The Compass
It was discovered that a sliver of a special kind of
rock would tend to point North (when set on a leaf in
a pool of water or balanced on a pin). This discovery
was worked into creating a compass.
The end of the sliver that pointed North was called
the North pointing pole, and the opposite end was
called the South pointing pole.
It was further discovered that a North pointing pole
repelled another North pointing pole and attracted
another South pointing pole. A South pointing
pole did repel another South pointing pole.
The middle name of “North pointing pole” is largely
not used, and we simply call this the “North pole”.
Magnetism
We are probably all familiar with the facts that
magnets (what we call that material used in the compass)
repel and attract each other. This is similar to
electric charges. This suggests that we can
propose the following law (like Newton’s Law of
Gravity and Coulomb’s Law for Electricity):
Fmagnetic = X p1 p2 / r122
where X would be the magnetic constant, analogous
to G and k, that describes the strength of the
magnetic force.
Magnetic Poles
The “p” in the previous equation stands for
poles. In magnetism, we have two kinds of
poles, and we call them North and South.
Like poles repel and unlike poles attract,
just as electric charges do.
However, unlike charges, we always have
two poles! If we break a magnet (which
has two poles) in half, we have not
separated the two poles, rather we have two
(smaller) magnets that both have two poles!
Magnetic Poles
N
N
S
S
N
S
Break one bar magnet in half, and you have
two smaller bar magnets!
Magnet in the Earth?
If the magnet in the compass has its North
pole pointing North, does that mean the
Earth acts like it has a big magnet inside it?
If so, that means that a South pointing pole
(South magnetic pole) must be near the
North rotational pole and a North magnetic
pole must be near the South rotational pole.
Weird – but to be consistent, that is what we
have!
Magnetism
Since we cannot seem to isolate one magnetic
pole like we could electric charges, the
force equation that is similar to Newton’s
and Coulombs Law turns out to be not very
useful.
We do have a more useful alternative, though.
It turns out that charges experience a force
when moving near magnets!
Magnetic Force
Just like charges set up an electric field, and other
charges in the vicinity feel an electric force due to
that electric field (Fel = qE), we can work with the
idea that magnets set up a magnetic field in space,
and charges moving through that field experience
a magnetic force. There are two things to note:
1. The charges must be moving.
2. The direction of the force is “weird”:
the direction of the magnetic force is
perpendicular to the velocity of the
moving charge, and perpendicular to the
magnetic field direction!
Magnetic Force Law
magnitude:
Fmagnetic = q v B sin(qvB)
direction: right hand rule:
thumb = hand  fingers
Point your right hand in the direction of v,
curl you fingers in the direction of B, and
the force will be in the direction of your
thumb; if the charge is negative, the
force direction is opposite that of your
thumb (or use you left hand).
Units
Fmagnetic = q v B sin(qvB)
This law effectively defines the magnetic
field, B (just like Felectric=qE defined E).
The MKS units of B are:
Tesla = Nt-sec / Coul-m .
This unit turns out to be a very large one. We
have a smaller unit:
10,000 Gauss = 1 Tesla .
Example
A proton is moving at a speed of 3 x 104 m/s
towards the West through a magnetic field
of strength 500 Gauss directed South. What
is the strength and direction of the magnetic
force on the proton at this instant?
qproton = +e = 1.6 x 10-19 Coul.
v = 3 x 104 m/s, West
B = 500 Gauss * 1 Tesla/10,000 Gauss
= .05 T, South
Example, cont.
magnitude:
Fmagnetic = q v B sin(qvB)
direction: right hand rule
magnitude: F = (1.6 x 10-19 Coul) * (3 x 104 m/s)
* (.05 T) * sin(90o) = 2.4 x 10-16 Nt.
direction: thumb = hand x fingers
= West x South = UP.
Note: although the force looks small,
consider the acceleration: a = F/m =
2.4 x 10-16 Nt / 1.67 x 10-27 kg = 1.44 x 1011 m/s2.
Magnetic Forces
We’ll play with magnetic forces on moving
charges in the Magnetic Deflection
experiment in lab. At that time we’ll also
discuss and experiment with the earth’s
magnetic field.
Magnetic Force and Motion
Since the magnetic field is perpendicular to
the velocity, and if the magnetic force is the
only force acting on a moving charge, the
force will cause the charge to go in a circle:
SF = ma, Fmag = q v B, and a = w2r = v2/r
gives: q v B = mv2/r, or r = mv/qB .
Mass Spectrograph
We can design an instrument in which we can
control the magnetic field, B. If we ionize
(almost always singly) the material, we
know the charge, q. We can use known
voltages to get a known speed for the ions,
v. We can then have the beam circle in the
field and hit a target, and from that we can
measure the radius, r. Hence, we can then
calculate the mass using r = mv/qB .
Example
To see if this is really feasible, let’s try using
realistic numbers to see what the radius for
a proton should be:
q = 1.6 x 10-19 Coul;
B = .05 Teslas (500 G)
m = 1.67 x 10-27 kg;
Vacc = 500 volts gives:
(1/2)mv2 = qV, or v = [2qV/m]1/2 = 3.1 x 105 m/s
r = mv/qB = (1.67 x 10-27kg)*(3.1 x 105 m/s) / (1.6
x 10-19 Coul)*(0.05 T) = .065 m = 6.5 cm.
Mass Spectrometer
Heavier masses will give bigger radii, but we
can shrink the radii if they become too big
by using bigger magnetic fields. Note that
by measuring quantities that we can easily
measure (charge, radius, Voltage, magnetic
field), we can determine very tiny masses!
In one of our experiments in lab (Charge to
Mass of the Electron), we will essentially
determine the mass of an electron!
Other uses
A cyclotron is an instrument used to
accelerate charged particles to very high
speeds. It uses magnetic fields to bend the
charges around in circles to keep them in
one place while they’re being accelerated.
Magnetic bottles can be used to contain high
energy plasmas in fusion research.
Magnetic Forces
The Computer Homework Vol. 4 #1,
Magnetic Deflection, deals with magnetic
forces on charged particles. We’ll discuss
strategies for completing this assignment in
class.
Creating Magnetic Fields
Gravitational fields acted on masses, and
masses set up gravitational fields:
Fgravity = mg where g = GM/r2
Electric fields acted on charges, and charges
set up electric fields:
Felectric = qE where E = kQ/r2
Magnetic fields acted on moving charges; do
moving charges set up magnetic fields?
Creating Magnetic Fields
Magnetic fields acted on moving charges; do
moving charges set up magnetic fields?
YES! From the gravitational and electric
cases, we can guess that we will need:
1. a constant that describes the strength
similar to G and k;
2. an inverse square relationship with
distance (1/r2); and
3. a dependence on what is acted upon
(like m and q) - in this case, qv.
Creating Magnetic Fields
But we also have the right hand rule in the
magnetic force equation, and we’ll need a
right hand rule in the field generating
equation also.
Magnetic Fields
B = (mo/4p) q v sin(qvr) / r2
direction: right hand rule
thumb = hand x fingers
Point your hand in the direction of v, curl you
fingers in the direction of r, and the field
will be in the direction of your thumb; if
the charge is negative, the field direction
is opposite that of your thumb.
Magnetic Constant
The constant (mo/4p) is a seemingly strange
way of writing a constant that serves the
same purpose as G and k, but that is exactly
what it does.
The value: (mo/4p) = 1 x 10-7 T*m*sec/Coul
(or 1 x 10-7 T*m/Amp).
In fact, the constant k is sometimes written as:
k = 1/(4peo).
[In both cases, the sub zero on m and e indicates the
field is in vacuum.]
Creating Magnetic Fields
A single moving charge does create a
magnetic field in the space around it, but
since both the charge and the constant are
very small, we usually don’t have to worry
about these effects - except in three cases:
1. magnetic materials (permanent magnets),
2. currents (electromagnets), &
3. inside the atom (chemistry energy levels).
Creating Magnetic Fields:
1. Permanent Magnets
Materials are made up of atoms that have
electrons moving around the central nucleus
(more on this in part 4). These moving
electrons create magnetic fields. However,
in almost all materials the magnetic fields
created by the orbiting electrons tend to
cancel - except in some materials like iron.
This is the basis of making magnets out of
iron bars.
Creating Magnetic Fields
2. Electromagnets
If we have a series of charges moving (which means
a current), then we can generate an appreciable
magnetic field.
A moving charge (qv) actually can be thought
of as a current over a small length:
qv = q(DL/Dt) = I DL
so that we have for each small length:
DB = (mo/4p) I DL sin(qIr) / r2
with direction: thumb(field) =
hand(current) x fingers(radius).
Special Cases
For the current over a complete circuit, the
field produced at any particular point in
space will depend on where the point is and
the shape of the circuit.
Loop of Current
DB = (mo/4p) I DL sin(qIr) / r2
For the field at the center of a current loop we
have the special equation:
Bat center of loop = mo N I / 2R
where N is the number of turns in the wire,
I is the current in each loop, and
R is the radius of the loop.
Example
What is the field strength at the center of a loop
that has 0.2 Amps running through 3400
turns, where the radius of the loop is 6 cm?
Bat center of loop = mo N I / 2R
B = (4p x 10-7 T*m/A) * 3400 * 0.2A / (2*.06m)
= .0071 T = 71 Gauss.
(As you move away from this center, the field will
decrease.)
Review
Basic equation for calculating magnetic fields:
B = (mo/4p) I DL sin(qIr) / r2
with direction: thumb(field) =
hand(current) x fingers(radius).
(where the IDL is interchangeable with qv).
Review - continued
Basic equation for calculating magnetic fields:
B = (mo/4p) I DL sin(qIr) / r2
with direction: thumb(field) =
hand(current) x fingers(radius).
• Bat center of loop = mo N I / 2R
• Bhelmholtz coil = 8moNI/[125]1/2 R
• Blong straight wire = mo I / 2pa
UNITS: Note all equations for B should have a
mo, an I, and a (1/distance).
Forces, Fields and Currents
If currents can create magnetic fields (as we
just saw), then currents should be acted
upon by fields. Another way of saying the
same thing is: fields act on moving charges
[F = qvB sin(qvB)], and currents consist of
moving charges [qv = IDL] , so we have:
Magnetic Force on Current
Fon current = I L B sin(qIB)
with direction: thumb(force) =
hand(current) x fingers(field).
Let’s now consider how this force will work
on complete current circuits, not just
individual lengths.
Forces on
rectangular current loop
Consider the situation in the figure below:
A current loop (with current direction going
counter-clockwise) is situated in a Magnetic
Field going from North to South poles.
N
B
I
S
Forces on
rectangular current loop
We need to consider the forces on each of the four
sides of the current loop.
The force on the top and bottom of the loop
are zero, since the field and current are either
parallel or anti-parallel (qIB = 0o or 180o).
N
 I Ftop=0
B
S
I  Fbottom=0
Forces on
rectangular current loop
The current on the left side is going down while
the field is directed to the right, so that means
the force is directed out of the screen, and the
magnitude is: Fleft = I L B sin(90o) = I L B.
N
F
I B
S
L
Forces on
rectangular current loop
The current on the right side is going up while
the field is directed to the right, so that means
the force is directed into the screen, and the
magnitude is: Fright = I L B sin(90o) = I L B.
N
B
F
 I S
L
Net Force
Ftop = 0
Fbottom = 0
Fleft = ILB out
Fright = ILB in
As we can see, the NET FORCE (S F) is
zero.
However, since the force is pushing out on the
left and in on the right, there is a Torque!
The loop will tend to rotate about an axis
through the center.
Torque on
rectangular current loop
Recall that torque is: t = r F sin(qrF). In the
figure below we can see that r = w/2. Thus
the Fleft gives a torque of (w/2)ILB, and the
Fright also gives a torque of (w/2)ILB.
r
F
 F
N I B
 I S
L
w
Torque on
rectangular current loop
Since both torques are trying to rotate the loop
in the same direction, the total torque is:
S t = wILB. We note that wL = A (width times
length = Area). Also, we can have several
loops (N) that will each give a torque.
N
F
 F
I B
 I S
w
r
L
Torque on a loop
The final result for this loop is:
t = N A I B sin(qIB) sin(qrF) .
In this orientation, qIB = 90o and qrF = 90o .
If the loop does rotate, we see that
qIB remains at 90o (the current still goes up and
down, the field still goes to the right),
but qrF changes as the loop rotates!
t = N A I B sin(qrF) .
Electric Meter
One of the early types of current meters is one
where we have such a loop of current in a
magnetic field, and we have a restraining
spring. As the current increases, the torque
increases, but we have a restraining spring
to keep it from rotating completely around.
The bigger the current, the bigger the
torque, and the loop will turn through a
bigger angle. Attach a pointer onto the
loop, and we have the (analogue) meter.
Electric Motor
To create an electric motor that will continue
to spin when a current is applied, we need
to keep the current going up the right side,
even when the side originally on the left
becomes the right side due to the spinning.
We can accomplish this by using a set of
brushes as indicated in the next slide.
Electric Motor
+
N
-
S
This diagram will be explained in class, since it
involves three dimensions. The two green “C’s”
are actually one ring that is split. The ring is in
and out of the screen.
Electric Motor
t = N A I B sin(qrF) .
Since we switch the current to make it always
run up whatever wire is on the left side, we
make sin(qrF) always positive.
To find the average torque, we need to
determine the average of sin(q) from 0o
through 180o. From the calculus, we find
that its average value is 2/p.
Average Torque
Thus we get for the average torque:
taverage = (2/p) N A I B .
But we usually describe motors by their
power, not by their torque. Recall that
Power is energy per time, and energy is
force through a distance. For rotations,
this becomes: energy is torque through
an angle, and power is torque through an
angle per time (but recall w = Dq/Dt).
Average Power
Putting all of this together gives:
Pavg = t w = w N A I B (2/p), and with w =
2pf, we have:
Pavg = 4 f N A I B.
Note that the power depends not only on the
details of the motor (N, A, B) and the
current (I), but also on how fast the motor
spins (f)!
Example
Design an electric motor that has a power of 1/2
hp when it rotates at a frequency of 120 Hz
(120 cycles/sec * 60 sec/min = 7200 rpm).
Pavg = 0.5 hp* (746 Watts/1 hp) = 373 Watts.
f = 120 Hz.
Design = ? (This means we specify N, A, I and
B such that P = 373 Watts when f=120 Hz.
We have some “free” choices!)
Example, cont.
Pavg = 4 f N A I B.
Pavg = 373 Watts, f = 120 Hz.
Let’s start by choosing some reasonable values:
choose an area (A) of 10 cm x 10 cm = .01 m2;
choose a magnetic field (B) of 1000 Gauss
(0.1 T). We can also choose a current (I) of
5 amps. This means that we can now solve for
the number of turns, N.
Example, cont.
Pavg = 4 f N A I B.
Pavg = 373 Watts, f = 120 Hz.
A = .01 m2 B = 0.1 T I = 5 amps.
N = P / [4 f A I B]
= 373 Watts / [4 * 120 Hz * .01 m2 * 5 Amps * .1 T]
= 155 turns.