.
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In the figure, the equivalent drift of positive charge carriers is in the direction of the
applied electric field, E. If we assume that these charge carriers all move with the
same drift speed vd and that the current density J is uniform across the wire’s crosssectional area A, then the number of charge carriers in a length L of the wire is nAL.
Here n is the number of carriers per unit volume.
Current Density, Drift Speed
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The total charge of the carriers in the length
L, each with charge e, is then
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The total charge moves through any cross
section of the wire in the time interval
is the current
Resistance and Resistivity
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We determine the resistance between any two points of a conductor by
applying a potential difference V between those points and measuring the
current i that results. The resistance R is then
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The SI unit for resistance that follows from Eq. 26-8 is the volt per ampere.
This has a special name, the ohm (symbol W):
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In a circuit diagram, we represent a resistor and a resistance with the
symbol
Resistance and Resistivity
Resistance and Resistivity
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The resistivity, r, of a resistor is
defined as:
This is a local property.
The SI unit for r is W m.
The conductivity s of a material is
the reciprocal of its resistivity:
Resistance and Resistivity, Variation with Temperature
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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:
r = r0 (1+ a (T -T0 ))
Resistance and Resistivity
Calculating Resistance from Resistivity
Ohm’s Law
Ohm’s Law
Power in Electric Circuits
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Charge dq moves through a decrease in
potential of magnitude V, and thus its
electric potential energy decreases in
magnitude by the amount
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The power P associated with that transfer
is the rate of transfer dU/dt, given by
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The unit of power is the volt-ampere (V
A).
Example, Rate of Energy Dissipation in a
Wire Carrying Current
Pumping Charges
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In order to produce a steady
flow of charge through a
resistor,
one
needs
a
“charge pump”, a device
that—by doing work on the
charge carriers—maintains a
potential difference between
a pair of terminals.
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Such a device is called an
emf device, which is said to
provide an emf. (emf stands
for electromotive force)
Pumping Charges
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A common emf device is the battery. The emf device that
most influences our daily lives is the electric generator, which,
by means of electrical connections (wires) from a generating
plant, creates a potential difference in our homes and
workplaces.
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Some other emf devices known are solar cells, fuel cells. An
emf device does not have to be an instrument—living
systems, ranging from electric ceels and human beings to
plants, have physiological emf devices.
Work, Energy, and Emf
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In any time interval dt, a charge dq
passes through any cross section of the
circuit shown, such as aa’. This same
amount of charge must enter the emf
device at its low-potential end and
leave at its high-potential end.
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The emf device must do an amount of
work dW on the charge dq to force it to
move in this way.
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We define the emf of the emf device in
terms of this work:
Work, Energy, and Emf
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An ideal emf device is one that has no internal resistance to
the internal movement of charge from terminal to terminal.
The potential difference between the terminals of an ideal emf
device is exactly equal to the emf of the device.
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A real emf device, such as any real battery, has internal
resistance to the internal movement of charge. When a real
emf device is not connected to a circuit, and thus does not
have current through it, the potential difference between its
terminals is equal to its emf.
However, when that device has current through it, the
potential difference between its terminals differs from its emf.
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Calculating the Current in a Single-Loop Circuit
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P =i 2R (dissipation in the
resistor)
The work that the battery does on
this charge, is
Calculating the Current in a Single-Loop Circuit
Potential Method
Calculating the Current in a Single-Loop Circuit
Potential Method
Other Single-Loop Circuits
Internal Resistance
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The figure above shows a real battery,
with internal resistance r, wired to an
external resistor of resistance R.
According to the potential rule
Other Single-Loop Circuits
Resistances in Series
Potential between two points
Going counterclockwise from a:
Going clockwise from a:
Potential across a real battery
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If the internal resistance r of the
battery in the previous case were
zero, V would be equal to the emf of
the battery—namely, 12 V.
However, since r =2.0W , V is less
than that emf.
The result depends on the value of
the current through the battery. If the
same battery were in a different
circuit and had a different current
through it, V would have some other
value.
Grounding a Circuit
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This is the same example as in the previous slide, except that battery
terminal a is grounded in Fig. 27-7a.
Grounding a circuit usually means connecting the circuit to a
conducting path to Earth’s surface, and such a connection means that
the potential is defined to be zero at the grounding point in the circuit.
In Fig. 27-7a, the potential at a is defined to be Va =0. Therefore, the
potential at b is Vb =8.0 V.
Power, Potential, and Emf
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The net rate P of energy transfer from the emf device to the charge carriers
is given by:
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where V is the potential across the terminals of the emf device.
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But
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But Pr is the rate of energy transfer to thermal energy within the emf device:
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The rate Pemf at which the emf device transfers energy both to the charge
carriers and to internal thermal energy is then
, therefore
Example, Single loop circuit with two real batteries:
Example, Single loop circuit with two real batteries, cont.:
Multi-loop Circuits
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At junction d in the circuit
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This rule is often called Kirchhoff’s
junction rule (or Kirchhoff’s current
law).
For the left-hand loop,
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For the right-hand loop,
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And for the entire loop,
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Multi-loop Circuits, Resistors in Parallel:
where V is the potential difference between a and b.
From the junction rule,
Multi-loop Circuits
Example, Resistors in Parallel and in Series
Example, Resistors in Parallel and in Series, cont.:
Example, Multi-loop circuit and simultaneous loop equations:
Ammeter and Voltmeter
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An instrument used to measure
currents is called an ammeter.
It is essential that the
resistance RA of the ammeter
be very much smaller than
other resistances in the circuit.
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A meter used to measure
potential differences is called a
voltmeter. It is essential that the
resistance RV of a voltmeter be
very much larger than the
resistance
of
any circuit
element across which the
voltmeter is connected.
RC Circuits
Charging a Capacitor
(# q =0, @ t =0)
RC Circuits
Time Constant
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The product RC is called the capacitive time constant of the
circuit and is represented with the symbol t:
At time t= t =( RC), the charge on the initially uncharged
capacitor increases from zero to:
RC Circuits
Discharging a Capacitor
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Assume that the capacitor of the figure
is fully charged to a potential V0 equal
to the emf of the battery.
At a new time t =0, switch S is thrown
from a to b so that the capacitor can
discharge through resistance R.
i=
dq
q
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= - 0 e-t/RC = - e-t/t
dt
RC
R
Fig. 27-16 (b) This shows the decline of the
charging current in the circuit. The curves are
plotted for R =2000 W, C =1 mF, and emf =10 V; the
small triangles represent successive intervals of one
time constant t.