Download Magnetic Forces and Fields

Magnetic Forces and Fields
Objective: TSW understand and apply the
concepts of a magnetic fields and forces by
predicting the path of a charged particle in a
magnetic field.
A Magnetic Field is a property of space
around a magnet causing a force on
other magnets.
• Every magnet has two poles, North and
South.
• Like poles repel and unlike poles attract.
• Magnetic fields are produced by moving
charge, such as current moving in a wire.
• The Earth has a magnetic field.
The next two slides contain
vocabulary and equations, you
should commit them to
memory
Vocabulary
electromagnet
A magnet with a field produced by an electric current.
law of poles
Like poles repel each other and unlike poles attract.
magnetic domain
Cluster of magnetically aligned atoms.
magnetic field
The space around a magnet in which another magnet or moving charge
will experience a force.
mass spectrometer
A device which uses forces acting on charged particles moving a
magnetic field and the resulting path of the particles to determine the
relative masses of the charged particles.
right-hand rules
Used to find the magnetic field around a current-carrying wire or the
force acting on a wire or charge in a magnetic field.
solenoid
A long coil of wire in the shape of a helix (spiral); when current is passed
through a solenoid it produces a magnetic field similar to a bar magnet.
Equations and symbols:
FB  qvB sin 
mv
r
qB
FB  BIL sin 
0 I
B
2r
where
B = magnetic field (T)
FB = magnetic force (N)
q = charge (C)
v = speed or velocity of a charge (m/s)
θ = angle between the velocity of a moving charge
and a magnetic field, or between the length of a
current-carrying wire and a magnetic field
r = radius of path of a charge moving in a magnetic
field, or radial distance from a current-carrying wire.
m = mass (kg)
I = current (A)
L = length of wire in a magnetic field (m)
μo = permeability constant
= 4π x 10-7 (T m) / A
Let’s Get Started!
By definition magnetic field lines go out of the north pole
and into the south pole. Here is two dimensional
representation of a the field lines around a bar magnet. A
compass will always point in the direction of the magnetic
field lines. (Toward the magnetic south.)
Magnetic Field due to two bar
magnets:
N
S
The Earth has a magnetic field. We don’t really know why.
The geographic north is the magnetic south. Solar
particles get trapped in the Earth’s magnetic field causing
the Aurora borealis (Northern Lights)
The Magnetic Force on a moving charge in
an external magnetic field.
• A moving charge creates a magnetic field, therefore it
will experience a magnetic force as it moves through an
external magnetic field.
• The direction of the force is given by right-hand rule #1.
• The magnitude of this force is given by the equation:
FB = magnetic force (N)
FB  qvBsin 
simulation
q = charge (C)
v = velocity (m/s)
B = magnetic field (Tesla)
θ = the angle between v and B
The direction is given by right-hand rule #1:
Fingers: Point in the direction of the field.
Thumb: Points in the direction of the charge velocity.
Palm of hand: Points in the direction of the force on a positive charge.
Back of hand: Points in the direction of the force on a negative charge.
F
F
N
S
v
B
I or v
B
Example: A magnetic field is directed to the right. Predict the direction and
magnitude of the magnetic force on the following charges moving through the
field.
B
v
+
FB  qvB sin( 90 )  qvB

Direction: Into the page.
Example: A magnetic field is directed to the right. Predict the direction and
magnitude of the magnetic force on the following charges moving through the
field.
B
+
v
FB  qvB sin( 0 )  0

Example: A magnetic field is directed to the right. Predict the direction and
magnitude of the magnetic force on the following charges moving through the
field.
B
FB  qvB sin( 40 )  .64qvB

40º
Direction: Into the page.
Example: A magnetic field is directed to the right. Predict the direction and
magnitude of the magnetic force on the following charges moving through the
field.
B
v
-
FB  qvB sin( 90 )  qvB

Direction: Out of the page.
To draw field lines perpendicular to the page
we will use the following representations:
A field going into the page will
be represented with X’s
A field going out of the page will be
represented with •’s
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Bin
Bout
More Examples:
Bin
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FB
X
X
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X
v
+
X
X
X
X
FB  qvB sin( 90 )  qvB

X
X
X
X
X
X
Direction: Upward toward the
top of the page.
X
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More Examples:
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Bout
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FB
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FB  qvB sin( 90 )  qvB

Direction: Left. Negative x
•
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•
direction.
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X
Let’s predict the path of a moving
charge in a magnetic field.
More Examples:
Bin
X
X
X
X
X
X
X
X
X
X
X
X
X
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XFB
X
v
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+
X
X
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X
r
X
The charge will follow
a circular path with a
X
X
X
constant speed.
Example 1
A proton enters a magnetic field B which is directed into the page. The proton
has a charge +q and a velocity v which is directed to the right, and enters the
magnetic field perpendicularly.
q = +1.6 x 10-19 C
v = 4.0 x 106 m/s
B = 0.5 T
Determine
(a) the magnitude and direction of the initial force acting on the proton
(b) the subsequent path of the proton in the magnetic field
(c) the radius of the path of the proton
(d) the magnitude and direction of an electric field that would cause the proton
to continue moving in a straight line.
B
q
v
The force on a current carrying wire in a
magnetic field.
• A current carry wire has charge moving through it.
• Each of the moving charges will experience a
force due to the magnetic field
• The individual magnetic forces on each charge
will produce a net force on the entire wire.
• The magnitude of the force is given by the
equation below:
FB  BIL sin 
The direction is given by right-hand rule #1:
Fingers: Point in the direction of the field.
Thumb: Points in the direction of the current flow.
Palm of hand: Points in the direction of the force on a positive charge.
F
F
N
S
B
I
I or v
B
X
X
X
X
X
X
X
X
X
X
X
X
FB  BIL sin( 90 )  BIL

X
X
X
X
X
X
Direction: Downward (-y direction)
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I
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FB
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FB  BIL sin( 90 )  BIL
FB
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Direction: To the left (-x direction)
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Example 2: A wire carrying a 20A current and having a
length of 0.10m is placed between the poles of two magnets
as shown below. The magnetic field is uniform and has a
value of 0.8T. Determine the magnitude and direction of the
magnetic force acting on the wire.
I
N
S
Example 2
A wire is bent into a square loop and placed completely in a
magnetic field B = 1.2 T. Each side of the loop has a length of
0.1m and the current passing through the loop is 2.0 A. The
loop and magnetic field is in the plane of the page.
(a) Find the magnitude of the initial force on
each side of the wire.
(b) Determine the initial net torque acting on the
loop.
B
I
Magnetic Fields produced by currents.
Ampere’s Law
• A current carry wire produces a magnetic field
around itself.
• The direction of this field is determined using
right-hand rule #2.
• The magnitude of the magnetic field is found
using the equation below:
o I
B
2 r
Right-hand Rule #2
current I
I
Magnetic Field B
r
I
o I
B
2 r
Example3: Find the magnetic field midway between the two
current carrying wires shown in the diagram. Will the wires
attract or repel each other?
40cm
I = 2A
I = 3A
Topic 12 Electromagnetic Induction
Electromagnetic induction is the process
by which an emf (voltage) is produced in a
wire by a changing magnetic flux.
• Magnetic flux is the product of the
magnetic field and the area through which
the magnetic field passes.
• Electromagnetic induction is the principle
behind the electric generator.
• The direction of the induced current due to
the induced emf is governed by Lenz’s
Law.
Here is a visual model of what we
did in chapter 19:
Loop of wire
Input
Current
and in an
External
magnetic field
Electric Motor
Output
Force
(wire moves)
Here is a visual model of what we
will do in chapter 20:
Input
Force
(move wire)
Loop of wire
in an
Output
External
Current
magnetic field
Generator
The next two slides contain
vocabulary and equations, you
should commit them to
memory
Important Terms
alternating current - electric current that rapidly reverses its direction
electric generator - a device that uses electromagnetic induction to
convert mechanical energy into electrical energy
electromagnetic induction - inducing a voltage in a conductor by
changing the magnetic field around the conductor
induced current - the current produced by electromagnetic induction
induced emf - the voltage produced by electromagnetic induction
Faraday’s law of induction - law which states that a voltage can be
induced in a conductor by changing the magnetic field around the
conductor
Lenz’s law - the induced emf or current in a wire produces a magnetic
flux which opposes the change in flux that produced it by
electromagnetic induction
magnetic flux - the product of the magnetic field and the area through
which the magnetic field lines pass.
motional emf - emf or voltage induced in a wire due to relative motion
between the wire and a magnetic field
Equations, Symbols, and Units
  BLv

I
R
  BA cos 

  N
t
P  IV
where
ε = emf (voltage) induced by
electromagnetic induction (V)
v = relative speed between a conductor
and a magnetic field (m/s)
B = magnetic field (T)
L = length of a conductor in a magnetic
field (m)
I = current (A)
R = resistance (Ω)
Φ = magnetic flux (Tm2 = Weber=Wb)
A = area through which the flux is
passing (m2)
= angle between the direction of the
magnetic field and the area through
which it passes
Magnetic Flux
Consider a rectangular loop of wire of height L and width x
which sits in a region of magnetic field of strength B. The
magnetic field is directed into the page, as shown below:
w
L
The magnetic flux is given by the following equation:
  BA cos 
Φ = The magnetic flux (Tm2 = Wb)
B = The magnetic field (T)
A = area of loop (m2)
Φ = angle between the field and the area
Faraday’s law states that an induced emf is
produced by changing the flux, but how
could the flux be changed?
• Turn the field off or on.
• Move the loop of wire out of the field
• Rotate the loop to change the angle
between the field and the area of the loop.
Here is Faraday’s Law in equation
form:

N
t
Where
Є = The induced emf (voltage) (V)
ΔΦ = The change in flux (Wb)
Δt = The change in time (s)
N = number of loops.
*Note that an emf is only produced if the flux changes. The quicker the
flux changes the larger the induced emf.
The induced emf in the wire will produce a current in the wire. The
magnitude of the induced current is found using Ohm’s Law:
V  IR
  IR
I

R
The direction of the induced current is found using Lenz’s Law (conservation of
energy).
Lenz’s Law – The induced current in a wire produces a
magnetic field such the flux of the produced magnetic
field opposes the original change in flux. In simple
terms the wire resists the change in flux and wants to
go back to the way things were.
It is helpful to use RHR#2 when using Lenz’s Law.
Example 1: A circular loop of wire with a resistance of 0.5Ω and
radius 30cm is placed in an external magnetic field of 0.2T. The
magnetic field is turned off in .02 seconds.
a) Calculate the original flux of the loop.
b) Calculate the induced emf.
c) Calculate the current induced in the wire.
d) What direction does the induced current have?
e) What other way could the same emf be induced without turning the
field off?
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Let’s do some examples
predicting the induced current
direction using Lenz’s Law
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• flux
Bout decreasing
CCW
Bout increasing flux
CW
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Bin decreasing flux
CW
Bin increasing flux
CCW
If you are really struggling applying Lenz’s
Law, then memorize the following table:
decreasing
Increasing
B out
CCW
CW
B in
CW
CCW
Example 2: A circuit with a total resistance of R is made using a set
of metal wires and a copper bar. The magnetic field is directed into
the page as shown in the diagram. The bar starts on the left and is
pulled to the right at a constant velocity.
a) Calculate the induced emf in the circuit.
b) Calculate the current induced in the wire.
c) What direction does the induced current have?
d) What is the magnitude and direction of the magnetic force that
opposes the motion of the bar?
X
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v
The last example led to the equation for the motional emf.
The motional emf is the voltage induced in a wire as it
moves in an external magnetic field. The induced emf will
produce a current in the wire, which will in turn result in a
force that opposes the motion of the wire. You don’t get
something for nothing.
Motional emf
Where
Є = The induced emf (V)
  BLv
B = The external magnetic field (T)
L = The length of the wire (m)
v = The velocity of the wire (m/s)
Example 3: A conducting rod of length 0.30 m and resistance 10.0 Ω
moves with a speed of 2.0 m/s through a magnetic field of 0.20 T which is
directed out of the page.
v
L
B (out of the
page)
a) Find the emf induced in the rod.
b) Find the current in the rod and the direction it flows.
c) Find the power dissipated in the rod.
d) Find the magnetic force opposing the motion of the rod.
Example 4: A square loop of sides a = 0.4 m, mass m =
1.5 kg, and resistance 5.0 Ω falls from rest from a
height h = 1.0 m toward a uniform magnetic field B
which is directed into the page as shown.
(a) Determine the speed of the loop just before it enters
the magnetic field.
As the loop enters the magnetic field, an emf ε and a
current I is induced in the loop.
(b) Is the direction of the induced current in the loop
clockwise or counterclockwise? Briefly explain how
you arrived at your answer.
When the loop enters the magnetic field, it falls through
with a constant velocity.
(c) Calculate the magnetic force necessary to keep the
loop falling at a constant velocity.
(d) What is the magnitude of the magnetic field B
necessary to keep the loop falling at a
constant velocity?
(e) Calculate the induced emf in the loop as it enters and
exits the magnetic field.
a
a
h
B
Electromagnetic induction
If a magnet is
moved inside a
coil an electric
current is
induced
(produced)
Generator/dynamo
A generator
works in this way
by rotating a coil
in a magnetic
field (or rotating a
magnet in a coil)
Motor = generator
If electric energy enters a motor it is
changed into kinetic energy, but if kinetic
energy is inputted (the motor is turned)
electric energy is produced!
The Motor Effect
When a current is placed in a magnetic
field it will experience a force (provided the
current is not parallel to the field). This is
called the motor effect.
Can you
copy this
sentence
into your
books
please.
Sample question
In this example, which way will the wire be
pushed? (red is north on the magnets)
Sample question
In this example, which way will the wire be
pushed? (red is north on the magnets)
Current
Field
Electromagnetic Induction
Imagine a wire moving with velocity v in a
Wire
magnetic field B out of the page.
L
moving
with
velocity v
v
Region of
magnetic
field B out of
page
The electrons in the wire feel a force (the
motor effect) which pushes the electrons
to the right. This creates a potential
difference in the wire.
L
v
Electrons
pushed this
way
ε = BLv
This means that a conducting wire of length L moving with
speed v normally to a magnetic field B will have a e.m.f.
of BLv across its ends. This is called a motional e.m.f.
L
Wire
moving
with
velocity v
v
Region of
magnetic
field B out of
page
Faraday’s Law
Faraday’s Law
Consider a magnet moving through a
rectangular plane coil of wire.
N
A
S
Faraday’s Law
A current is produced in the wire only
when the magnet is moving.
N
A
S
Faraday’s Law
The faster the magnet moves, the bigger
the current.
N
A
S
Faraday’s Law
The stronger the magnet, the bigger the
current.
N
A
S
Faraday’s Law
The more turns on the coil (same area),
the bigger the current.
N
A
S
Faraday’s Law
The bigger the area of the coil, the bigger
the current.
N
A
S
Faraday’s Law
If the movement is not perpendicular, the
current is less.
A
Magnetic Flux (Ф)
Imagine a loop of (plane) wire in a region
where the magnetic filed (B) is constant.
B
The magnetic flux (Ф) is defined as Ф = BAcosθ
where A is the area of the loop and θ is the
angle between the magnetic field direction and
the direction normal (perpendicular) to the plane
of the coil.
B
If the loop has N turns, the flux is given by
Ф = NBAcosθ in which case we call this the flux
linkage.
B
The unit of flux is the Weber (Wb) (= 1 Tm2)
It can help to imagine the flux as the number of
lines of magnetic field going through the area of
the coil. We can increase the flux with a larger
area, larger field, and keeping the loop
perpendicular to the field.
B
Faraday’s law (at last!)
As we seen, an e.m.f.
is only induced when
the field is changing.
The induced e.m.f. is
found using Faraday’s
law, which uses the
idea of flux.
Faraday’s law
The induced e.m.f. is
equal to the (negative)
rate of change of
magnetic flux,
ε = -ΔФ/Δt
Example question
The magnetic field through a single loop of
area 0.2 m2 is changing at a rate of 4 t.s-1.
What is the induced e.m.f?
“Physics for the IB Diploma” K.A.Tsokos (Cambridge University Press)
Example question
The magnetic field (perpendicular) through a single loop of area 0.2
m2 is changing at a rate of 4 t.s-1. What is the induced e.m.f?
Ф = BAcosθ = BA
E = ΔФ = ΔBA = 4 x 0.2 = 0.8 V
Δt
Δt
Another example question!
There is a uniform magnetic filed B = 0.40 T out of the
page. A rod of length L = 0.20 m is placed on a railing
and pushed to the right at a constant speed of v = 0.60
m.s-1. What is the e.m.f. induced in the loop?
L
v
The area of the loop is decreasing, so the
flux (BAcosθ) must be changing. In time Δt
the rod will move a distance vΔt, so the
area will decrease by an area of LvΔt
L
LvΔt
v
ε = ΔФ = BΔA = BLvΔt = BLv
Δt
Δt
Δt
E = 0.40 x 0.20 x 0.60 = 48 mV
L
LvΔt
v
An important
result, you may be
asked to do this!
Lenz’s Law
The induced current will be in such a
direction as to oppose the change in
magnetic flux that created the current
(If you think about it, this has to be so…….
Conservation of Energy)
Alternating current
A coil rotating in a magnetic field will
produce an e.m.f.
N
S
Alternating current
The e.m.f. produced is sinusoidal (for
constant rotation)
e.m.f.
V
Slip ring commutator
To use this e.m.f. to produce a current the
coil must be connected to an external
circuit using a split-ring commutator.
Slip-rings
lamp
Increasing the generator
frequency?
e.m.f.
V
Root mean square voltage and current
It is useful to define an “average” current
and voltage when talking about an a.c.
supply. Unfortunately the average voltage
and current is zero!
To help us we use the idea of root mean
square voltage and current.
Root mean square voltage
e.m.f.
V
Root mean square voltage
First we square the voltage to get a
quantity that is positive during a whole
cycle.
e.m.f.
V
Root mean square voltage
Then we find the average of this positive
quantity
e.m.f.
V
Root mean square voltage
We then find the square root of this
quantity.
e.m.f.
V
Root mean square voltage
We then find the square root of this
quantity.
e.m.f.
V
This value is called the
root mean square
voltage
Root mean square voltage
We then find the square root of this
quantity.
e.m.f.
V
Emax
Erms = Emax/√2
Transformers
What can you
remember about
transformers from last
year?
Transformers
Vp
Np
turns
Vs
Ns
turns
Primary coil
Iron core
“Laminated”
Secondary coil
Transformers
How do they work?
Vp
Np
turns
Vs
Ns
turns
Primary coil
Iron core
Secondary coil
An alternating current in the primary coil
produces a changing magnetic field in the
iron core.
Vp
Np
turns
Vs
Ns
turns
Primary coil
Iron core
Secondary coil
The changing magnetic field in the iron
core induces a current in the secondary
coil.
Vp
Np
turns
Vs
Ns
turns
Primary coil
Iron core
Secondary coil
It can be shown using Faraday’s law that:
Vp/Vs = Np/Ns and VpIp = VsIs
Vp
Np
turns
Vs
Ns
turns
Primary coil
Iron core
Secondary coil
Power transmission
When current passes through a wire, the
power dissipated (lost as heat) is equal to
P = VI across the wire
Since V = IR
Power dissipated = I2R
Power transmission
Power dissipated = I2R
Since the loss of power depends on the square
of the current, when transmitting energy over
large distances it is important to keep the current
as low as possible.
However, to transmit large quantities of energy
we therefore must have a very high voltage.
Power transmission
Electricity is thus transmitted at very high voltages using
step up transformers and then step down transformers.
220 V
Step-down
250,000 V
15,000 V
Step-up
15,000 V
Step-down
Dangerous?