III

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III3 Applications of Magnetism
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Main Topics

• Magnetic Dipoles
• The Fields they Produce
• Their Behavior in External Magnetic Fields
• Calculation of Some Magnetic Fields
• Solenoid
• Toroid
• Thick Wire with Current

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Magnetic Dipoles I

• In electrostatics we defined electric dipoles. We
  can imagine them as solid rods which hold one
  positive and one negative charge of the same
  absolute values some distance apart. Although
  their total charge is zero they are sources of
  fields with special symmetry which decrease
  faster than fields of point sources. External
  electric field is generally trying to orient and
  shift them.

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Magnetic Dipoles II

• Their analogues in magnetism are either thin flat
  permanent magnets or loops of current. These also
  are sources of fields with a special symmetry
  which decrease faster than fields from straight
  currents and by external magnetic field they are
  affected similarly as electric dipoles. Later we
  shall describe magnetic behavior of matter using
  the properties of magnetic dipoles.

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Magnetic Dipoles III

• Let us have a circular conductive loop of the
  radius a and a current I flowing in it. Let us
  describe the magnetic field at some distance b on
  the axis of the loop.
• We can cut the loop into little pieces
• dl ad? and vector add their contribution to
  the magnetic induction using the Biot-Savart law.

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Magnetic Dipoles IV

• For symmetry reasons the direction of B is the
  same as the direction of the axis z -
  perpendicular to the loop and integration in this
  case means only to add the projections dBz dB
  sin? . And from the geometry
• sin? a/r ? 1/r2 sin2? /a2
• r2 a2 b2
• Let us perform the integration.

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Magnetic Dipoles V

• Since magnetic dipoles are sources of magnetic
  fields they are also affected by them.
• In uniform magnetic field they will experience a
  torque.directing them in the direction of the
  field.
• We shall illustrate it using a special case of
  rectangular loop a x b carrying current I.

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Magnetic Dipoles VI

• Form the drawing we see that forces on the sides
  a are trying to stretch the loop but if it is
  stiff enough they cancel.
• Forces on the sides b are horizontal and the
  upper acts into the blackboard and the lower from
  the blackboard. Clearly they are trying to
  stretch but also rotate the loop.

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Magnetic Dipoles VII

• To find the contribution of each of the b sides
  to the torque we have to find the projection of
  the force Fb perpendicularly to the loop T/2
  Fbsin? a/2
• Since both forces act in the same sense
• T BIabsin?
• We can generalize this using the magnetic dipole
  moment m Iabm0
• T m x B

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Magnetic Field of a Solenoid I

• Solenoid is a long coil of wire consisting of
  many loops.
• In the case of finite solenoid the magnetic field
  must be calculated as a superposition of magnetic
  inductions generated by all loops.
• In the case of almost infinite we can use the
  Amperes law in a very elegant way.

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Magnetic Field of a Solenoid II

• As a closed path we choose a rectangle whose two
  sides are parallel to the axis of the solenoid.
• From symmetry we can expect that the field lines
  will be also parallel to the axis direction.
• Since the closed field lines return through the
  whole Universe outside the solenoid we can expect
  they are infinitely diluted.

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Magnetic Field of a Solenoid III

• Only the part of the path along the side inside
  the solenoid will make non-zero contribution to
  the loop integral.
• If the rectangle encircles N loops with current I
  and its length is l then
• Bl ?0NI
• And if we introduce the density of loops
• n N/l ? B ?0nI

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Magnetic Field of a Solenoid IV

• For symmetry reason we didnt make any
  assumptions about how deep is our rectangle
  immersed in the solenoid. We didnt have to since
  the magnetic field in the long solenoid is
  expected to be uniform or homogeneous.
• A reasonably uniform magnetic field can be
  obtained if we shorten thick solenoid and cut it
  into halves - Helmholtz coils.

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Magnetic Field of a Toroid I

• We can think of the toroid as of a solenoid bent
  into a circle. Since the field lines cant escape
  we do not have to make any assumptions about the
  size.
• If the toroid has a radius R to its central field
  line and N loops of current I, we can simply show
  that all the field is inside and what is the
  magnitude on a particular field line.

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Magnetic Field of a Toroid II

• Lets us choose the central filed line as our
  path then the integration simplifies and
• B(2?r) ?0NI ? B(r) ?0NI/2?r
• His is also valid for any r within the toroid.
• The field
• is not uniform since it depends on r.
• is zero outside the loops of the toroid

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Magnetic Field of a Thick Wire I

• Lets have a straight wire of a diameter R in
  which current I flows and let us suppose that the
  current density is constant.
• We use Amperes law. We use circular paths one
  outside and one inside the wire.
• Outside the field is the same as if the wire was
  infinitely thin.
• Inside we get linear dependence on r.

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Magnetic Field of a Thick Wire II

• If we take a circular path of the radius r inside
  the wire we get
• B(2?r) ?0Ienc
• The encircled current Ienc depends on the area
  surrounded by the path
• Ienc I ?r2/?R2 ?
• B ?0Ir/2?R2

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Homework

• No homework today!

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Things to read

• Chapter 27 5 and 28 4, 5

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Circular Loop of Current I
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Circular Loop of Current II
A ?a2 is the area of the loop and its normal
has the z direction. We can define a magnetic
dipole moment m IA and suppose that we are far
away so bgtgta. Then
Magnetic dipole is a source of a special magnetic
field which decreases with the third power of the
distance.

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Circular Loop of Current II
A is the area of the loop and its normal has the
z direction. We can define a magnetic dipole
moment ? IA and suppose that we are far away so
bgtgta then we can write
and the formula to calculate the field, which is
the Biot-Savart law
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Circular Loop of Current I
That is the reaso
an
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Circular Loop of Current II
That is the reaso
and the formula to calculate the field, which is
the Biot-Savart law
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The vector or cross product I

• Let ca.b
• Definition (components)

The magnitude c
Is the surface of a parallelepiped made by a,b.
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The vector or cross product II
The vector c is perpendicular to the plane made
by the vectors a and b and they have to form a
right-turning system.
?ijk 1 (even permutation), -1 (odd), 0 (eq.)