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Chapter 26
DC Circuits
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Midterm 1 - Recap
Statistics:
Average = 78.9
σ
= 16.3
Problem questions:
Three identical capacitors are connected in series to a battery. If a total
charge of Q flows from the battery, how much charge does each capacitor
carry?
A solid non-conducting sphere of radius R carries a uniform charge
density. At a radial distance r1 = R/4 the electric field has a magnitude E0.
What is the magnitude of the electric field at a radial distance r2 = 2R?
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26-5 Circuits Containing Resistor
and Capacitor (RC Circuits)
When the switch is
closed, the
capacitor will begin
to charge. As it
does, the voltage
across it increases,
and the current
through the resistor
decreases.
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I
26-5 RC Circuits
To find the voltage as a function of time, we
write the equation for the voltage changes
around the loop:
Q t 
0 
 I t  R
C
Since I = dQ/dt, we can integrate to find the
charge as a function of time:
Q  t   C  1  e  t /RC 
 I  t   C  e
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 t / RC
 1    t /RC
  RC   R e
26-5 RC Circuits
The voltage across the capacitor is VC = Q/C:
VC  t     1  e
 t / RC

The quantity RC that appears in the exponent
is called the time constant of the circuit:
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26-5 RC Circuits
Example 26-11: RC circuit,
with emf.
The capacitance in the circuit shown
is C = 0.30 μF, the total resistance is
20 kΩ, and the battery emf is 12 V.
Determine (a) the time constant, (b)
the maximum charge the capacitor
could acquire, (c) the time it takes
for the charge to reach 99% of this
value, (d) the current I when the
charge Q is half its maximum value,
(e) the maximum current, and (f) the
charge Q when the current I is 0.20
its maximum value.
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26-5 RC Circuits
If an isolated charged
capacitor is
connected across a
resistor, it discharges:
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26-5 RC Circuits
Once again, the voltage and current as a
function of time can be found from the
charge:
and
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26-5 RC Circuits
Example 26-12: Discharging RC circuit.
In the RC circuit shown, the battery has fully charged
the capacitor, so Q0 = CE. Then at t = 0 the switch is
thrown from position a to b. The battery emf is 20.0 V,
and the capacitance C = 1.02 μF. The current I is
observed to decrease to 0.50 of its initial value in 40
μs. (a) What is the value of Q, the charge on the
capacitor, at t = 0? (b) What is the value of R? (c) What
is Q at t = 60 μs?
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26-5 RC Circuits
Conceptual Example 26-13: Bulb in RC circuit.
In the circuit shown, the capacitor is originally
uncharged. Describe the behavior of the lightbulb
from the instant switch S is closed until a long time
later.
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Summary of Chapter 26
• A source of emf transforms energy from
some other form to electrical energy.
• A battery is a source of emf in parallel with an
internal resistance.
• Resistors in series:
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Summary of Chapter 26
• Resistors in parallel:
• Kirchhoff’s rules:
1. Sum of currents entering a junction
equals sum of currents leaving it.
2. Total potential difference around closed
loop is zero.
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 26
• RC circuit has a characteristic time constant:
  RC
V0  t /
Charging: I  t   e
& VC  t   V0  1  e  t / 
R
V0  t /
Discharging: I  t   e & VC  t   V0e  t /
R
• To avoid shocks, don’t allow your body to
become part of a complete circuit.
Copyright © 2009 Pearson Education, Inc.
Chapter 27
Magnetism
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Units of Chapter 27
• Magnets and Magnetic Fields
• Electric Currents Produce Magnetic Fields
• Force on an Electric Current in a Magnetic
Field; Definition of B
• Force on an Electric Charge Moving in a
Magnetic Field
• Torque on a Current Loop; Magnetic Dipole
Moment
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Units of Chapter 27
• Applications: Motors, Loudspeakers,
Galvanometers
• Discovery and Properties of the Electron
• The Hall Effect
• Mass Spectrometer
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27-1 Magnets and Magnetic Fields
Magnets have two
ends – poles – called
north and south.
Like poles repel;
unlike poles attract.
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27-1 Magnets and Magnetic Fields
However, if you cut a magnet in half, you don’t
get a north pole and a south pole – you get two
smaller magnets.
Demo
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27-1 Magnets and Magnetic Fields
Magnetic fields can be visualized using
magnetic field lines, which are always closed
loops.
(Except for one April Fool’s Day in California … but that’s another story.)
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Demo
27-1 Magnets and Magnetic Fields
The Earth’s magnetic field is similar to that of a
bar magnet.
Note that the Earth’s
“North Pole” is really
a south magnetic
pole, as the north
ends of magnets are
attracted to it.
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27-1 Magnets and Magnetic Fields
A uniform magnetic field is constant in
magnitude and direction.
The field between
these two wide poles
is nearly uniform.
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27-2 Electric Currents Produce
Magnetic Fields
Experiment shows that an electric current
produces a magnetic field. The direction of the
field is given by a right-hand rule.
Demo
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27-2 Electric Currents Produce
Magnetic Fields
Here we see the
field due to a
current loop;
the direction is
again given by
a right-hand
rule.
Demo
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27-3 Force on an Electric Current in
a Magnetic Field; Definition of B
A magnet exerts a
force on a currentcarrying wire. The
direction of the force
is given by a righthand rule.
Demo
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27-3 Force on an Electric Current in
a Magnetic Field; Definition of B
The force on the wire depends on the
current, the length of the wire, the magnetic
field, and its orientation:
This equation defines the magnetic field B
B.
In vector notation:
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27-3 Force on an Electric Current in a
Magnetic Field; Definition of B
Unit of B: the tesla, T:
1 T = 1 N/A·m.
Another unit sometimes used: the gauss (G):
1 G = 10-4 T.
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27-3 Force on an Electric Current in
a Magnetic Field; Definition of B
Example 27-1: Magnetic Force
on a current-carrying wire.
A wire carrying a 30-A
current has a length l = 12
cm between the pole
faces of a magnet at an
angle θ = 60°, as shown.
The magnetic field is
approximately uniform at
0.90 T. We ignore the field
beyond the pole pieces.
What is the magnitude of
the force on the wire?
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27-3 Force on an Electric Current in
a Magnetic Field; Definition of B
Example 27-3: Magnetic Force
on a semicircular wire.
A rigid wire, carrying a
current I, consists of a
semicircle of radius R and two
straight portions as shown.
The wire lies in a plane
perpendicular to a uniform
magnetic field B
B0. Note choice
of x and y axis. The straight
portions each have length l
within the field. Determine the
net force on the wire due to
the magnetic field B
B0.
Copyright © 2009 Pearson Education, Inc.
27-4 Force on an Electric Charge
Moving in a Magnetic Field
The force on a moving charge is related to
the force on a current since a current is
just a bunch of moving charges:
Once again, the
direction is given by
a right-hand rule.
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27-4 Force on an Electric Charge
Moving in a Magnetic Field
Conceptual Example 27-4: Negative
charge near a magnet.
A negative charge -Q is placed at rest
near a magnet. Will the charge begin
to move? Will it feel a force? What if
the charge were positive, +Q?
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27-4 Force on an Electric Charge
Moving in a Magnetic Field
Example 27-5: Magnetic force on a proton.
A magnetic field exerts a force of 8.0 x 10-14 N toward the
west on a proton moving vertically upward at a speed of
5.0 x 106 m/s (a). When moving horizontally in a
northerly direction, the force on the proton is zero (b).
Determine the magnitude and direction of the magnetic
field in this region. (The charge on a proton is q = +e =
1.6 x 10-19 C.)
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27-4 Force on an Electric Charge
Moving in a Magnetic Field
If a charged particle is
moving perpendicular
to a uniform magnetic
field, its path will be a
circle.
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27-4 Force on an Electric Charge
Moving in a Magnetic Field
Example 27-7: Electron’s path in a
uniform magnetic field.
An electron travels at 2.0 x 107 m/s in a
plane perpendicular to a uniform
0.010-T magnetic field. Describe its
path quantitatively.
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27-4 Force on an Electric Charge
Moving in a Magnetic Field
Problem solving: Magnetic fields – things to
remember:
1. The magnetic force is perpendicular to the
magnetic field direction.
2. The right-hand rule is useful for determining
directions.
3. Equations in this chapter give magnitudes
only. The right-hand rule gives the direction.
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ConcepTest 27.1c Magnetic Force III
A positive charge enters a
uniform magnetic field as
shown. What is the direction of
the magnetic force?
1) out of the page
2) into the page
3) zero
4) to the right
5) to the left


v

q

ConcepTest 27.1c Magnetic Force III
A positive charge enters a
uniform magnetic field as
shown. What is the direction of
the magnetic force?
1) out of the page
2) into the page
3) zero
4) to the right
5) to the left
Using the right-hand rule, you can
see that the magnetic force is
directed into the page. Remember
that the magnetic force must be
perpendicular to BOTH the B field
and the velocity.


v

 q
F

ConcepTest 27.3 Magnetic Field
A proton beam enters a
magnetic field region as shown
below. What is the direction of
the magnetic field B?
1) + y
2) – y
3) + x
4) + z (out of page)
5) – z (into page)
y
x
ConcepTest 27.3 Magnetic Field
A proton beam enters a
magnetic field region as shown
below. What is the direction of
the magnetic field B?
1) + y
2) – y
3) + x
4) + z (out of page)
5) – z (into page)
The picture shows the force acting
in the +y direction. Applying the
y
right-hand rule leads to a B field
that points into the page. The B
field must be out of the plane
because B  v and B  F.
Follow-up: What would happen to a beam of atoms?
x
27-4 Force on an Electric Charge
Moving in a Magnetic Field
Conceptual Example 27-9: A helical path.
What is the path of a charged particle in a
uniform magnetic field if its velocity is not
perpendicular to the magnetic field?
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27-4 Force on an Electric Charge
Moving in a Magnetic Field
The aurora borealis (northern lights) is caused
by charged particles from the solar wind
spiraling along the Earth’s magnetic field, and
colliding with air molecules.
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