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ppt document - CBU

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									                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              S

   N    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:
     a constant that describes the strength
 similar to G and k;
     an inverse square relationship with
 distance (1/r2); and
     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 two cases:
1. magnetic materials (permanent magnets)
                  and
2. currents (electromagnets).
     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).
UNITS: Note all equations for B should have
  a mo, an I, and a (distance/distance2) or
  (1/distance).
• Bat center of loop = mo N I / 2R
• Bhelmholtz coil = 8moNI/[125]1/2 R
• Blong straight wire = mo I / 2pa
    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           S

                I 
              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).
                     I Ftop=0
       N            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.

            F
       N   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.

                          F
       N         B        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.
               F           F
       N I B              I S      L
                  w     r
             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.

								
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