Magnetism
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Magnetic fields can be caused in three different ways
1. A moving electrical charge such as a wire with current flowing
in it
2. By electrons or protons which act like little bar magnets
3. By some magnetic atoms such as iron (ferromagnetism)
All of the appearances of magnetism can be attributed to the
movement of electrical charges or the magnetic moment from
elementary particles as a result of their spin.
In order to describe the phenomena of magnetism, we need the
concept of a magnetic field. Magnetic fields can be caused by
1. magnetic material such as a permanent magnet
2. electric current such as current flowing through an inductor
3. the change in time of an electric field.
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Lorentz Force
The magnetic field has an effect on the force of moving
charges. This is called the Lorentz force. The Lorentz force is
the force on a point charge due to electromagnetic fields. It
takes effect perpendicular to the field lines of the magnetic field
and perpendicular to the direction of the movement of the
charge.
where
F is the force (in newtons)
E is the electric field (in volts per metre)
B is the magnetic field (in teslas)
q is the electric charge of the particle (in coulombs)
v is the instantaneous velocity of the particle (in metres per
second)
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Biot-Savart Law
Consider a current I flowing through a wire. Break the wire into
little pieces of length ds. The magnetic field due to this little
piece of current is then found experimentally to be
or
This is the law of Biot-Savart. r is the distance from the current
element I ds to the field point P where we wish to find the
magnetic field B. is a unit vector pointing along r.
is a
constant of nature, the permeability of free space.
The relationship between ds, er, and dB is shown in the figure.
To determine the direction of
use your right hand and
point your fingers along ds and curl them toward er. Your right
thumb will point along dB.
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To find the total magnetic field due to a conductor, we add up the
contributions from each current element by integrating over the
entire conductor. Thus
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Problem
Determine the magnetic field at the center of a circular loop of
radius R carrying current I.
where ds = R µ and
The magnet field of a small current loop is just like that of a small
bar magnet with B lines emerging from an imaginary north pole and
looping back to end in an imaginary south pole. Thus the field of a
small current carrying loop is that of a magnetic dipole. See figure
on the next slide.
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Ampere‘s Law
A magnet field proportional to the current is generated and
follows the right hand rule. The current is represented by
the curl of the fingers and the thumb points in the direction
of the north pole. The integral form is written
Ampere‘s Law is only useful for determining the magnet
field for simple cases such as when one assumes that the
magnet field of an inductive coil is over all homogenous
which is only the case for an infinitely long coil.
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:
If we consider an ideal coil with N windings per meter, we can
insert a rectangular frame through the coil. The top side of the
coil lies on the same axis as the length of the coil and the right
and left sides are infinitely long. The magnetic field is
assumed to lie perpendicular to the sides, therefore the
components in the direction of the frame are zero. The other
side is infinitely far away so the magnet field must be zero
there.
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Consider the line integral shown in the figure. The path
integral of the magnetic field around this integration path
is equal to
where L is the horizontal length of the integration path.
The current enclosed by the integration path is equal to
where N is the number of turns enclosed by the
integration path and I0 is the current in each turn of the
solenoid. Using Ampere's law we conclude that
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or
where n is the number of turns of the solenoid per unit length.
The equation shows that the magnetic field B is independent
of the position inside the solenoid. We conclude that the
magnetic field inside an ideal solenoid is uniform.
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Problem
The figure below shows two views of a circular loop of radius
4 cm placed within a uniform magnetic field B of magnitude
0.3 T.
a.) What is the magnetic flux through the loop?
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b.) What would be the magnetic flux through the loop if the
loop were rotated
?
c.) What would be the magnetic flux through the loop if the
loop were rotated
?
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a.) Since B is parallel to A, the magnetic flux is equal to BA:
b.) Since the angle between B and A is
through the loop is
c.) If the angle between B and A is
through the loop is zero since
, the magnetic flux
, the magnetic flux
.
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Dynamo on your Bicycle
A dynamo works to light up your bike using induction.
When you pedal your bike, you rotate a magnet in an iron
core and produce a current in an inductor coil.
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