Ammeter and Voltmeter
 A device used to measure current is called an ammeter
 A device used to measure voltage is called a voltmeter
 To measure the current, the ammeter must be placed in the
circuit in series
 To measure the voltage, the voltmeter must be wired in
parallel with the component across which the voltage is to be
measured
Voltmeter in parallel
Ammeter in series
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Tuesday, February 25, 2014
iClicker
Two resistors, R1 = 3.00 Ω and R2 = 5.00 Ω, are connected
in series with a battery with Vemf = 8.00 V and an
ammeter with RA = 1.00 Ω, as shown in the figure. What
is the current measured by the ammeter?
a) 1/2 A
b) 3/4 A
c) 8/9 A
d) 1 A
e) 3/2 A
Tuesday, February 25, 2014
Two resistors, R1 = 3.00 Ω and R2 = 5.00 Ω, are connected
in series with a battery with Vemf = 8.00 V and an
ammeter with RA = 1.00 Ω, as shown in the figure. What
is the current measured by the ammeter?
a) 1/2 A
b) 3/4 A
c) 8/9 A
d) 1 A
e) 3/2 A
V
i=
= 8/9
R 1 + R 2 + R3
Tuesday, February 25, 2014
This is NOT the current that
flows in the circuit without
ammeter
iClicker
To measure the voltage, the voltmeter must be
wired in parallel with the component across which
the voltage is to be measured.
Connecting ammeter and voltmeter changes the
circuit!
Question: what should be resistances of ammeter and voltmeter?
A: ammeter high R, voltmeter low R
B: ammeter low R, voltmeter low R
C: ammeter low R, voltmeter high R
D: ammeter high R, voltmeter high R
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Tuesday, February 25, 2014
To measure the voltage, the voltmeter must be
wired in parallel with the component across which
the voltage is to be measured.
Connecting ammeter and voltmeter changes the
circuit!
Question: what should be resistances of ammeter and voltmeter?
A: ammeter high R, voltmeter low R
B: ammeter low R, voltmeter low R
C: ammeter low R, voltmeter high R
D: ammeter high R, voltmeter high R
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Tuesday, February 25, 2014
RC Circuits (1)
 So far we have dealt with circuits containing sources of emf
and resistors
 The currents in these circuits did not vary in time
 Now we will study circuits that contain capacitors as well as
sources of emf and resistors
 These circuits have currents that vary with time
 Consider a circuit with
• a source of emf, Vemf,
• a resistor R,
• a capacitor C
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Tuesday, February 25, 2014
 If R=0, at the moment the
circuit is closed, very large
current, all the power in the
capacitor is quickly
dissipated: short circuit.
 For finite resistance, there is
potential drop over the
resistor V = i R
 Current cannot be too large:
no short circuit
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Tuesday, February 25, 2014
RC Circuits (2)
 We then close the switch, and current begins to flow in the
circuit, charging the capacitor
 The current is provided by the
source of emf, which maintains
a constant voltage
When the capacitor is fully
charged, no more current flows in the circuit
 When the capacitor is fully charged, the voltage across the
plates will be equal to the voltage provided by the source of
emf and the total charge qtot on the capacitor will be qtot =
CVemf
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Tuesday, February 25, 2014
Capacitor Charging (1)
 Going around the circuit in a counterclockwise direction we can write
 We can rewrite this equation
remembering that i = dq/dt

dq q − V C
+
=0
dt
RC
q̂ = q − V C
dq̂
q̂
+
=0
dt
RC
−t/RC
q̂ = q̂0 e
q = V C + q̂0 e
−t/RC
at time t=0, q = 0 → q̂0 = −V C
The term Vc is negative since
the top plate of the capacitor is
connected to the positive higher potential - terminal of
the battery. Thus analyzing
counter-clockwise leads to a
drop in voltage across the
capacitor!
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Tuesday, February 25, 2014
Capacitor Charging (1)
 Going around the circuit in a counterclockwise direction we can write
 We can rewrite this equation
remembering that i = dq/dt
 The solution is
 where q0 = CVemf and τ = RC
The term Vc is negative since
the top plate of the capacitor is
connected to the positive higher potential - terminal of
the battery. Thus analyzing
counter-clockwise leads to a
drop in voltage across the
capacitor!
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Tuesday, February 25, 2014
Capacitor Charging (2)
 We can get the current flowing in the circuit by
differentiating the charge with respect to time
Math Reminder:
 The charge and current as a function of time are shown here
(τ = RC)
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Tuesday, February 25, 2014
Capacitor Discharging (1)
 Now let’s take a resistor R and a fully charged capacitor
C with charge q0 and connect them together by moving the switch
from position 1 to position 2
 In this case, current will flow in the circuit until the capacitor is
completely discharged
 While the capacitor is discharging we can apply the Loop Rule around
the circuit and obtain
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Tuesday, February 25, 2014
Capacitor Discharging (2)
 The solution of this differential equation for the
charge is
 Differentiating charge we get the current
 The equations describing the time dependence of the charging and
discharging of capacitors all involve the exponential factor e-t/RC
 The product of the resistance times the capacitance is defined as the
time constant τ of an RC circuit
 We can characterize an RC circuit by specifying the time constant of the
circuit
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Tuesday, February 25, 2014
Example: Time to Charge a Capacitor (1)
 Consider a circuit consisting of a 12.0 V battery,
a 50.0 Ω resistor, and a 100.0 µF capacitor wired in series.
 The capacitor is initially uncharged.
Question:
How long will it take to charge the capacitor in this circuit to 90% of its
maximum charge?
Answer:
The charge on the capacitor as a function of time is
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Tuesday, February 25, 2014
Example: Time to Charge a Capacitor (2)
 We need to know the time corresponding to
 We can rearrange the equation for the charge on the
capacitor as a function of time to get
Math Reminder: ln(ex)=x
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Tuesday, February 25, 2014
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Tuesday, February 25, 2014
Magnetism
 A very strange force...
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Tuesday, February 25, 2014
Magnetism
 Magnetic force/field is a very strange force...
 It is related to electric: Electro-magnetic force
 Magnetic field acts on moving charges only (and spins quantum effect)
 Force is perpendicular to velocity, still depends on charge
FB = q v × B
Lorentz force
 Full electromagnetic force
FB = q (E + v × B)
Coulomb + Lorentz force
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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Tuesday, February 25, 2014
Magnetic (Lorentz) force
F = qv × B
 Depends on velocity
 Acts in a direction perpendicular both to velocity and
B-field
 Depends on the electric charge
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Tuesday, February 25, 2014
Right Hand Rule (1)
 The direction of the cross product (v x B ) is given by the
right hand rule
 To apply the right hand rule
• Use your right hand!
• Align thumb in the direction of v
• Align your index finger with the magnetic field
• Your middle finger will point in the direction of the cross
product v x B
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Tuesday, February 25, 2014
Magnitude of Magnetic Force
 The magnitude of the magnetic force on a moving charge is
… where θ is the angle between the velocity of the charged
particle and the magnetic field.
 Do you see that there is no magnetic force on a charged
particle moving parallel to the magnetic field?
(Because θ is zero and sin(0)=0)
 Do you see when the magnetic force is most strong?
(Max. force is for θ = 90 degrees; then F = qvB)
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Tuesday, February 25, 2014
Units of Magnetic Field Strength
 The magnetic field strength has received its own named unit,
the tesla (T), named in honor of the physicist and inventor
Nikola Tesla (1856-1943)
Check unit
consistency:
F=qvB
N = C (m/s) T
 A tesla is a rather large unit of the magnetic field strength
 Sometimes you will find magnetic field strength stated in
units of gauss (G), (not an official SI unit)
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Tuesday, February 25, 2014
Recall electricity: Action at distance:
through “fields”
 Charge -> E-field -> force on another charge
Force
another charge
Electric force along E-field
electric field
electric charge (source of field)
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Tuesday, February 25, 2014
Magnetic field
B
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Tuesday, February 25, 2014
No magnetic charge!
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Maxwell’s equations
∇ · E = ρ/�0
∇·B=0
electric charge produces E-field
no magnetic charge
∇ × B = µ0 J + µ0 � 0 ∂ t E
∇ × E = −∂t B
changing B-field also
produce E-field
∇ = {∂x , ∂y , ∂z }
electric current and
changing E-field
produce B-field
No magnetic charge
B-field can be due to
current or changing
E-field
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Tuesday, February 25, 2014
How to make B-field?
 ??? -> B-field -> force on another charge
∇ × B = µ0 J + µ0 � 0 ∂ t E
B-field can be created by
- Currents
- Changing E-field
B - magnetic field (magnetic induction)
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Tuesday, February 25, 2014
Two types of current
 Moving charges (eg wire with DC current)
A
A
 Spins of elementary particles
Most elementary particles
are like small current loops.
Think of a rotating charges sphere
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Tuesday, February 25, 2014
Magnetic field of a moving charge
µ0 q v × r
B=
3
4π r
V
q
r
B
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Tuesday, February 25, 2014
But... velocity depends on observer
E
+q
FB = q (E + v × B)
v=0, FL= e vxB =0
Only E-field
+q
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Tuesday, February 25, 2014
But... velocity depends on observer
E
+q
FB = q (E + v × B)
v=0, FL= e vxB =0
Only E-field
+q
V
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Tuesday, February 25, 2014
But... velocity depends on observer
E
+q
FL = q (−v × B) �= 0
v=0, FL= e vxB =0
Only E-field
FB = q (E + v × B)
+q
V
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Tuesday, February 25, 2014
But... velocity depends on observer
E
+q
FL = q (−v × B) �= 0
v=0, FL= e vxB =0
Only E-field
FB = q (E + v × B)
+q
V
µ0 q v × rE-field and B-field!
B=
3
4π r
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Tuesday, February 25, 2014
 Separation of Electromagnetic field into
electric and magnetic fields depends on the
observer.
 Observer moving past a stationary charge will
see both electric and magnetic fields.
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Tuesday, February 25, 2014
Magnetic Field Lines (1)
 In analogy with the electric field, we define a magnetic field
to describe the magnetic force
 As we did for the electric field, we may represent the
magnetic field using magnetic field lines
 The magnetic field direction is always tangent to the
magnetic field lines
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Tuesday, February 25, 2014
Magnetic dipole
 Similar to E-field, when total charge was zero, it’s the
dipole moment that became important.
 For B-field, (total magnetic charge is zero) dipole is
the first component
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Tuesday, February 25, 2014
Permanent magnets: alignment of
elementary dipole
N
S
Spin= circular current
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Tuesday, February 25, 2014
Permanent magnet: aligned dipoles
N
N
S
S
N
N
S
N
S
N
S
S
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Tuesday, February 25, 2014
Magnetic Field Lines (2)
 The magnetic field lines from a permanent bar magnet are
shown below
Two dimensional computer calculation
Three dimensional real-life
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Tuesday, February 25, 2014
Permanent Magnets - Poles
 Magnets exert forces on one
another --- attractive or repulsive
depending on orientation.
 If we bring together two permanent
magnets such that the two north
poles are together or two south
poles are together, the magnets will
repel each other
 If we bring together a north pole
and a south pole, the magnets will
attract each other
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Tuesday, February 25, 2014
Broken Permanent Magnet
 If we break a permanent
magnet in half, we do not
get a separate north pole
and south pole
 When we break a bar
magnet in half, we always
get two new magnets, each
with its own north and south pole
 Unlike electric charge that exists as positive (proton) and negative (electron)
separately, there are no separate magnetic monopoles (an isolated north pole or an
isolated south pole)
 Scientists have carried out extensive searches for magnetic monopoles; all results are
negative
 Magnetism is not caused by magnetic particles! Magnetism is caused by electric
currents
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Tuesday, February 25, 2014
The Earth’s Magnetic Field
 The Earth itself is a magnet
 It has a magnetic field sort of like a bar
magnet (but not really like a bar magnet)
 The poles of the Earth’s
magnetic field are not aligned
with the Earth’s geographic
poles defined as the endpoints
of the axis of the Earth’s rotation
 The Earth’s magnetic field is not a simple dipole
field because it is distorted by the solar wind
 Protons from the Sun moving at 400 km/s
 The magnetic field inside the Earth is very
complex
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Tuesday, February 25, 2014
Van Allen Radiation Belts
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Tuesday, February 25, 2014
Earth’s Magnetic Poles
 The north and south magnetic poles are not exactly located at
the north and south geographic poles
• The magnetic north pole is located in Canada
• The magnetic south pole is located on the edge of Antarctica
 The magnetic poles move around, at a rate of 40 km per year
• By the year 2500 the magnetic north pole will be
located in Siberia
• There are indications that the Earth’s magnetic field reverses
↔S) on the time scale of 1 million years or so.
(N
A compass needle points toward the magnetic north pole rather than
true north
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Tuesday, February 25, 2014
Earth’s Magnetic Field Strength
 The strength of the Earth’s magnetic field at the
surface of Earth varies between 0.25 G and 0.65 G
•
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Tuesday, February 25, 2014