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III

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If the toroid has a radius R to its central field line and N ... a Toroid II ... also valid for any r within the toroid. The field: is not uniform since it ... – PowerPoint PPT presentation

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Title: III


1
III3 Applications of Magnetism
2
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

3
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.

4
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.

5
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.

6
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.

7
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.

8
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.

9
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

10
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.

11
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.

12
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

13
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.

14
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.

15
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

16
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.

17
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

18
Homework
  • No homework today!

19
Things to read
  • Chapter 27 5 and 28 4, 5

20
Circular Loop of Current I
21
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.

22
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
23
Circular Loop of Current I
That is the reaso
an
24
Circular Loop of Current II
That is the reaso
and the formula to calculate the field, which is
the Biot-Savart law
25
The vector or cross product I
  • Let ca.b
  • Definition (components)

The magnitude c
Is the surface of a parallelepiped made by a,b.
26
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.)
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