The force on a current-carrying wire

1
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.
• Using q I t, this becomes
• But a velocity multiplied by a time is a length
  L, so this can be written
• 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.

2
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

3
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
4
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.

5
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

6
The force on a current-carrying loop

• The net force is always zero on a
  current-carrying loop in a UNIFORM magnetic
  field.

7
Is there a net anything on the loop?

• Lets 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?

8
Is there a net anything on the loop?

• Lets 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.

9
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 ( ) about an
  axis running through the center of the loop is

10
The torque on a current loop

• ab is the area of the loop, so the torque here is
  .
• 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

where q is the angle between the area vector, A,
(which is perpendicular to the plane of the loop)
and the magnetic field, B.
11
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 counter-clockwise during the
  other half. To keep the torque (and the rotation)
  going the same way, a DC motor usually has a
  "split-ring commutator" that reverses the current
  every half rotation. Simulation

12
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.

13
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
• , the permeability of free space, is

14
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.

15
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

16
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.

17
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

18
The net magnetic field

• We add the individual fields to find the net
  field, which is directed right.

19
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

20
A loop and a wire

• The long straight wire creates a non-uniform
  magnetic field, pictured below.

21
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.

22
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
L
23
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 well 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?

24
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. 1
• 2. 2
• 3. 3
• 4. 4
• 5. 0 or more than 4

25
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
26
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.

27
Four wires

• Now well 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?

28
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. 1
• 2. 2
• 3. 3
• 4. 4
• 5. 0 or more than 4

29
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
30
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.

31
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

32
The field from a solenoid

• We can make this simpler by using n N/L as the
  number of turns per unit length, to get
  .
• 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).

33
A bar magnet and a solenoid

• A bar magnet field looks like the field of a
  solenoid. Why?

34
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.

35
Magnetic Resonance Imaging
Gradient field coils
Homogeneous field coils
RF trans-mitter coils
System with 3 tesla superconducting magnet
Body-part-specific RF pickup coils not shown.
Diagram and photo from Wikipedia
36
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.

37
Magnetism on the atomic level

• An electron in a circular orbit looks like a
  current loop, with a current
• A current loop has a magnetic moment µ, so for
  the orbiting electron we get
• 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

38
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
  , 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
• Another contribution to an atom's magnetic moment
  comes from electron spin. The magnetic moment
  associated with electron spin is

39
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.

40
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.

41
Ferromagnetism

• When a ferromagnetic material is not magnetized,
  the domains have random magnetization directions.
  If an external field is turned on, two things
  happen.
• Domains aligned with the field grow at the
  expense of domains aligned against the field.
• The magnetization direction within each domain
  tends to shift towards the direction of the
  applied field.

42
Ferromagnetism

• When a ferromagnetic material is not magnetized,
  the domains have random magnetization directions.
  If an external field is turned on, two things
  happen.
• Domains aligned with the field grow at the
  expense of domains aligned against the field.
• The magnetization direction within each domain
  tends to shift towards the direction of the
  applied field.

43
Whiteboard