Transcript Using the “Clicker” - Boston University: Physics

The force on a current-carrying wire
A magnetic field exerts a force on a single moving charge, so
it's not surprising that it exerts a force on a current-carrying
wire, seeing as a current is a set of moving charges.
F  qvB sin 
Using q = I t, this becomes: F  IvtB sin 
But a velocity multiplied by a time is a length L, so this can be
written:
F  ILB sin 
The direction of the force is given by the right-hand rule, where
your fingers point in the direction of the current. Current is
defined to be the direction of flow of positive charges, so your
right hand always gives the correct direction.
The right-hand rule
A wire carries current into the page in a magnetic field
directed down the page. In which direction is the
force?
1. Left
2. Right
3. Up
4. Down
5. Into the page
6. Out of the page
7. The net force is zero
Three wires
Consider three wires carrying identical currents between
two points, a and b.
The wires are exposed to a uniform magnetic field.
Wire 1 goes directly from a to b. Wire 2 consists of two
straight sections, one parallel to the magnetic field and
one perpendicular to the field. Wire 3 takes a meandering
path from a to b. Which wire experiences more force?
1. Wire 1
2. Wire 2
3. Wire 3
4. equal for all three
Three wires
The force is equal for all three. What matters is the
displacement perpendicular to the field, and that's equal for
all wires carrying equal currents between the same two points
in a uniform magnetic field.
The force on a current-carrying loop
A wire loop carries a clockwise current in a uniform
magnetic field directed into the page. In what
direction is the net force on the loop?
1. Left
2. Right
3. Up
4. Down
5. Into the page
6. Out of the page
7. The net force is zero
The force on a current-carrying loop
The net force is always zero on a current-carrying loop in a
UNIFORM magnetic field.
Is there a net anything on the loop?
Let’s change the direction of the uniform magnetic field. Is the
net force on the loop still zero? Is there a net anything on the
loop?
Is there a net anything on the loop?
Let’s change the direction of the uniform magnetic field. Is the
net force on the loop still zero? Is there a net anything on the
loop?
The net force is still zero, but there is a net torque that tends
to make the loop spin.
The torque on a current loop
The magnetic field is in the plane of the loop and parallel to
two sides. If the loop has a width a, a height b, and a current
I, then the force on each of the left and right sides is F = IbB.
The other sides experience no force because the field is
parallel to the current in those sides. Simulation
The torque (   rF sin  ) about an axis running through
the center of the loop is:
a
a
  F F
2
2
 aF
 IabB
The torque on a current loop
  IabB
ab is the area of the loop, so the torque here is   IAB .
This is the maximum possible torque, when the field is in the
plane of the loop. When the field is perpendicular to the loop
the torque is zero. In general, the torque is given by:
  IAB sin 
where  is the angle
between the area
vector, A, (which is
perpendicular to the
plane of the loop) and
the magnetic field, B.
A DC motor
A direct current (DC) motor is one application of the torque
exerted on a current loop by a magnetic field. The motor
converts electrical energy into mechanical energy.
If the current always went the same way around the loop, the
torque would be clockwise for half a revolution and counterclockwise during the other half. To keep the torque (and the
rotation) going the same way, a DC motor usually has a "splitring commutator" that reverses the current every half rotation.
Simulation
Producing a magnetic field
Electric fields are produced by charges.
Magnetic fields are produced by moving charges.
In practice, we generally produce magnetic fields from
currents.
The magnetic field from a long straight wire
The long straight current-carrying wire, for magnetism, is
analogous to the point charge for electric fields.
The magnetic field a distance r
from a wire with current I is:
0 I
B
2 r
 0 , the permeability of free space, is:
7
0  4 10 Tm/A
The magnetic field from a long straight wire
Magnetic field lines from a long straight current-carrying wire
are circular loops centered on the wire.
The direction is given by another
right-hand rule.
Point your right thumb in the
direction of the current
(out of the screen in the
diagram, and the fingers on
your right hand, when you curl
them, show the field direction.
The force between two wires
A long-straight wire carries current out of the page. A
second wire, to the right of the first, carries current
into the page. In which direction is the force that the
second wire feels because of the first wire?
1. Left
2. Right
3. Up
4. Down
5. Into the page
6. Out of the page
7. The net force is zero
The force between two wires
In this situation, opposites repel and likes attract!
Parallel currents going the same direction attract.
If they are in opposite directions they repel.
The net magnetic field
In which direction is the net magnetic field at the origin in
the situation shown below? All the wires are the
same distance from the origin.
1. Left
2. Right
3. Up
4. Down
5. Into the page
6. Out of the page
7. The net field is zero
The net magnetic field
We add the individual fields to find the net field, which is
directed right.
A loop and a wire
A loop with a clockwise current is placed below a long
straight wire carrying a current to the right. In which
direction is the net force exerted by the wire on the
loop?
1. Left
2. Right
3. Up
4. Down
5. Into the page
6. Out of the page
7. The net force is zero
A loop and a wire
The long straight wire creates a non-uniform magnetic field,
pictured below.
A loop and a wire
The forces on the left and right sides cancel, but the forces on
the top and bottom only partly cancel – the net force is
directed up, toward the long straight wire.
A loop and a wire
The forces on the left and right sides cancel, but the forces on
the top and bottom only partly cancel – the net force is
directed up, toward the long straight wire.
I1
a
I2
b
Fnet
L
0 I1I 2 L  1
1 
 I 2 L( Btop  Bbottom ) 
 

2  a a  b 
Five wires
Four long parallel wires carrying equal currents perpendicular
to your page pass through the corners of a square drawn on
the page, with one wire passing through each corner. You get
to decide whether the current in each wire is directed into the
page or out of the page.
First we’ll have a fifth parallel wire, carrying current into the
page, that passes through the center of the square. Can you
choose current directions for the other four wires so that the
fifth wire experiences a net force directed toward the top right
corner of the square?
How many ways?
You can choose the direction of the currents at each
corner. How many configurations give a net force on
the center wire that is directed toward the top-right
corner?
1.
2.
3.
4.
5.
1
2
3
4
0 or more than 4
How many ways?
First, think about the four forces we need to add to get a net
force toward the top right. How many ways can we create this
set of four forces?
Note: if the length of each
side is d, and the currents
are all I, the net force per
unit length here is:
Fnet 2 0 I  I

L
2 d / 2
2 0 I

d
2
How many ways?
How many ways can we create this set of four forces?
Two. Wires 1 and 3 have to
have the currents shown.
Wires 2 and 4 have to
match, so they either both
attract or both repel.
Currents going the same
way attract; opposite
currents repel.
Four wires
Now we’ll remove the fifth wire and focus on the net magnetic
field at the center of the square because of the other four
wires. Can you choose current directions for the four wires so
that the net magnetic field at the center is directed toward the
top right corner of the square?
How many ways?
You can choose the direction of the currents at each
corner. How many configurations give a net magnetic
field at the center that is directed toward the top-right
corner?
1.
2.
3.
4.
5.
1
2
3
4
0 or more than 4
How many ways?
First, think about the four fields we need to add to get a net
field toward the top right. How many ways can we create this
set of four fields?
Note: if the length of each
side is d, and the currents
are all I, the net field is:
Bnet
2 0 I

2 d / 2
2 0 I

d
How many ways?
How many ways can we create this set of four fields?
Two. Wires 2 and 4 have to
have the currents shown.
Wires 1 and 3 have to
match, so their fields cancel.
The right-hand rule:
Point your thumb in the
direction of the current,
and your curled fingers
show the direction of the field.
The field from a solenoid
A solenoid is simply a coil of wire with a current going through
it. It's basically a bunch of loops stacked up. Inside the coil,
the field is very uniform (not to mention essentially identical to
the field from a bar magnet).
For a solenoid of length L, current I, and total number of turns
N, the magnetic field inside the solenoid is given by:
B
0 NI
L
The field from a solenoid
B
0 NI
L
We can make this simpler by using n = N/L as the number of
turns per unit length, to get: B  0 nI .
The magnetic field is almost uniform - the solenoid is the
magnetic equivalent of the parallel-plate capacitor. If we put a
piece of ferromagnetic material (like iron or steel) inside the
solenoid, we can magnify the magnetic field by a large factor
(like 1000 or so).
A bar magnet and a solenoid
A bar magnet field looks like the field of a solenoid. Why?
A bar magnet and a solenoid
A bar magnet field looks like the field of a solenoid. Why?
The currents associated with the atoms mostly cancel inside the
bar magnet, but they add together around the outside, giving
something that looks remarkably like a solenoid.
Magnetic Resonance Imaging
Gradient
field coils
Homogeneous
field coils
RF
transmitter
coils
Body-part-specific
RF pickup coils
not shown.
System with 3 tesla
superconducting magnet
Diagram and photo from Wikipedia
Magnetism on the atomic level
Currents in wires produce magnetic fields. What produces the
magnetic field from a bar magnet, where there are no wires?
Why does that field look like the field of a solenoid?
Consider the Bohr model of the atom, where electrons travel
in circular orbits around the nucleus. An electron in a circular
orbit looks like a current loop, so it produces a magnetic field.
In some materials (ferromagnetic materials) the magnetic
moments associated with the atoms align, leading to a large
net magnetic field.
Magnetism on the atomic level
An electron in a circular orbit looks like a current loop, with a
current:
e
ev
I
T

2 r
A current loop has a magnetic moment μ, so for the orbiting
electron we get:
ev
evr
2
  IA 
 r 
2 r
2
If we multiply top and bottom by m, the electron mass, we get
mvr in the numerator. This is the orbital angular momentum of
the electron, L. This gives:
e(mvr ) eL


2m
2m
Magnetism on the atomic level
The orbital magnetic moment of the electron is
proportional to its orbital angular momentum, which
is quantized in multiples of  h /(2 ) , where h is
Planck's constant. A better analysis using quantum
mechanics shows that the smallest non-zero value
of the electron's orbital magnetic moment is:
2e

2m
Another contribution to an atom's magnetic moment
comes from electron spin. The magnetic moment
associated with electron spin is:
e

2m
Magnetism on the atomic level
The net magnetic moment of an atom is the
vector sum of its orbital and spin magnetic
moments. Many materials are not magnetic
(i.e., they don't act like bar magnets) because
the magnetic moments completely or mostly
cancel. In materials you can make bar
magnets out of, however, neighboring atoms
interact in such a way that their magnetic
moments are aligned. In other words, the
material acts like one big current loop,
producing a magnetic field.
Ferromagnetism
Examples of ferromagnetic materials include
iron, cobalt, nickel, and an alloy called Alnico.
The atoms in these materials have permanent
magnetic moments, and a phenomenon called
exchange coupling takes place in which the
magnetic moments of nearby atoms line up
with one another. This forms domains, small
neighborhoods where the magnetic moments
are aligned. Typical dimensions of domains
are 0.1 to 1 mm.
Ferromagnetism
When a ferromagnetic material is not
magnetized, the domains have random
magnetization directions. If an external field is
turned on, two things happen.
1. Domains aligned with the field grow at the
expense of domains aligned against the field.
2. The magnetization direction within each
domain tends to shift towards the direction of
the applied field.
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