32-1 (SJP-phys1120)
Magnetism: In 1110 and 1120 so far we’ve seen 2 fundamental forces
of nature: gravity and electrical forces. Electrical force depends on the
existence of charge – charges make E fields, and then E fields in turn
exert forces on other charges, F = qE. There is another kind of force in
the world, called magnetism (attracting “rocks” were found in Magnesia
> 2,000 years ago). You’ve surely played with kitchen magnets. They
stick to some materials but not others. E.g. magnets don’t stick to
aluminum. Magnetism is not equal to Electricity! They are different
forces!
E.g.: Hold a magnet near the electroscope (which is very sensitive to
even tiny amounts of electric charge). Nothing happens!
E.g.: Hold a magnet near those electric dipole seeds we used to demo Efields. You’ll see nothing.
E.g.:Charge up a balloon, hold a magnet near it. Nothing.
Magnetic forces are new, a different force than electrostatics.
Phenomenology of Magnets:
Play with magnets a little! Some attract, and some repel. 1
In fact, all magnets seem to have 2 “sides” or “poles”.
Once you’ve labeled the poles, you’ll notice they act like this:
1
2
1
Attract
2
Opposite poles attract and
like poles repel:
1
2
2
2
Repel
This is a bit like electricity, where we also had two charges:
opposites attracted while likes repelled. But this is not electrical!
So let’s avoid naming the magnetic “charges” + and -.
Here’s another name: “N” and “S” (North and South). We’ll label one
(arbitrarily) and then we can figure out all the others in the world.
Unlike electricity, you’ll never see:
N (impossible) N
You always have
called a "dipole" magnet, because
it has two (different) poles.
N
S
If you break a magnet, you DON'T get one "N-only” and one "S-only"
magnets, instead you simply get two smaller
N S N S
dipole magnets!
1
32-2 (SJP-phys1120)
There is a magnetic field which (like E-fields) extends through space. It
exerts a force on other magnetic objects. (It’s a vector associated with
every point in space)
We can use little “test magnets” to map out
a B field (just like little “test charges”
mapped E-fields for us.)
E.g. Iron filings, or a small compass, near a
magnet.
N
S
The compass can define the direction of
those lines. We can draw arrows on field
lines (pointing where the compass does).
(Looks rather like an electric dipole E-field pattern!)
N
S
Remember, opposites attract, and a
compass needle’s tip is (by definition) “N”,
so the compass points towards (is attracted
to) the “S” pole of other magnets.
The Earth is a giant magnet:
A compass points towards the geographic
“North” of the planet, so the magnetic
“S” pole (of the giant hidden magnet) sits
up near the planet’s geographic N pole!
(It's a little strange, think about this
picture until you understand the
conventions)
S
N
32-3 (SJP-phys1120)
Some key questions to ask now:



What makes/causes magnetic fields (call them B-fields)?
Can we quantify the strength of B-fields?
Can we quantify the effects of B-fields?
Lots of experiments were done (1800’s) to figure this out. E.g.:
1) (Oersted discovered) B-fields are always created by currents, i.e.
by moving electrical charges! (So although B-fields and E-fields
are very different, they are also related too)
2) B-fields always exert forces on any other currents.
So what about regular magnets? (Where’s the current in a kitchen
magnet? You don’t need to buy batteries for them, right?!)
Answer: All atoms have tiny currents around them,
all of the time! (Just the electrons in orbit.)
But normally, atoms are randomly oriented, so there’s no
net effect. (Magnetic fields of different atoms cancel)
+
-
But if the atomic currents all line up (which happens only in unusual and
special materials, like ferromagnets!) then they act magnetic.
This happens in E.g. iron (Fe), Nickel, Cr, not too much else.
32-4 (SJP-phys1120)
The “rules” of magnetism we’re about to discuss cannot be derived, they
are experimental facts. They look crazy, in fact, but this is how the
world apparently works!
Rule #1: Given a current, what is the B field?
Currents (I) always spontaneously form B-fields around themselves.
(Compare with the old rule “ charges make E-fields around themselves”)
We'll discuss the formula for the strength of B in a few pages.
But for now, lets just
look at the pattern:
Side view
top view
Current, I
B
The B field lines form
CIRCLES around the
wire (or current)
•
x
• I
(stronger)
(weaker)
The direction of the Bfield is found with the
B
Right Hand Rule #1. (RHR1):
Take your right thumb, point it along the current direction, I .
Your fingers naturally curl around the current the same direction B does.
(Try it, see if you understand the directions in the pictures above)
Note: the pictures are meant to be 3-D. I use a standard convention:
•
means the field is pointing AT you (out of the paper)
X means the field is pointing AWAY from you (into the paper).
To remember this, I sometimes think of the X as saying “dig here, buried
treasure.”
Other people think of an arrow.
If it points towards you, you only see the tip.
If it’s running away from you, you see the tail feathers... X
32-5 (SJP-phys1120)
Example: A current flows around a ring (a loop). What does the B
field look like?
Answer: We can't yet exactly derive the answer, but you can see it
intuitively just from the previous rule. Think about the B field produced
by little pieces of the wire, and then imagine “superposing” them,
building up the total B field. Here’s my sketch - think about it a little,
does it make any sense to you?
Side view
B
top view
x
x
I
x
x
x
x
I
x
B
x
B
x I
x
x
x
x
x
Look again at this figure. Some people introduce another “Right Hand
Rule” for this situation, which we might call “Right Hand Rule #1b”
RHR1b: To find the B field near a current loop, rather than a long wire:
If your right hand fingers curl with the current in a current loop, your
thumb points in the direction of the B through the center of the loop.
B
I
B
This is different than the RHR#1
(where your thumb pointed with I, and
your fingers pointed like B! )
So don’t mix them up! You never
B x
absolutely need RHR #1b, but I just
I
find it much quicker and easier when
you have current loops to deal with. (Which we will, often.)
32-6 (SJP-phys1120)
How STRONG is this B field: A moving charge (or a current) makes B
fields. This is the Biot-Savart law (named after 2 people!)
It's a purely empirical law, I can't derive it, it's just an accurate
description of the world! Here it is, the formula for the magnitude of B,
caused by a charge q moving with velocity v.
B =
2
µ0
q (v # rˆ ) /r .
4"
That's the Biot-Savart formula.
µ0 is a constant of nature, it has the value
!
!
#7
µ0
= 10 (in SI units)
4"
rˆ (called "rhat") is a UNIT vector (magnitude ONE) that points from the
charge q, towards the point you're interested in finding B at.
Units for B is the Tesla (which is !
a lot, e.g. the earth's field is 5E-5 T!)
_______________________
(I want to remind you of Coulomb's law from the start of the term:
E=
!
2
1
q rˆ /r . It's quite similar! Both are inverse square laws. Both
4 "#0
involve a constant of nature. Both depend on the "unit vector" that points
FROM the source of the field (the charge, or moving charge) TO the
point in space where you'd like to know the resulting field. )
The hard part about Biot-Savart is that "cross product".
I hope you remember that from a math class, but here's a quick review.
If you have two vectors C and D, you can "multiply them" to get a new
vector, called the cross product. The result has magnitude
|C " D|= |C| |D| sin(θ), (where θ is the angle between C and D).
!
The direction of the cross product is perpendicular to
BOTH C and D. The direction has an ambiguity (in
the picture, up or down?) and is chosen by the
RIGHT HAND RULE FOR CROSS PRODUCTS:
Point your FINGERS in the direction of C (the first
vector), curl your fingers towards the direction of D, and look at your
thumb. It is perpendicular to both C and D, it points in the desired "cross
product" direction)
Try it (right now, beside that little picture!!) to see how it works, if
you've forgotten.
Note that the sin(θ) term means that if two vectors are perpendicular,
they have a maximal cross product.
But if they are parallel, the cross product vanishes!
32-7 (SJP-phys1120)
Here's a little pictorial example of the Biot-Savart law.
I want to find the direction and magnitude of a B field at a point (P) in
empty space, caused by charges moving through a wire.
Let's consider just the chunk of field, dB,
caused ONLY by charges flowing past one
particular spot on the wire (the spot labeled by
dL in the figure)
(Later we'll have to sum up dB from all the
OTHER charges moving everywhere else in
the wire!)
Biot-Savart tells us the B field caused by individual charges q, moving
with speed v. But current tells you about moving charges: I=q/time.
So I would argue q v (which we need in the Biot-Savart law) must be
the same as I dL. (Reason: v is just distance/time = dL/t, so
q v = q dL/t = (q/t) dL = I dL)
Thus you can rewrite Biot-Savart as
B =
!
2
µ0
I (dL # rˆ ) /r .
4"
OK, look at the picture again. I am focusing on just one small piece of
wire (labeled dL): that piece alone creates a B field EVERYWHERE.
We're figuring out dB at the point labeled P, a distance r away from this
chunk of current... You first need to think of the direction of the "radius
vector", rˆ . I drew it in the picture, it points away from the current chunk,
towards the place in space (P) I'm interested in.
The right hand rule says dL x r points perpendicular to both dL and r.
!
That must be perpendicular to the page! But which way, in or out?
Remember the game: point your fingers along dL. Now curl your fingers
towards r. (You will have to twist a little!) Your thumb should point
INTO THE PAGE at the point P we're looking at, indicated by the X on
the figure. Convince yourself, you must be able to do this yourself!)
This direction does makes a lot of sense to me, because I'm also
visualizing B field lines as "curling around" that current, going into the
page on the right side of the current, where my point is. (Again,
convince yourself!)
32-8 (SJP-phys1120)
Example of computing a B field using the Biot-Savart law:
Let's find the magnetic field right at the center of a loop of current.
(Note that we've sketched the field a couple of pages back.
Also note that a current loop is not too hard to make, just add a tiny
battery to keep the current flowing around in a circle)
The little chunk of wire labeled dL will
produce a small chunk of magnetic field, dB, at
the center of the loop. Look at the picture (and
think about it!) to convince yourself that here,
dL and rhat both lie in the plane of the loop, so
dB is UP, as shown. (Use the RHR for cross
products, try to do the "finger game" to see that
dL x r really does point that way at the center)
Now remember, Biot-Savart says
dB =
2
µ0
I (dL # rˆ ) /r
4"
We have just figured out the direction, and since the magnitude of rhat
is ONE (it's a unit vector!) and dL is always exactly perpendicular to
! rhat everywhere,
dL x rhat is just given by |dL| |rhat| sin(90) = dL (*1*1)
Thus the magnitude of dB is just
dB =
2
µ0
I (dL) /r
4"
But wait. r=R is constant all the way around the loop. And so "adding
up" dB around the loop (to account for ALL the chunks dL all the way
! around) just adds up dL around a loop, which is the circumference (do
you see why? Just add little lengths all the way around the circle!)
So summing this, the dL's give us 2 pi R, and we end up with
B(center of loop of current ) = µ0 I /2R
(Do that one step of algebra yourself, to see what happened with the pi's
and R's and stuff!)
!
32-9 (SJP-phys1120)
We now know the pattern of B created by currents. (RHR#1)
We also know the formula for B created by a current, I ("Biot-Savart")
Remember at the start of the term, how we first had Coulomb's law
(which gave us E, if we were given charge) and then we moved to a
deeper, more fundamental law, Gauss' law (which also related E to
charges, but a little more indirectly).
The same thing happens here: Biot-Savart says B comes from current,
but there is a deeper, more fundamental law relating B to currents...
It was discovered by Ampere. The math is a little tough (“vector
calculus”.) But here's the basic statement:
If you have a current, draw an imaginary loop around the wire.
It can be any size or shape, we call it an "Amperian loop".
(There might be nothing physical there - it's just a path, a loop )
Look at the B field at every point on that path, and
add up the component of B parallel to the path,
times the path length.
That is, add up B • dL, where dL is your little path length.
When you do this, your sum (really an integral, right?) is given by
! Law:
Ampere's
" B • dL = µ I(thru)
0
Here, the little circle on the integral sign just reminds me that we're
going around a closed loop, adding up B • dL along the way.
!
(It's called a "line integral")
On the right side, what I mean by
! I(thru) is the total amount of current
which pokes through that Amperian loop. (In the figure above, just I)
This formula (along with Gauss' law!) is one of the master formulas of
all of electricity and magnetism. It tells you that B fields come from
currents, and the more current you have, the stronger the B field will be.
It also tells you (though it may not be so obvious if you've never worked
with line integrals) that B fields "curl around" wires.
32-10 (SJP-phys1120)
OK, wait a minute. What just happened? Where is this law coming
from? Why is it true? And why would we ever think to do this crazy
thing (Multiplying B • dL and summing, what does that even mean?)
I wish I had simple answers for you, but Ampere's law is just this
remarkable, true result. It can be mathematically derived from Biot!
Savart (and vice-versa!) but it turns out to be more fundamental than
Biot-Savart. We won't go there (the math is a little nasty) but will take it
from experiment. You can measure B, and compute that integral... and it
ALWAYS comes out right. Ampere's law is ALWAYS TRUE for
situations that are not changing with time (we'll fix it up later to account
for what happens if things are varying with time)
It tells you that the more current you have, the strong B will be. (And it's
linear: double current => double B!)
It is helpful in calculating B if you have a lot of symmetry, so e.g. if you
can make an argument that B is constant and always parallel to dL,
then, you can pull B OUT of the integral and solve for it. That would be
the case for a LONG straight wire, where B makes lovely circles around
it: let's work that case out. Ampere's laws says " B • dL = µ0I(thru) .
Let's pick an (imaginary) loop a distance r from the wire, and figure out
B there. So we're looking for B(r) (because B may depend on distance!)
All the way around the loop, B(r) is the same
(by symmetry!) and
!
parallel to dL everywhere (it loops the same way as the dashed circle!),
so " B • dL is just B(r) times the length of the circular loop, giving:
B(r) 2 π r = µ0 I, or solving for B(r):
!
|B| =
µ0 I
2! r
where µ0 = 4π 10^-7 (Tm/A) is a constant of nature,
and r is the distance you are away from the long wire.
The B field is proportional to current, I, but decreases with distance.
(Notice that it's not 1/r^2 like Coulomb's law was, but maybe that makes
sense, because we don't have a "point current", but an infinitely long line
of current!)
OK, so now we can find B... what do we DO with it?
32-11 (SJP-phys1120)
Rule #2: Given a B field, what force does a current feel?
B-fields exert a force on currents! If you put a current I into some
external B-field, the current is pushed sideways.
B
B
The size of the force has been measured in many exp'ts,
and the following formula summarizes the data:
!
I
Magnitude of force:
F=I L B sinθ
Here, I = current (in Amps)
B = strength of B-field
Resulting force on I
L = length of the wire in B-field
is out of the page.
θ is the Angle between I and B
Direction of this force: Right Hand Rule #2” (RHR2)
 Point your right-hand fingers in the direction of I.
 Orient so your fingertips can curl naturally towards B.
 Stick out your thumb. It points in the direction of F.
 This is the same as saying F = I " L (see previous page!)
In the picture above, the force on the wire points out of the page:
The wire in the picture will get pushed towards you, out of the page.
Convince Yourself!! The!RHR's take some practice.
(Note: F is always perpendicular to B, and to I) You must think in 3-D.
We have not derived any of these results, it's simply what exp'ts tell us...
If θ = 90, F is max: Fmax = ILB (into page, here):
Convince Yourself!
I
90
B
So
B = Fmax /IL .
This in fact defines the numerical value of B!
Recall Units [B] = N/(A*m) = 1 Tesla = T
B
Resulting Force direction
(into the page)
1 Tesla is a lot. The Earth’s natural field is B about 0.5 * 10^-4 T.
 An old unit for B is “Gauss,” 1 Gauss = 10^-4 T. That means
Bearth = 0.5 Gauss, roughly.
 Kitchen magnets are around 50G or 5*10^-3 T = 5 mT.
 Industrial magnets = 1 or 2 T (that’s a natural limit for Iron)
(This is a typical NMR field. It's a strong magnetic field)
 Superconducting magnets might reach 10 T
(I think 14.7 T is the
current world record?) http://enews.lbl.gov/Science-Articles/Archive/14-tesla-magnet.html
32-12 (SJP-phys1120)
Example: Take a “horseshoe magnet” (Which generates a big B-field
between its poles, but nowhere else. ) Remember B points from N
towards S. Run a small current, I, into the page through this big B field.
RHR#2 says F on the wire is up
(convince yourself!)
This wire gets "sucked up" towards the
horseshoe magnet.
F (on wire)
I (into page)
S
B
x
B
N
The magnitude of the force is F = I L B
(L is the the length of the wire that is in
the B-field)
NOTE: the B field here is "external", created by the horseshoe magnet.
The wire itself will also create its own little B field (looping around it),
but we're not concerned about that here.
Example: Take that horseshoe magnet and reorient it so “N” is in front
of the page, “S” is behind. As shown, B now points into the page.
Now lower a current carrying square loop of
B
wire into this B-field. .
S
I
(The B-field is localized
in the region between
N
L
the two poles. It's zero
x x x x x x
pretty much everywhere
x x x x x x
else in space.)
There will only be forces
on the parts of the wire
that are in the B-field.
x
x
x
x
x
x
x
x
x
x
x
x
(B=0 outside)
front view
Use the 2nd RHR, and convince yourself which way the forces are on
the various "segments" of the current loop. I've drawn them here:
I
F=IDB D
L
F=ILB
F=IDB
The forces on the left & right
cancel, but because of the lower
segment, there remains a net force,
down.
As shown, the current loop is “sucked in” to this particular B-field. If the
current I ran the other way, it would be "shoved out", can you see why?
B
32-13 (SJP-phys1120)
The force on the loop in that last example is very real - it could be
measured with a scale - a way to deduce the strength of the B field with
a perfectly ordinary mechanical scale. E.g.,
if
I=20 A, (Typical large current in household wires)
L=10 cm, (Typical large horseshoe magnet width)
B=1 T (A very strong magnet!)
then F=I L B=(20 A) (0.1 m)(1 N/Am) = 2 N.
Not huge, but you could easily feel this, and accurately measure it.
This force has nothing to do with the metal or material of the wire itself.
It is really the moving charges themselves, the current, that feel(s) this
force. If you turn off the current, this force will disappear.
E.g. A beam of protons flying through B would also feel the magnetic
force, even though there’s no wire anywhere at all.
F= I L B sinθ
Our formula for a wire, or current.
F=q v B sinθ
The new formula for individual particles.
Where v=velocity of the particle, q is the charge.
B
[These are really one and the same formula!
Why? If you're interested, consider this:
I = #charges/sec = Nq/t
v = dist/time =L/t
So I L B = (Nq/t)(L)(B) = Nq*vB
That means all N charges feel a total force of NqVB,
so each charge feels qVB. ]
B
v
!
+q
Resulting Force direction
(out of the page)
The direction is just as before: Use the exact same RHR#2.
Except be careful if the q’s are negative, then the direction of F must be
reversed!
32-14 (SJP-phys1120)
Example: Two charged particles enter a region of uniform B which
points into the page, as shown.
B
x
x
x
The force F on +q is up (e.g. protons)
+q v
The F on –q is down (e.g. electrons)
x
x
x
x
x
x
-q
B
(Because of these "sideways" forces, their
path will curve, as we'll see)
These formulas are Bizarre! This is all bizarre! Look:
A proton at rest feels zero force, no force.
 But if it’s moving right, it feels a force up.
+q
 If it’s moving left, it feels a force down.
x
x
x
 If it’s moving up, it feels a force left.
 If it moves into (or out of) the page, it feels no
x
x
x B
force!! (In that case v&B are parallel (or antiparallel), θ=0 (or 180), and sin(θ)=0
Bizarre but completely true. Nature has come up with a curious force!
x
x
x
B
Example: Uniform B pointing out of page:
Proton, initially moving with vi, to the right.
So F is initially straight down.
v right, F down means it will curve down.
New velocity v’ is “down-and-right.”
New F’ is shown.
B
vi
Fi
F'
Note: F is always perpendicular to v. (By RHR#2)
So the work done by the B field = F d cosθ = 0.
This force does not speed up the proton, it is a centripetal force!
We’ve seen this before, Ch. 5, it’s uniform circular motion.
F is qVB(constant), and acceleration is v^2/R: Newton II gives
qvB=(mv2)/R or R = (mv)/qB
The proton moves in a circle of radius R. Bigger B => tighter circle
There was nothing "special" about the proton in this problem - any
charged particle in a uniform magnetic field will travel in circles.
(An electron, however, will curve the opposite direction because of its
negative charge). This effect allows particle physicists to identify many
properties of charged particles, even tiny ones, in "bubble chamber"
pictures.
v'
32-15 (SJP-phys1120)
I
B
r
Let me remind you of a result we got (from Ampere's
law) for the B field around a wire:
µ I
|B| = 0
2! r
where µ0 = 4π 10^-7 (Tm/A) is a constant of nature,
and r is the distance you are away from the long wire.
Consequence: What if you stick a second wire into the B field of the
first wire? There will be a force on the second wire (it's a current in a B
field, after all.) Let's figure out the magnitude and direction:
Wire #1
I1
Wire #2
I2
B1 due to
wire #1.
L2
d
We know the force on wire 2 due
to any B field is
|F | = I2 L2 B
We also know how big B is over
there (where wire #2 is) from our
formula above:
µ I1
|B|= 0
2! d
Plug this in, to get
µ I1
F (on wire 2, of length L2) = I2 L2 0
2! d
µ0 I1"I2
=
L2
2! d
The force per unit length" on wire 2 is just µ0 I1I2 /2π d.
What is the direction of the force?
RHR#2 says it is left (convince yourself!)
By symmetry, wire #1 is also attracted to wire #2.
That means these parallel currents attract!
(This is magnetic attraction, not electric)
Remember, "like charges" repel. But here, when
magnetism is involved, "like currents" attract!
I2
I1
F on 1 F on 2
by 2
by 1
32-16 (SJP-phys1120)
What if the currents go opposite directions?
If you flip I1, this reverses the direction of B1
I1
I2
( i.e. the B field due to current #1, over there
F on 1
F on 2 at the place where wire 2 is),
which in turn flips the direction of the force
by 2
by 1
on 2 by 1.
Convince yourself of both those statements.
It's good practice of both RHR's.
That means opposite currents repel.
Note: The formula itself, F/unit length = [µ0 I1I2 /2π d]
is totally symmetric, if you replace "1" with "2", so
F( on 1by 2) = F( on 1by 2)
(Newton III is still true, even for bizarre forces like magnetism!)
Example: Wires in your house are always really two wires like this (to
make a circuit, current goes in, it must
I
also go back out again!)
We just found the formula for the
(sideways) force on the wires themselves:
Light bulb
F = [µ0 I1I2 /2π d](L)
If you have a 2 m long wire to a lamp (L=2 m), and the wires are
wrapped together in a single cable, as they generally are, let's say 1 mm
apart (r = .001 m),. with a typical current of I = 1 Amp (e.g. that's what
you have for a single 120W bulb) then
F = [(4π10^-7)/(2π)]{Tm/A}[(1A)2/(10-3m)](2m)
= 4*10-4 Tm/A = 4 *10-4 N
It’s a small, barely noticeable (repulsive) force.
A big nuclear physics spectrometer electromagnet might have I~50A,
r~10-4m, length ~ 1 km of wire,
F = [(4π10^-7)/(2π)][(50A)2/(10-4m)](103) = 5000N
= 1/2 ton of repulsive force! You have to build those things strong, or
the magnetic forces between wires will blow up your device!
32-17 (SJP-phys1120)
Imagine wrapping many wire loops next to each other:
I
I
Each loop makes a B like the one
on p. 20-6, and these superpose.
B
B
You get a strong and very uniform
B field inside the loops. (It's very
weak outside.)
We call this wire shape a solenoid.
From the outside, it looks rather a lot like an ordinary bar magnet!
S
N
Indeed, that's basically what a bar magnet is
like, microscopically, The wire loops,
however, are not connected like in the
solenoid- they are little circular atomic
currents!
Side view
In this “head on” picture of a
magnet (like eg the one above)
the "I"'s everywhere in the
middle all cancel each
I I
other out.
I effective
Head on view:
I
I
But at the outside edges, the
"I"'s add up to look a lot like a
single, big ring of circulating curren.
(Thus, like the outside edge of a solenoid.)
In ferromagnetic materials, the atomic currents tend to line up like
this, causing a large magnetic field to be created naturally.
x
32-18 (SJP-phys1120)
This explains why bar magnets attract N to S (as we saw at the start of
this section).
S
N
S
N
These are parallel currents - they attract!
S
N
N
S
These are opposite currents - they repel!
Iron naturally has small spatial regions (domains) that each act like
small magnets, like on the previous page. They tend to be randomly
oriented. So iron is not normally a magnet. (E.g, a normal nail doesn't
stick to the fridge). But if you put iron into a strong external B field, this
will tend to line up all those little domains, and you can magnetize iron
this way. (See text for some sketches)
If you put iron inside a solenoid, the B field from the solenoid lines up
the iron magnetic domains, which means the iron becomes itself highly
magnetized, adding to the solenoid field. Result: a very powerful
magnet. This is an electromagnet. Turning on the solenoid current lines
up the domains => big B field. Turning off the solenoid current => the
domains re-randomize, and the magnet gets much weaker.
But, why do the domains line up with the external B?
Or, why do magnets "like" to line up with each other?
(Answer on next page...)
32-19 (SJP-phys1120)
Consider a current loop (or a
"domain") inside an external B field.
B(external)
I
F on the top wire is 0 (because θ =0).
F on the left wire is INTO the page.
F on the right wire is OUT of the page.
Result: F_net = 0, but there is a torque, a twist on the loop...
F
Top view
F
(top of wire loop)
x
I
B(ext)
which wants to twist
into this position...
I
B(ext)
x
F
F
Note what happened: that loop had its own little B field (I didn’t draw it,
because it would clutter the diagram, but it points into the page in the
figure at the top) originally. After it twists, that internal B field (which I
still haven’t drawn, but you can figure it out-) now points the same
direction as the big B external!
Convince yourself of all this, using RHR #1 and #2.
Here, the current loop is basically a little magnet (all current loops are
basically little magnets!) and we just worked out that it wants to twist to
line up its own B field with the external one.
It's just what I claimed, magnets prefer to line up with their B fields
parallel.
In an external B, the little atomic current loops themselves are twisted
until the natural (internal) B points in the same direction as B external.
It’s not much different from the example above with a wire current loop.
32-20 (SJP-phys1120)
A galvanometer is just the previous example in practical action:
You run a current "I" through a wire loop,
Pointer
Coil spring
(which is perpendicular to the page, see
(resists twist)
previous figures) and a coil spring resists the
resulting twisting. More "I" means more
x
I
torque, the needle moves farther.
You can measure "I" this way, if B is fixed and known.
Or, if you know "I", you can measure "B" this way. It's a "B-meter".
D.C. motors work on this principle too. Imagine a loop in a B (external)
Top view
(top of wire loop)
F
x
I
B(ext)
F
you then turn off "I".
As shown, the loop starts to twist.
Suppose, when the loop just
“Equilibrium”
reaches its "equilibrium"
F
position, (the orientation
where it feels no more net
B(ext)
I
twist, or torque...)
x
F
The loop had been turning, and now it suddenly feels no forces.
It'll just coast. When it has coasted back to the original orientation again
you could now turn "I" back on. It'll twist again. If you keep repeating
this, it goes around and around. You have used a current, I, to give you
physical motion. This is a simple motor! You could turn a drill bit, or
car wheels,... (This simple invention has completely changed the world
we live in. Think about everything done by electric motors!)
The practical trick is turning "I" on and off at just the right times.
(Or, better yet, reversing the direction of "I", so that instead of coasting
it continues to twist itself the same way all the time) Real motors use
commutators or brushes to turn the current on and off as the motor
rotates.
32-21 (SJP-phys1120)
Crossed E and B fields:
- - - - - - - - - E (up)
Imagine having both E and B fields together
in one place. E.g, picture a capacitor (with a
q
v
big E field) inside a magnet whose B field
was perpendicular to E.
B (out of page)
What would happen if you ran a charge "+q"
+ + + + + + + +
through this region?
In this picture, we have an UPWARD electric force F(E) = qE.
We have a DOWNWARD magnetic force F(B) = q v B. (from RHR #2)
If F(E) ! F(B), the particle will curve, eventually hitting the plate.
But If F(E)=F(B), the particle goes straight, because F_net=0.
So if qE = qvB, or in other words if v = E/B, it goes straight.
If you send in a bunch of particles of different charges, different v's,
different masses.... only those with the particular velocity v = E/B
make it through out the far side. This device is a velocity selector.
Note: It doesn't matter what q is, not even what sign! It doesn't matter
what "m" is either.
What can you do with a velocity selector?
Here's one example. Take that beam (so now you know v. By tuning E
or B, you can pick the v you want!) and run it into a region of uniform B.
Remember, particles in uniform B fields
follow a simple circular path.
v
R
They go in a circle with radius
R = mv / (q B).
Where the beam comes out at the bottom tells
B (out of page) you the radius R. You can pick B. We already
chose v. So this device will "read off" m/q.
A mass spectrometer is basically just like the above device.
It's fantastically useful to identify what "stuff" you have in your beam.
(m/q is pretty much unique for every different type of atom or particle).
This device is used by chemists, physicists, biologists, forensicists...
Think of how useful it is to identify even single atoms of some material!
32-22 (SJP-phys1120)
Some final comments on this chapter:
1) If you run past static charges, you see (in your reference frame) a
current running towards you. That means you see a B field. However, a
stationary observer nearby sees only static charges. They see no B field!
Who is right? Is there a B field there or not? Einstein thought a lot about
this puzzle... (more on this later! Think about it yourself for now.)
2) E and B are clearly different. But they are also clearly related.
Charges make, and feel, E. Currents make, and feel B. But current are
just moving charges! James Clerk Maxwell in the late 1800's
discovered/explained that there is really only one fundamental force at
work here - electromagnetism - and one field, the electromagnetic
field. The math involved is a little hairy for Phys 2020, but Maxwell's
work is a beautiful unification of the two forces of electricity and
magnetism. It's one of the grand and glorious achievements in physics,
nearly unrivalled in mathematical beauty and practical significance.
3) All of classical mechanics is based primarily on Newton's laws. (3 of
them, plus his law of gravity)
All of electromagnetism is based primarily on Maxwell's equations. (4
of them)
Together, mechanics and electromagnetism describe a vast amount of
phenomena - the "classical" world we live in! Relativity and quantum
mechanics finish off the story as we know it today, but at its heart, all
the complexity and diversity of the physical world boils down to this
tiny handful of simple laws of nature!
4) Parting thought for this chapter: we've learned that E fields can cause
currents, which make B fields. Can you go the other way? Can a B field
somehow make currents flow, or E fields (or voltage differences)? This
was a hot question in the 1800's, and the answer (the subject of Ch. 33 of
Knight) once again changed the world we live in radically, because it's
the basis for electric power generation!