Download Electric Current and Circuits

Chapter 21
Electric Current and Direct-Current Circuits
Current and Resistance
Whenever there is a net movement of charge, there exists
an electrical current. If a charge Q moves
perpendicularly through a “surface” of area A in a time
t, then there is a current I:
I 
Q
t
The unit of current is the Ampere (A): 1 A = 1 C/s.
By convention, the direction of the current is the direction
of the flow of positive charges. The actual charge carriers
are electrons; hence they move in the opposite direction
to I.
Batteries and Electromotive Force (emf)
Any device which increases the potential
energy of charges which flow through it is
called a source of emf, e.
SI unit for emf : Volt (V)
The emf may originate from a chemical
reaction as in a battery or from mechanical
motion such as in a generator.
A battery is a device that uses chemical
reactions to produce a potential difference
between its two terminals.
Water flow as analogy
for electric current
Resistance and Ohm’s Law
In order for a current I to flow there must be a
potential difference, or voltage V, across the
conducting material. We define the resistance, R, of a
material to be:
V
R
I
The unit of resistance is Ohms (W): 1 W  1 V/A
For many materials, R is constant (independent of V).
Such a material is said to be ohmic, and we write
Ohm’s Law:
V  IR
Resistivity
An object which provides resistance to current flow is
called a resistor. The actual resistance depends on:
• properties of the material
• the geometry (size and shape)
The symbol for a resistor is
For a conductor of length L and area A, the resistance
is
L
R
A
where  is called the resistivity of the material.
Temperature Dependence and Superconductivity
In general, the resistivity of most materials will depend on the
temperature. For most metals, resistivity increases linearly
with temperature:
   0 [1   (T  T0 )]
Some materials, when very cold, have a resistivity which
abruptly drops to zero. Such materials are called
superconductors.
A bird lands on a bare copper wire carrying a current of 32 A.
The wire is 8 gauge, which means that its cross-sectional area is
0.13 cm2. (a) Find the difference in potential between the bird’s
feet, assuming they are separated by a distance of 6.0 cm. (b)
Will your answer to part (a) increase or decrease if the separation
between the bird’s feet increases?
Direct Current (DC)
Circuits
A circuit is a loop comprised
of elements like resistors
and capacitors around which
current flows.
battery
For current to continue to
flow in a circuit, there must
be an energy source such as
a battery.
The light bulb in this circuit is
the resistor. Connecting
wires are assumed to have
zero resistance.
battery
Battery as emf in DC Circuits
+ terminal at higher potential than – terminal
I
-
+
I
Electric potential increases by e
Electric potential decreases by IR
R
Imagine positive charges
moving clockwise around the
circuit. The electric potential
increases by 12 V across the
battery and decreases by 12 V
across the resistor.
Energy and Power in Electric Circuits
Resistance is like an internal friction; energy is dissipated.
The energy dissipated per unit time is the power P:
P =U/ t =(Q/t)V = IV
SI unit: watt, W
Using Ohm’s Law, V=IR, power can be rewritten as:
P = I2R = V2/R
Energy Usage:
1 kilowatt-hour = (1000 W)(3600 s) = (1000 J/s)(3600 s) = 3.6106 J
It costs 2.6 cents to charge a car battery at a voltage of 12 V and a
current of 15 A for 120 minutes. What is the cost of electrical
energy per kilowatt-hour at this location?
A 75-W light bulb operates on a potential difference of 95 V. Find
the current in the bulb and its resistance.
Resistors in Series and Parallel
Any two circuit elements can be
combined in two different ways:
• in series - with one right after the other;
the same current must flow through both
elements.
R1
R2
Series
Combination
R1
• in parallel – connected across the same
potential difference; the current is divided
into two paths.
R2
Parallel
Combination
Equivalent Resistance
R1
R1
R2
a
b
Series
Combination
The current I is the same in both
resistors, so the voltage Vba must
satisfy:
Vba= IR1 + IR2 = I(R1 + R2)
Req = R1+ R2
a
R2
b
Parallel
Combination
The current may be different in
each resistor, but the voltage
Vba is the same across each
resistor and the total current
is conserved: I = I1 + I2
1
1
1


Req R1 R2
Resistors in series
(a) Three resistors, R1, R2, and
R3, connected in series. Note
that the same current I flows
through each resistor.
(b) The equivalent resistance,
Req = R1 + R2 + R3
has the same current flowing
through it as the current I in
the original circuit.
Resistors in parallel
(a) Three resistors, R1, R2, and R3,
connected in parallel. Note that
each resistor is connected
across the same potential
difference, e.
(b) The equivalent resistance,
1
1
1
1



Req R1 R2 R3
has the same current flowing
through it as the total current I
in the original circuit.
Conceptual Question
Consider the circuit shown in the figure, in which three lights, each
with a resistance R, are connected in parallel. What happens to the
intensity of light 3 when the switch is closed? What happens to the
intensities of lights 1 and 2?
Analyzing a complex circuit of resistors
All resistors are the same in Figure (a).
(a) The two vertical resistors are in
parallel with one another, hence
they can be replaced with their
equivalent resistance, R/2.
(b) Now, the circuit consists of three
resistors in series. The equivalent
resistance of these three resistors
is 2.5 R.
(c) The original circuit reduced to a
single equivalent resistance.
Walker Problem 26, pg. 710
What is the equivalent resistance?
Walker Problem 44, pg. 711
The current in the 13.8 W resistor is 0.750 A.
Find the current in the other resistors in the circuit.
Kirchhoff’s Rules
Often what seems to be a complicated circuit can be reduced
to a simple one, but not always. For more complicated circuits
we must apply Kirchhoff’s Rules:
• Junction Rule:
follows from
conservation of charge
• Loop Rule:
follows from
conservation of energy
The sum of currents entering a junction
equals the sum of currents leaving a
junction.
I  0
The sum of the potential difference
across all the elements around any
closed circuit loop must be zero.
V  0
Kirchhoff’s junction rule
Kirchhoff’s junction rule states that the sum of the currents
entering a junction must equal the sum of the currents leaving the
junction. In this case, for the junction labeled A:
I1 = I 2 + I3
or
I1 – I2 – I3 = 0
A specific application of Kirchhoff’s junction rule
Applying Kirchhoff’s junction rule to the junction A:
I1  I2  I3 = 0
I3 = (2.0  5.5) A = 3.5 A
The minus sign indicates that I3 flows opposite to the direction
shown; that is, I3 is upward.
Kirchhoff’s loop rule
Kirchhoff’s loop rule states that as one moves around a closed loop
in a circuit the algebraic sum of all potential differences must be
zero. The electric potential:
• increases as one moves from the minus to the plus plate of a battery
• decreases as one moves through a resistor in the direction of the
current
Analyzing a simple circuit
Junction Rule: I1 = I2 + I3
Loop Rule: Use any two of these three loops
e  I3R  I1R=0
I3R  I2R = 0
What is the equation?
Walker Problem 52, pg. 711
How much current flows through each battery when the
switch is (a) closed and (b) open? (c) With the switch
open, suppose that point A is grounded. What is the
potential at point B?
B
A
Circuits containing Capacitors
Capacitors are used in electronic circuits. The symbol
for a capacitor is
+

We can also combine separate capacitors into one
effective or equivalent capacitor. For example, two
capacitors can be combined either in parallel or in
series.
Series
Parallel
Combination
C1
C2
Combination
C1
C2
Capacitors in parallel
(a) Three capacitors, C1, C2, and
C3, connected in parallel.
Note that each capacitor is
connected across the same
potential difference, e.
(b) The equivalent capacitance,
Ceq = C1 + C2 + C3
has the same charge on its
plates as the total charge on
the three original capacitors.
Capacitors in series
(a) Three capacitors, C1, C2, and C3,
connected in series. Note that
each capacitor has the same
magnitude of charge on its
plates.
(b) The equivalent capacitance,
1
1
1
1



Ceq C1 C2 C3
has the same charge as the
original capacitors.
Parallel vs. Series Combination
Parallel
Series
• charge Q1 , Q2
• charge on each is Q
• total Q = Q1 + Q2
• total charge is Q
• voltage on each is V
• voltage V1 , V2
• Q1= C1V
• Q = C1V1
• Q2= C2V
• Q = C2V2
• Q = CeffV
• Q = Ceff(V1+V2)
• Ceff = C1+C2
• 1/Ceff = 1/C1+1/C2
Walker Problem 54, pg. 711
A 15 V battery is connected to three capacitors in series.
The capacitors have the following capacitance: 4.5 F,
12 F, and 32 F. Find the voltage across the 32 F
capacitor.
RC Circuits
We can construct circuits with more than just a resistor.
For example, we can have a resistor, a capacitor, and a
switch:
R
e
C
S
When the switch is closed the current will change.
The capacitor acts like an open circuit: no charge flows across
the gap. However, when the switch is closed, current can flow
from the negative plate of the capacitor to the positive plate.
A typical RC circuit
(a) Before the switch is closed
(t < 0) there is no current in the
circuit and no charge on the
capacitor.
(b) After the switch is closed
(t
> 0) current flows and the
charge on the capacitor builds
up over a finite time. As t
increases without limit, the
charge on the capacitor
approaches Q = Ce.
Capacitor Charging
Assume that at time t = 0, the
capacitor is uncharged, and we
close the switch. It can be shown
that the charge on the capacitor
at some later time t is:
Charge versus time
for an RC circuit
q = qmax(1 – e-t/t)
The time constant t = RC, and
qmax is the maximum amount of
charge that the capacitor will
acquire:
qmax=Ce
The current is given by
I = (e/R)e-t/t
Current versus
time for an RC
circuit
What happens after the switch is closed?
The capacitor is initially uncharged.
Walker Problem 62, pg. 712
The capacitor in an RC circuit (R = 120 W, C = 45 F) is
initially uncharged. Find (a) the charge on the capacitor and (b)
the current in the circuit one time constant (t = RC) after the
circuit is connected to a 9.0 V battery.
Walker Problem 78, pg. 713
Consider the circuit shown below. (a) Is the current flowing through
the battery immediately after the switch is closed greater than, less
than, or the same as the current flowing through the battery long after
the switch is closed? (b) Find the current flowing through the battery
immediately after the switch is closed. (c) Find the current in the
battery long after the switch is closed.
Discharging a capacitor
(a) A charged capacitor is
connected to a resistor. Initially
the circuit is open, and no
current can flow.
(b) When the switch is closed
current flows from the + plate of
the capacitor to the - plate. The
charge remaining on the
capacitor approaches zero after
several time units, RC.
Capacitor Discharging
Consider this circuit with a
charged capacitor at time t = 0:
R
C
+Q
-Q
S
It can be shown that the charge
on the capacitor is given by:
q(t) = Qe-t/t
The time constant t = RC.
Current versus time in an RC circuit
Measuring the current in a circuit
An ammeter is device for measuring currents in electrical circuits.
To measure the current
flowing between points A
and B in (a) an ammeter is
inserted into the circuit, as
shown in (b).
An ideal ammeter would
have zero resistance.
Measuring the voltage in a circuit
A voltmeter measures voltage differences in electrical circuits.
The voltage difference between points C
and D can be measured by connecting a
voltmeter in parallel to the original circuit.
An ideal voltmeter would have infinite
resistance.