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Magnetism
A Brief Historical Interlude
Greek scholars and scientists discovered certain types of rocks, called lodestones, which
would attract small pieces of iron; this demonstrated the existence of a magnetic force.
These lodestones were discovered in a province of Greece known as Magnesia. It is
from the name of this region that we get the term magnet.
At first, magnets served as an interesting conversation piece and a cool “magic” trick.
However, people soon started to see some practical uses for these strange pieces of
rock. It was discovered by the Chinese that small fragments of these lodestones would
tend to rotate and orient themselves in a particular direction: North(ish). By harnessing
these tiny fragments, Chinese sailors invented the first navigational compasses in the
12th century.
The question, though, was why? Why do these small pieces of rock (and eventually
metal) turn and point to the North? It was suggested in the 16th century by the British
physician William Gilbert that a compass will always point North because the Earth itself
has some inherent magnetic properties. {This should make sense; lodestones are rocks
found in the Earth. Isn’t it likely that, if pieces of the Earth could be magnetic, then the
Earth itself could be magnetic as well?}
Electromagnetism
During the early stages of experimentation with magnetism, it was seen as a wholly new
phenomenon that was unrelated to other discoveries. In fact, experiments were being
conducted in magnetism and electricity concurrently, and the two fields developed
independently until 1820, when…
One of the great discoveries in the field of magnetism was made by the Danish
professor Hans Christian Oersted as he was preparing a lecture for his students. He was
attempting to show that electricity and magnetism were unrelated by showing that a
compass would be unaffected by an electric current. However, much to his surprise,
when a current was directed near a compass, the compass needle was seen to move!
Oersted had discovered, completely contrary to his initial intentions and to his total
surprise, that an electric current could indeed affect a compass.
Evidence began to show up from other scientists confirming Oersted’s discovery.
Eventually, the French physicist Andre-Marie Ampere (after whom the unit for electric
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current is named) proposed that all magnetic phenomena are the result of electric
currents. {It is imperative to recall at this time that an electric current is simply the flow
of charged particles from one location to another}
Forces and Fields
When talking about Electrostatics, we said that there exists a force between any two
charged objects. Furthermore, we defined this force as the interaction of two electric
fields. (Recall that an Electric Field is created due to the presence of a net charge.)
The Magnetic Force (FB) is a result of the interaction of
two magnetic fields (B).
In electrostatics, we used a very simple rule for determining the direction of the
Electrostatic Force: Opposites Attract (Paula Abdul, anyone?). With magnetism,
there is a similar rule, although there is a key difference. In magnetism, we refer to
the “magnetic poles” as being North and South; much like electric charges, the opposite
poles of a magnet will attract, while the like poles will repel. {Note: the use of the
term ‘pole’ is almost certainly a result of the geographic term.}
In electrostatics, it is perfectly acceptable and very common to find a solitary positive
charge or negative charge. In magnetism, it is IMPOSSIBLE to locate a solitary North or
South pole. In other words, there is no such thing as a magnetic monopole. So every
magnet, regardless of its size or strength, must have a North and a South pole; in fact,
many magnets have multiple North and South poles. {Recent studies and research have
hinted at the possible existence of magnetic monopoles; however, none have currently
been isolated or identified.}
In electrostatics, the direction of the electric field is defined such that it points away
from a positive charge and towards a negative charge. Similarly, the direction of
Magnetic Field lines always point in such a manner that they go away from the North
Pole and towards the South Pole.
Magnetic fields are produced by moving charges. This could be either a single charge q
moving with a speed v, or an electric current. When either of these two scenarios takes
place, a magnetic force can be observed.
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The S.I. unit for measuring the strength of the magnetic field is called the Tesla
(T). It is named after one of the most interesting personalities in all of Physics,
Nikola Tesla
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The Earth’s magnetic field strength is approximately BEarth
= 5 E-5 T.
A Moving Charge in a Magnetic Field
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q moves with a velocity v through a magnetic field B, it will
experience a magnetic force FB. The magnitude of this force is given by
When a charge
where θ is the angle between the direction of v and B. This angle is often (but
not always) 90 degrees. We will find that the maximum amount of force will be
found when the angle = 90 (or 270) degrees; conversely, we find that there is
zero force when the angle is 0 (or 180) degrees.
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The direction of this force is determined in a very peculiar manner known as
the “Right Hand Rule.” Assuming the charge is positive, you point the fingers of
your right hand in the direction of the charge’s velocity (v). While doing this, you
face your palm in the direction of the magnetic field (B) [this can be envisioned
as the direction in which your fingers will bend]. When you stick out your thumb
(your hand should now look like you are trying to shake hands with someone) it
points in the direction of the magnetic force (FB). To recap:
Fingers
Palm/Bend
Fingers
Thumb
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v
B
FB
Velocity
Magnetic
Field
Magnetic
Force
What this indicates is that the plane the magnetic force resides in is
perpendicular to the plane that contains the velocity and magnetic field vectors.
In other words, we are dealing with a three-dimensional situation. {Note: If the
moving charge happens to be a negative charge, we use the same rule but use
our left hands.}
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An Electric Current in a Magnetic Field
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When a current-carrying wire of length L carries a current I through a magnetic
field B, the wire will feel a magnetic force FB. The magnitude of this force is
given by
where θ is the angle between the direction of I and B. This angle is often (but
not always) 90 degrees. We will find that the maximum amount of force will be
found when the angle = 90 (or 270) degrees; conversely, we find that there is
zero force when the angle is 0 (or 180) degrees.
○
The direction of this force is determined by the “Right Hand Rule.” You point
the fingers of your right hand in the direction of the current (I). While doing
this, you face your palm in the direction of the magnetic field (B) [this can be
envisioned as the direction in which your fingers will bend]. When you stick out
your thumb (your hand should now look like you are trying to shake hands with
someone) it points in the direction of the magnetic force (FB). To recap:
Fingers
Palm/Bend
Fingers
Thumb
I
B
FB
Current
Magnetic
Field
Magnetic
Force
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In order to express vectors in three dimensions (because we have a threedimensional situation) we need a way to show a vector that indicates directions
of depth (i.e., away and towards). To do this, we naturally turn to… archery.
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Picture an arrow. It has a pointed end, and it has a fletched (or feathered) end.
If someone shot an arrow at you, as you watched it approach, you would see
a dot coming towards you… and then it would be embedded in your forehead.
However, you would see a dot. If you shot an arrow away from you, you would
see the criss-cross of the fletching, and it would appear like an ‘x’. As a result,
we use the following system to represent these directions:
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So now, we must ask ourselves: where do these magnetic fields come from? Well, it can
be shown that, due to a process known as Induction, all magnetic fields are the result of
electric currents.
Magnetic Field produced by an Electric Current
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When an electric current I flows through a wire, the flowing charges will produce
a magnetic field B that circles around the wire (like water circling a drain) and is
weaker the farther R away from the wire you get. This magnetic field is given by
where μ0 is a constant known as the Permeability of Free Space; it has a value
of μ0 = 4π E -7 T*m/A.
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The S.I. unit of the magnetic field is the Tesla (T). Recall that the magnetic field
strength of the Earth is just 5 E-5 T.
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To determine the direction of this magnetic field, we have a 2nd “Right Hand
Rule”: point your right thumb along the wire in the direction that the current is
flowing. Then wrap your fingers around the wire as if you were trying to grasp
it; the magnetic field will circle the entire wire in the direction that your fingers
want to curl. {See diagrams below.}
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Note: the following two diagrams are describing the exact same magnetic fields
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Magnetic Field inside a Solenoid
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A Solenoid is formed when a long straight wire is looped around a cylindrical
center (think of a Slinky™). As current flows through the wire, a magnetic field
is established that circles the wire according to what we have previously seen.
However, due to the loop nature of the solenoid, what we will find is that, for
each successive loop, the magnetic field will pass directly through the center of
the solenoid. This essentially creates a linear magnetic field.
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Assume that we have a solenoid with a length L that is formed by a number N of
individual loops of wire. Assuming that L is much greater than the radius of the
loop of the solenoid, when the solenoid carries a current I we can determine the
strength of the magnetic field inside the solenoid as
where μ0 = 4π E -7 T*m/A.
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The direction of this magnetic field is also found via the Right Hand Rule.
Magnetic Force between two parallel current-carrying wires
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In the diagram above, Wire 1 is parallel to Wire 2. Wire 1 carries a current I1 and
Wire 2 carries a current I2. The wires each have a length L and are a distance d
apart.
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As shown earlier in the notes, each current will generate a magnetic field at
the location of the other wire; i.e., there is a magnetic field at wire 2 due to the
current through wire 1, and vice versa.
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When a current flows through a magnetic field, as we have seen, there will be a
magnetic force. So, each wire feels a magnetic force due to the presence of the
other. The magnitude of the magnetic force that each wire feels is
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The direction of this force is easy to remember: when the currents flow in
the same direction, the force is attractive; when the currents flow in opposite
directions, the force is repulsive.
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Often, the length of wire will be divided out of the previous equation, giving an
expression for the Force per unit Length (second equation avove) acting on a
wire; this is in many cases a more useful expression.
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**It is from this expression that we get our modern definition of the unit of
current; the Ampere. The definition states that 1 A of current is defined as
the amount of current flowing through two long parallel wires 1 m apart that
results in a force of exactly 2 E-7 N per meter of length of the wire.
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