Lesson 15 - Magnetic Fields II
I.
Permanent Magnets
A.
Over 2000 years ago in Turkey, people discovered that rocks of iron ore would
apply a force on each other. These rocks were called magnets due to their
discovery near the town of magnesia.
B.
Several interesting facts concerning these rocks were discovered:
1.
If a rod of iron touched these rocks, then it would become a magnet and
begin interacting with these rocks. The rod was said to be have been
magnetized.
2.
If the magnetized rod was placed on a leaf and floated on water, the rod
would always align itself so that it pointed approximately in the North/South
direction on the Earth.
We now know that this is because the Earth is a big magnet due to currents
inside the Earth. Thus, people were using compasses for navigation long
before we had any fundamental understanding of the cause of magnetic
fields. For instance, "Columbus sailed the ocean the ocean blue in 1492," but
the theory of magnetism dates from the 1800's.
3.
After marking the side of the rod that pointed in the northern direction as
north and the side that pointed in the southern direction as south, people
discovered that
a) the north side (pole) of the magnet repelled the north pole and attracted
the south pole of another magnet.
b) the south pole of a magnet repelled the south pole of another magnet.
S
N
N
S
N
S
N
S
N
S
S
N
Question: From the discussion above, what type of magnetic pole is near the Earth's
North Pole?
4.
Cutting a magnet in half will produce two complete magnets. This discovery
is probably far more intriguing today than it was long ago since it implies
that the cause of magnetism is not a point source (like a point charge) but an
extended source (a current loop). The search for a point source of magnetism
(magnetic monopole) is an active field of physics research. We will discuss
this further when we get to Gauss' Law for Magnets.
Summary: A considerable amount of empirical information about magnetism was
discovered using permanent magnets. This allowed for the construction of some useful
devices using permanent magnets long before any theory of magnetism existed.
However, the connection between the phenomena of electricity (discovered in Greece)
and magnetism (discovered in Turkey) prevented people from developing motors,
generators, stronger magnets (electromagnets), and electrical power. It was the
development of the battery in the late 1700's that would begin the age of electricity.
Thus, many of the conventions concerning magnets (using N/S instead of +/-) are
historical in nature.
II.
Magnetic Field and Magnetic Field Lines
A.
The concept of the magnetic field can be developed in a manner similar to the
way we developed the electric field. The magnitude of the magnetic field at a
particular point in space is found by using a moving charged particle and
determining the maximum force exerted on the particle for a given speed.
B F
qv
MAX
The magnetic field vector lies along the direction in which the moving charged
particle experiences no additional force due to its motion (no force if we remove
electric fields, gravity, etc.) with the direction of the vector given by the right
hand rule.
B.
We can build a graphical picture of the magnetic field (magnetic field lines) in a
manner similar to our work with electric fields. However, there are some
important differences due to the nature of the vector cross product that you need
to understand.
Similarities To Electric Field Diagrams:
1.
Magnetic field lines leave the north pole (like electric field lines
from a positive charge) and terminate on the south pole of a magnet (like
electric field lines on a negative charge). Thus, the direction of the magnetic
field at any point is the direction pointed to by a compass (assuming that the
Earth's magnetic field is negligible).
2.
Magnetic field lines can never cross (the magnetic field has a single value at
every point is space),
3.
The density of magnetic field lines is proportional to the strength of the
magnetic field.
The easiest way to map the magnetic field is to use iron fillings. These act like
little magnets and align with the field. A compass can then be used to determine
the direction of the arrow. Also, the strength of the magnetic field is obtained
since more iron filings will be attracted to regions of higher magnetic field.
Differences:
1.
A graph of the magnetic field lines doesn't completely specify the force on a
charged particle. This is because the force also depends on the velocity of the
particle. Stationary particles experience no magnetic force at all!
2.
A magnetic field line DOES NOT point in the direction of the force applied
on a moving charged particle. If fact, a charged particle moving the direction
of the magnetic field line experiences NO magnetic force. The non-zero
force experienced by any moving charge will always be perpindicular to the
magnetic field lines.
3.
We find that the graphs of the magnet field lines always show circulation
(rotation) and never flow! More evidence that their is no point source of
magnetism.
EXAMPLE: The following can not be a graph of the magnetic field because it has
flow and no circulation.
B
EXAMPLE: The magnetic field diagram for a magnetic dipole (the simplest magnetic
source known).
N
S
I
Current Loop
III.
Gauss' Law of Magnetism
A.
Magnetic Flux - B
Permanent Magnet
The flux of the magnetic field is defined in the same manner as the electric flux.
The units of magnetic flux are _____________________ which are defined as
B.
Experimentally, it appears that there is no point source for magnetism (ie you can
break a magnet apart and get only the north or south pole). This fact is
mathematically expressed by Gauss' Law of Magnetism:
Gauss' Law of Magnetism is one of the four fundamental equations of
electromagnetic theory known as Maxwell's Equations. These are the PHYS2424
equivalents of Newton's Laws. It says that "Magnetic Fields "Never Flow."
IV.
A.
Magnetic Force On A Current Carrying Wire
Theory
B - field

dS
I
wire
dq
The magnetic force on a differential amount of moving charge, dq, in the wire is

dF 

 ds
Using the definition of velocity, v  , we have
dt

dF 
By Calculus, we have
   
dq d s    dq d s
dt   dt 





Substituting this result and using the definition of current, we have

 
dF I d s B
The total force on the wire is then found by summing up the force on each element of the
wire.
 B 
F   dF 
A
Since the elements of the wire are in series, they have the same current (ie I constant with

respect to integration variable dS .
 B  
F  I  d s B
A
where I is the current in the wire

B is the magnetic field vector

ds is the displacement vector point in the direction of I.
B.
Problem Solving Strategy
Step 1:


Break up the wire into segments in which the angle between ds and B and
the direction of the resulting cross product DOES NOT CHANGE.
Step 2:


You can disregard all wire segments in which I {ie. ds } is parallel to B as
Step 3:

For each remaining wire segment, find F by

 
d s B 0  F  0
A.
 
Finding d s  B
B. Find the force on this wire segment using

 
F  I  d s B
segment
Step 4:
Sum segment forces using vector addition!

F
T OT AL

 F
i
i
EXAMPLE: What is the force exerted by a 5.00 T magnetic field in the +y-direction
upon the 25.0 meter long wire shown below when 2.00 A of current is flowing in the
wire.

B
6m
y
6m
x
12 m
4m
2A
3m
Solution:
EXAMPLE 2: The unusual wire shape shown below is placed in a uniform magnetic
field pointing out of the page. Derive the relationship between the current, I, the wire

parameter, R, the magnetic field strength B, and the force, F , upon the wire.

B
y
R
R
x
45
R
Solution:
R
I