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Chapter 25The Eye
Simple MagnifiersAngular
MagnificationThe Telescope
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We’re going to take a quick, 1-lecture look atthe material in this chapter. There’s little newhere--just an application of what we alreadyknow. But here is a look at the practical sideof optics.
Optical Instruments
The human eye is really just another,relatively simple optical system. Theeye takes advantage of refraction tofocus light from objects in our field ofview onto the retina.
Incoming light rays pass through thecornea and pupil of the eye, where theythen intersect the lens, which focusesthe light onto the retina.
NormalEye
As you are no doubt aware, two commondefects plague human vision:
Hyperopia (or farsightedness)
Myopia (or nearsightedness)
Each of these problems results in blurredvision and is a result of the inability ofthe eye to properly focus light on theretina.
The normal eye is able to comfortably focusobjects at distances from the far point (infinity)to as close as the near point (about 25 cm).
Hyperopia
For the farsighted person, the lens ofthe eye focuses the light at a point beyondthe back of the retina.
Objects at a great distance from the observerappear fairly clear, but objects close to theobserver are blurred.
Myopia
For the nearsighted person, the lens ofthe eye focuses the light at a point in frontof the retina.
Objects close to the observer appear fairlyclear, but objects far from the observer areblurred.
We can correct these optical defects quitesimply by prescribing corrective lenses tobe placed in front of the eye.
Hyperopia
What type of lens will we want to put infront of an eye suffering from hyperopia?
converging lens
Helps out the eye with hyperopiaby starting to bring the lightto a focus before it encountersthe lens of the eye.
diverging lens
Helps out the eye with myopiaby separating the incoming lightbefore it encounters the abnormallystrong focusing lens of the eye.
Optometrists and opthalmologists usuallyprescribe lenses by describing the powerof the corrective lens.
Pf
1
[ ][ ]
Pf m
1 1
= 1 diopter
The angular magnification of an objectis simply the ratio of the angle subtendedby the object at your eye when magnifiedto that angle when unmagnified and placedat the near point.
A diagram will help here...
25 cmobject atthe near
point
h0
25 cmimage atthe near
point
h’
pObject just outsideof the focal point ofthe converging lens
Our simple magnifier has a focal lengthless than 25 cm and will produce an imagewith an angular magnification given by:
m 0
This ratio achieves its maximum valuein the geometry of the last slide (wherethe image appears at the near point).
In that case, the object should be placed at
p
25
25
f
fwhere p and f are both
measured in cm.
p
25
25
f
f
When the object is placedat the distance p in frontof the simple magnifier, themaximum angularmagnification occurs.
mf
125
The simple magnifier is a useful device.Not only do we find it in magnifyingglasses, but they also serve as theeyepieces for microscopes andtelescopes!
Let’s look at how a telescopeworks a little more closely.
There are two principle types of opticaltelescopes: reflecting and refracting.
Reflecting telescopes use a mirror tobring distant light to a focus. A lens(the eyepiece) is then used to magnifythe image formed by the primary mirror.
Refracting telescopes use a lens to bringdistant light to a focus and a second lens(the eyepiece) to magnify the image formedby the first lens.
Let’s look in more detail at a refractingtelescope (though everything we derivehere will also be applicable to the reflectingtelescope as well).
In general, the objects at which we peerthrough telescopes are a LONG distancefrom us. So, let’s assume an object distanceof infinity.
light from the object objective
eyepiece
0
fo fe
light from the object objective
eyepiece
0
fo fefinalimage
object
ive
imag
e
hei
ght h’
m 0
angularmagnification
fo
fe
h’o
mh f
h f
f
fe
o
o
e
0
'/
'/
We’re now going to start down the path ofexamining forces whose origin is not visibleto us. We call the quality of matter responsiblefor these forces “charge,” a substance neverdirectly observed in the history of the humanrace!
Electrical Forces and Fields
Amber (elektron in Greek) attracts straw/feathers when rubbed (observed by Thales of Miletus ~ 600 B.C.).
Iron ore from the country ofMagnesia seemed to have anatural affinity for metals.
When released, all objectsseem to fall toward the ground.
Attraction in Nature
Electrostatic and Magnetic Forces
William Gilbert (1540 - 1603, English physician)- clarifies the difference between the attraction
of amber and that of magnetic iron ore;- shows that many materials besides amber
exhibit electrical attraction- showed that the behavior of a compass needle
results from the magnetic field of the Earthitself!
Stephen Gray (early 1700’s) shows that the electrostaticattractive and repulsive forces can be transferredthrough contact alone…Metals didn’t need to berubbed.
Charles Du Fay (1698 - 1739) first postulated the existenceof two distinct kinds of electricity: vitreous (theglass rod) and resinous (the silk). But thought theyexisted together in most matter and when separatedthrough friction, resulted in an electrical force.
Benjamin Franklin (1706 - 1790) hypothesized a one-fluidmodel of electricity: charge is transferred from onebody to another (e.g. through rubbing); but the totalcharge on the two bodies combined remains thesame. This theory is known as the…
Conservation of Charge.
Two Kinds of Charge
Franklin decided to call the materials which he believed had excess charge “positively” charged materials. Those with a deficiency of charge he called “negatively” charged.
Parenthetical Remark:
Unfortunately, as we would later learn, it is the electrons (negative charge carriers) that are mobile while the protons (positive charge carriers) generally remain fixed in the nucleus of atoms. So materials with excess electrons appear negatively charged. Nevertheless, Franklin’s convention has stuck with us…quite literally!
Hence, the glass rod (vitreous) was positive when rubbed with silk while the amber (resinous) was negative when rubbed with wool.
Charge is Quantized
Robert Millikan (1868 - 1953) in a very cleverexperiment showed that electrical chargecame in quantized units. In other words,charge of 0, +/- 1e, +/- 2e, +/- 3e,...+/- ne (where n is an integer) couldbe observed, but never a charge of1.5e (see section 15.7).
An electron carries a charge of -1e.A proton carries a charge of +1e.
•Electrons and protons carry charge•They are responsible for the electrical
forces we encounter•Atoms are made up of electrons and
protons•Matter is composed of atoms
So why doesn’t every object we encounter exert an electric force on every other object around it?
P+ElectronCloud
e-
Hydrogen atom
Neutral Space
Electrically Neutral...
Most objects in nature are electricallyneutral (i.e. they contain an equal number of protons and electrons).
Ne = Np
Therefore most objects exert no electrical force on the objects around them.
Atoms in which Ne< Np or Ne> Np arecalled ions.
Insulators and Conductors
Objects on which electrons move freely areknown as conductors. Most metals are goodconductors. Electrical wires are made ofgood conductors.
Objects on which charges do not move freelyare known as insulators. Glass, amber,rubber, silk and cloth are all examples ofgood insulators.
If metals are good conductors, why is it hard to charge them by rubbing them with wool?
If rubber is such a bad conductor, why is it so easy to put a charge on rubber by rubbing it with wool?
GROUND
RUBBER
The charges remain near the end of the rubber rod--right where we rubbed them on!
GROUND
COPPER
Rub charges on here
They move down theconductor toward our hand
Eventually ending up in the ground.
GROUND
COPPER
Rub charges on here
A good conductor distributesthe charge uniformly over itssurface.
Rubber glove insulates copper rodfrom us and therefore the ground.
So, we can charge up a conductor!
And human beings must be fairly goodconductors of electricity, too.
Notice that the Earth’s surface (ground)acts as a vast source and sink for electricalcharge. Touching a conductor to groundwill neutralize the charge on the conductor:If the conductor is positively charged, electronsflow from the ground to the conductor. If theconductor is negatively charged, electrons flowoff the conductor into the Earth.
So when we talk about an object beinggrounded, we literally mean that it isconnected via a conductor to the Earth’ssurface.
All electrical outlets now have a groundprong. And most electrical devices usea 3-prong plug that requires the groundconnection.
Well….just in case the device malfunctions,it’s nice to be able to siphon off the excesselectrical charges to ground rather thanallowing them to accumulate in the device.
If they build up in the device, they willeventually find their way to ground. If a person comes in contact with the device, the resulting flow of charges through the body can be deadly.
The ground prong provides a nice safety feature.
We can take advantage of the Earth’sability to accept and provide charges toplace a net charge on a conductor….
Here’s how you do it!
GROUND
COPPER Bring negatively charged rubber ball close to the a copper rod. The copper rod is initially neutral.
Negative charges on the copper runaway from the rubber ball and into theground.
Rubber
GROUND
COPPER The copper rod is now positively charged. The electrons originally on it were forced away into the ground by the negative charges on the rubber ball.
Rubber
+++
++
++
GROUND
COPPER
Put a rubber glove on your hand to insolate the copper rod from ground.
Rubber
+++
++
++
Finally, remove the rubber ball...
GROUND
COPPER+
++
++
++
The excess positive charge is trapped on the copper rod with no path to ground. It redistributes itself uniformly over the
copper rod. We have taken an initially neutral copper rod and induced a positive charge on it!
Note on quantifying charges…
The mks unit for charge is the Coulomb,named after Charles Coulomb in honorof his many contributions in the field ofelectricity.
1e = 1.6 X 10-19 C
Joseph Priestly and Charles Coulomb set out toquantify the electrical force in the late 1700’s.
Let’s see if we can figure it out an expression forthe electric force ourselves...
Distance
Charge on Body 1
Charge on Body 2
Anything else?
1/r2 (units of 1/m2)
q1 (units of C)
q2 (units of C)
Just a constant ofproportionality…Let’s call it k.
F = ???
What do we have so far...
F
kq q
r1 22
r
F The arrow indicates that F is a vector
quantity (i.e., to specify F, you need both magnitude and direction)!
r Indicates a direction radially away fromthe center of our coordinate system. It couldbe the x-direction, the y-direction, orsomething in between.
[F] = [k] [q1] [q2] / [r]2
N = [k] C2/m2
F
kq q
r1 22
r
[k] = N m2/C2!
The value of k is determinedexperimentally to be…
k = 8.99 X 109 N m2/C2
We have now determined a quantitativeexpression for the electrostatic force!
Let’s call it 9 X 109 N m2/C2
Notes on Coulomb’s Law:
Applies only to point charges, particlesor spherical charge distributions.
Obeys Newton’s 3rd Law.
The electrical force, like gravity, is a“field” force…that is, a force is exertedat a distance despite lack of physicalcontact.
The electrostatic force obeys the
Superposition Principle
This implies that to solve problems withmultiple charges, we may consider eachtwo charge system separately and combinethe results at the end.
Remember, force is a vector quantity,so you must use vector addition!
Example:q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
1) Start by examining the force exerted by q1 on q2.
r Points in the directionfrom 1 to 2!
= (9 X 109 N m2 / C2)(-1 C)(-2 C)/(1 m)2
= +1.8 X 10-2 N (i.e., in the +x direction).
The plus sign indicates the force is repulsive.
x
F1 2
Example (con’t):
q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
2) Then examine the force exerted by q3 on q2.
rPoints in the directionfrom 3 to 2!
= (9 X 109 N m2 / C2)(+1 C)(-2 C)/(2 m)2
= -4.5 X 10-3 N The minus sign indicates the
force is attractive…..therefore it’s in the +x direction.
F3 2
Example (con’t):q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
3) Finally, carefully add together the results.
F2 = F12 + F32 = 1.8 X 10-2 N + 4.5 X 10-3 N
= 2.25 X 10-2 N (i.e., in the +x direction).
x xx
•Gravitational force•Electromagnetic force•Strong nuclear force•Weak nuclear force
How different would life be if therewere more ions around?
We’re used to living in a world dominated by gravity. Let’s compare the force of gravity between two protons to the electrostatic force:
Fg = G mp mp / r2
Fe = k e e / r2
*Note how similar these two look!
Let’s now take the ratio of theelectrostatic force to the gravitationalforce:
Fe k e2 / r2
= Fg G mp
2 / r2
(9 X 109 N m2/C2)(1.6 X 10-19C)2
= (6.7 X 10-11N m2/kg2)(1.67X10-27kg)2
Note: this shouldbe a unit-lessquantity. Right?
Fe 2.3 X 10-28 C2/ C2
= Fg 1.9 X 10-64 kg2/ kg2
= 1.2 X 1036 !!!!!!
Good thing most objects are neutral, eh?
We said that like gravity, the electric force is a
Field Force
The Earth has a gravitational field. Weexperience its effects on a daily basis.In fact, we describe it with the quantity
g = 9.81 m/s2
Does the Earth’s gravity extend to theMoon’s surface?
But is that familiar expression for theEarth’s gravitational field still valid atthe Moon’s surface?
That expression has been derived for conditions at the Earth’s surface. The more general expression for the Earth’s gravitational field is given by...
Ge
e = -Gmr
r2
The minus sign indicates that it is an attractivefield that points toward the Earth’s center...
Ge
e = -Gm
rr2
The familiar g is only one possible value of Ge.
While G and me are constant, if we move away from the Earth (toward the moon, for example),
the magnitude of Ge decreases like 1/r2.
Gravity is a field force -- that is, a force that acts at a distance without requiring physical contact.
Now that we have an expression forthe gravitational field, we can determinethe force on a given mass (like ourselves)from the expression:
F = mG
are you spending so much time on gravity? Didn’t we cover that last semester?
Everything we just did with gravity we can do with electricity, too!
(and I think our intuition about gravity is better…)
An object with a charge (q) produces anelectric field around it given by
E =
k
r 2
q r
So, when we bring a second charge (q) into the neighborhood of an existing charge, the second charge will feel a force due to the electric field of the first charge.
That force is given by….
F = qE
The superposition principle applies to the electric field, too!
So…. E E E Etotal 1 2 3 ...
The proof is rather straightforward…if youbelieve that the electrostatic force obeysthe superposition principle...
Since we have a hard time visualizinga field, it is useful to develop the conceptof field lines. Electric Field Lines indicatethe direction and magnitude of the electricfield at any point in space.
1) Begin on positive charges2) End on negative charges3) Point from positive to negative charge4) Are most dense where the electric field
is the strongest.5) Are the least dense where the electric
field is the weakest.
+ -
Field linesFar apart. Field lines
close together.
Looking at the electric field lines givesus a way to come to understand the 1/r2
nature of the expression for electrostaticforce…
Let’s start in the 2 dimensional world first..
What is the density of lines passing through theblue circle? 8/2rblue
What about the green circle?
8/2rgreen
So in flat-land, the density of lines goesdown by 1/distance away from theconvergence point of the lines.
In the 3-D world we live in, we replacethe circles of flat-land with spheres.
The density of lines passing throughsurrounding shells will decrease like1/distance2, since the surface area ofa sphere is
4r2
Example:q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
1) Start by calculating the electric field of charge q1 at the location of charge q2.
r Points in the directionfrom 1 to 2!
E1 2@
kq
r
(9 10 Nm / C )( 1 C)
(1 m)
= - 9 10 N/ C
1
12
9 2 2
2
3
r r
r
(i.e., in the -x direction)
Example (con’t):
q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
2) Then examine the electric field of q3 at q2.
rPoints in the directionfrom 3 to 2!
E3 2@
kq
r
(9 10 Nm / C )( 1 C)
(2 m)
= + 2.25 10 N/ C
3
22
9 2 2
2
3
r r
r
(i.e., in the -x direction)
Example (con’t):q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
3) Next, use the superposition principle to carefully add together the results.
E E Etot 1 2 3 2
3 3
3
@ @9 10 N/ C 2.25 10 N/ C
= - 11.25 10 N/ C
x x
x (i.e., the electric field at the location of q2 points in the -x direction)
Example (con’t):q1 q2 q3
q1 = -1 C q2 = -2 C q3 = +1 Cr1 = 1 m r2=2 m
What is the Total Force on q2?
r1 r2
y
x
4) Finally, multiply by q2 to get the force...
F q ( 2 C)(-11.25 10 N/ C )
= 2.25 10 N tot 2 tot
3
-2
= E
x
x (i.e., the electric force at the location of q2 points in the +x direction)
What happens to a conductor when youplace it in an electric field and allowit to come to equilibrium?
(Remember, charges are free to movearound on the surface of conductors.)
Econductor
1) no electric field exists inside theconductor.
What if it did?
Then an electrical force would be exerted onthe charges present in the conductor. In agood conductor, charges are free to movearound, and will when a force is exerted on them.If charges are moving around, we are not inequilibrium.
2) Excess charges on an isolatedconductor are found entirelyon its surface.
The 1/r2 nature of the electrostatic repulsiveforce is responsible for this one. The excesscharges are trying to get as far away from oneanother as possible. It turns out, therefore, theyall end up on the surface of the conductor.
3) The electric field just outside of aconductor must be perpendicularto the surface of the conductor.
Again, what if this were not the case?
Then a component of the electric field wouldexist along the conductor’s surface. This wouldyield an electrical force along the surface. As agood conductor, charges would move around inthe presence of the force...
4) On an irregularly shaped conductor,charges build up near the points(regions with smallest curvature).
E
Charges build up here: not as much room for them to move apart!
-- - --
--
-
--
-
--
In both cases, you are advised to THINK about the direction!
|F|k|q ||q |
r1 2
2
| |k|q|
r2E
E =
k
r 2
q r
F
kq q
r1 22
r
What is ?r
To determine the direction of the , simply join point A to point B. The sign of their charges does not matter!
r
AB
r
Just connect the dots! pointsfrom the charge creating the fieldtoward the charge of interestregardless of their signs!
r
If you use the book’s formulation, you must take the absolute values of the signs and THINK about the direction of the vector quantities you are calculating!!!
Use whatever works for you!
When you use (and only when you use this formulation) the signs of the charges become mathematically meaningful in the formulae you apply to the problems!
r
