Title: III
1III3 Applications of Magnetism
2Main Topics
- Magnetic Dipoles
- The Fields they Produce
- Their Behavior in External Magnetic Fields
- Calculation of Some Magnetic Fields
- Solenoid
- Toroid
- Thick Wire with Current
3Magnetic 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.
4Magnetic 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.
5Magnetic 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.
6Magnetic 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.
7Magnetic 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.
8Magnetic 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.
9Magnetic 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
10Magnetic 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.
11Magnetic 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.
12Magnetic 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
13Magnetic 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.
14Magnetic 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.
15Magnetic 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
16Magnetic 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.
17Magnetic 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
18Homework
19Things to read
20Circular Loop of Current I
21Circular 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.
22Circular 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
23Circular Loop of Current I
That is the reaso
an
24Circular Loop of Current II
That is the reaso
and the formula to calculate the field, which is
the Biot-Savart law
25The vector or cross product I
- Let ca.b
- Definition (components)
The magnitude c
Is the surface of a parallelepiped made by a,b.
26The 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.)