Chapter 25 The Eye Simple Magnifiers Angular Magnification

Chapter 25 The Eye Simple Magnifiers Angular Magnification The Telescope

Optical Instruments We’re going to take a quick, 1 -lecture look at the material in this chapter. There’s little new here--just an application of what we already know. But here is a look at the practical side of optics.

The human eye is really just another, relatively simple optical system. The eye takes advantage of refraction to focus light from objects in our field of view onto the retina. Normal Eye Incoming light rays pass through the cornea and pupil of the eye, where they then intersect the lens, which focuses the light onto the retina.

The normal eye is able to comfortably focus objects at distances from the far point (infinity) to as close as the near point (about 25 cm). As you are no doubt aware, two common defects plague human vision: Hyperopia (or farsightedness) Myopia (or nearsightedness) Each of these problems results in blurred vision and is a result of the inability of the eye to properly focus light on the retina.

Hyperopia For the farsighted person, the lens of the eye focuses the light at a point beyond the back of the retina. Objects at a great distance from the observer appear fairly clear, but objects close to the observer are blurred.

Myopia For the nearsighted person, the lens of the eye focuses the light at a point in front of the retina. Objects close to the observer appear fairly clear, but objects far from the observer are blurred.

We can correct these optical defects quite simply by prescribing corrective lenses to be placed in front of the eye. What type of lens will we want to put in front of an eye suffering from hyperopia? Hyperopia

converging lens Helps out the eye with hyperopia by starting to bring the light to a focus before it encounters the lens of the eye.

diverging lens Helps out the eye with myopia by separating the incoming light before it encounters the abnormally strong focusing lens of the eye.

Optometrists and opthalmologists usually prescribe lenses by describing the power of the corrective lens. = 1 diopter

The angular magnification of an object is simply the ratio of the angle subtended by the object at your eye when magnified to that angle when unmagnified and placed at the near point. A diagram will help here. . .

h q 0 25 cm object at the near point p h’ q image at the near point 25 cm Object just outside of the focal point of the converging lens

Our simple magnifier has a focal length less than 25 cm and will produce an image with an angular magnification given by: This ratio achieves its maximum value in the geometry of the last slide (where the image appears at the near point). In that case, the object should be placed at where p and f are both measured in cm.

When the object is placed at the distance p in front of the simple magnifier, the maximum angular magnification occurs. The simple magnifier is a useful device. Not only do we find it in magnifying glasses, but they also serve as the eyepieces for microscopes and telescopes!

Let’s look at how a telescope works a little more closely. There are two principle types of optical telescopes: reflecting and refracting. Reflecting telescopes use a mirror to bring distant light to a focus. A lens (the eyepiece) is then used to magnify the image formed by the primary mirror. Refracting telescopes use a lens to bring distant light to a focus and a second lens (the eyepiece) to magnify the image formed by the first lens.

Let’s look in more detail at a refracting telescope (though everything we derive here will also be applicable to the reflecting telescope as well). e light c e e v from i i t p the o c e e bjec y j e b t o q 0 fo fe In general, the objects at which we peer through telescopes are a LONG distance from us. So, let’s assume an object distance of infinity.

light from the o bjec t e g a m e i h’ c e e e i it v ght p c c i e e he y je j e b b o o q 0 final image q fo fe angular magnification

h’ qo q fo fe

Electrical Forces and Fields We’re now going to start down the path of examining forces whose origin is not visible to us. We call the quality of matter responsible for these forces “charge, ” a substance never directly observed in the history of the human race!

Attraction in Nature Amber (elektron in Greek) attracts straw/feathers when rubbed (observed by Thales of Miletus ~ 600 B. C. ). Iron ore from the country of Magnesia seemed to have a natural affinity for metals. When released, all objects seem to fall toward the ground.

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 Earth itself! Stephen Gray (early 1700’s) shows that the electrostatic attractive and repulsive forces can be transferred through contact alone…Metals didn’t need to be rubbed.

Two Kinds of Charge Charles Du Fay (1698 - 1739) first postulated the existence of two distinct kinds of electricity: vitreous (the glass rod) and resinous (the silk). But thought they existed together in most matter and when separated through friction, resulted in an electrical force. Benjamin Franklin (1706 - 1790) hypothesized a one-fluid model of electricity: charge is transferred from one body to another (e. g. through rubbing); but the total charge on the two bodies combined remains the same. This theory is known as the… Conservation 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. Hence, the glass rod (vitreous) was positive when rubbed with silk while the amber (resinous) was negative when rubbed with wool. 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!

Charge is Quantized Robert Millikan (1868 - 1953) in a very clever experiment showed that electrical charge came in quantized units. In other words, charge of 0, +/- 1 e, +/- 2 e, +/- 3 e, . . . +/- ne (where n is an integer) could be observed, but never a charge of 1. 5 e (see section 15. 7). An electron carries a charge of -1 e. A proton carries a charge of +1 e.

• 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?

Hydrogen atom P+ e- Electron Cloud Neutral Space

Electrically Neutral. . . Most objects in nature are electrically neutral (i. e. they contain an equal number of protons and electrons). Ne = N p Therefore most objects exert no electrical force on the objects around them. Atoms in which Ne< Np or Ne> Np are called ions.

Insulators and Conductors Objects on which electrons move freely are known as conductors. Most metals are good conductors. Electrical wires are made of good conductors. Objects on which charges do not move freely are known as insulators. Glass, amber, rubber, silk and cloth are all examples of good 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?

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 the conductor toward our hand Eventually ending up in the ground. GROUND

COPPER Rub charges on here A good conductor distributes the charge uniformly over its surface. Rubber glove insulates copper rod from us and therefore the ground. GROUND

So, we can charge up a conductor! And human beings must be fairly good conductors of electricity, too. Notice that the Earth’s surface (ground) acts as a vast source and sink for electrical charge. Touching a conductor to ground will neutralize the charge on the conductor: If the conductor is positively charged, electrons flow from the ground to the conductor. If the conductor is negatively charged, electrons flow off the conductor into the Earth.

So when we talk about an object being grounded, we literally mean that it is connected via a conductor to the Earth’s surface. All electrical outlets now have a ground prong. And most electrical devices use a 3 -prong plug that requires the ground connection.

Well…. just in case the device malfunctions, it’s nice to be able to siphon off the excess electrical charges to ground rather than allowing them to accumulate in the device. If they build up in the device, they will eventually 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’s ability to accept and provide charges to place a net charge on a conductor…. Here’s how you do it!

Rubber COPPER Bring negatively charged rubber ball close to the a copper rod. The copper rod is initially neutral. Negative charges on the copper run away from the rubber ball and into the ground. GROUND

Rubber + ++ 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. GROUND

Rubber + ++ Finally, remove the COPPER + + rubber ball. . . + + Put a rubber glove on your hand to insolate the copper rod from ground. GROUND

The excess positive charge is trapped on the copper rod with + no path to ground. It COPPER + + + redistributes itself + uniformly over the + + copper rod. We have taken an initially neutral copper rod and induced a positive charge on it! GROUND

Note on quantifying charges… The mks unit for charge is the Coulomb, named after Charles Coulomb in honor of his many contributions in the field of electricity. 1 e = 1. 6 X 10 -19 C

Joseph Priestly and Charles Coulomb set out to quantify the electrical force in the late 1700’s. Let’s see if we can figure it out an expression for the electric force ourselves. . .

F = ? ? ? Distance 1/r 2 (units of 1/m 2) Charge on Body 1 q 1 (units of C) Charge on Body 2 q 2 (units of C) Anything else? Just a constant of proportionality… Let’s call it k.

What do we have so far. . .

The arrow indicates that F is a vector quantity (i. e. , to specify F, you need both magnitude and direction)! Indicates a direction radially away from the center of our coordinate system. It could be the x-direction, the y-direction, or something in between.

[F] = [k] [q 1] [q 2] / [r]2 N = [k] C 2/m 2 [k] = N m 2/C 2!

The value of k is determined experimentally to be… k = 8. 99 X 109 N m 2/C 2 Let’s call it 9 X 109 N m 2/C 2 We have now determined a quantitative expression for the electrostatic force!

Notes on Coulomb’s Law: Applies only to point charges, particles or spherical charge distributions. Obeys Newton’s 3 rd Law. The electrical force, like gravity, is a “field” force…that is, a force is exerted at a distance despite lack of physical contact.

The electrostatic force obeys the Superposition Principle This implies that to solve problems with multiple charges, we may consider each two charge system separately and combine the results at the end. Remember, force is a vector quantity, so you must use vector addition!

Points in the direction from 1 to 2! Example: q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 1) Start by examining the force exerted by q 1 on q 2. = (9 X 109 N m 2 / C 2)(-1 m. C)(-2 m. C)/(1 m)2 = +1. 8 X 10 -2 N (i. e. , in the +x direction). The plus sign indicates the force is repulsive.

Points in the direction from 3 to 2! Example (con’t): q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 2) Then examine the force exerted by q 3 on q 2. = (9 X 109 N m 2 / C 2)(+1 m. C)(-2 m. 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.

Example (con’t): q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 3) Finally, carefully add together the results. F 2 = F 12 + F 32 = 1. 8 X 10 -2 N + 4. 5 X 10 -3 N = 2. 25 X 10 -2 N (i. e. , in the +x direction).

• Gravitational force • Electromagnetic force • Strong nuclear force • Weak nuclear force

How different would life be if there were 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 m p mp / r 2 Fe = k e e / r 2 *Note how similar these two look!

Let’s now take the ratio of the electrostatic force to the gravitational force: Fe k e 2 / r 2 = Fg G m p 2 / r 2 Note: this should be a unit-less quantity. Right? (9 X 109 N m 2/C 2)(1. 6 X 10 -19 C)2 = (6. 7 X 10 -11 N m 2/kg 2)(1. 67 X 10 -27 kg)2

Fe 2. 3 X 10 -28 C 2/ C 2 = Fg 1. 9 X 10 -64 kg 2/ kg 2 = 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. We experience its effects on a daily basis. In fact, we describe it with the quantity g = 9. 81 m/s 2 Does the Earth’s gravity extend to the Moon’s surface?

But is that familiar expression for the Earth’s gravitational field still valid at the 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. . . The minus sign indicates that it is an attractive field that points toward the Earth’s center. . .

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/r 2. Gravity is a field force -- that is, a force that acts at a distance without requiring physical contact.

Now that we have an expression for the gravitational field, we can determine the force on a given mass (like ourselves) from the expression: 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 an electric field around it given by

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….

The superposition principle applies to the electric field, too! So…. The proof is rather straightforward…if you believe that the electrostatic force obeys the superposition principle. . .

Since we have a hard time visualizing a field, it is useful to develop the concept of field lines. Electric Field Lines indicate the direction and magnitude of the electric field at any point in space.

1) Begin on positive charges 2) End on negative charges 3) Point from positive to negative charge 4) Are most dense where the electric field is the strongest. 5) Are the least dense where the electric field is the weakest.

+ Field lines Far apart. - Field lines close together.

Looking at the electric field lines gives us a way to come to understand the 1/r 2 nature of the expression for electrostatic force… Let’s start in the 2 dimensional world first. . What is the density of lines passing through the blue circle? 8/2 prblue What about the green circle? 8/2 prgreen

So in flat-land, the density of lines goes down by 1/distance away from the convergence point of the lines. In the 3 -D world we live in, we replace the circles of flat-land with spheres. The density of lines passing through surrounding shells will decrease like 1/distance 2, since the surface area of a sphere is 4 pr 2

Points in the direction from 1 to 2! Example: q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 1) Start by calculating the electric field of charge q 1 at the location of charge q 2. (i. e. , in the -x direction)

Points in the direction from 3 to 2! Example (con’t): q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 2) Then examine the electric field of q 3 at q 2. (i. e. , in the -x direction)

Example (con’t): q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 3) Next, use the superposition principle to carefully add together the results. (i. e. , the electric field at the location of q 2 points in the -x direction)

Example (con’t): q 1 r 1 q 2 r 2 q 3 y x q 1 = -1 m. C q 2 = -2 m. C q 3 = +1 m. C r 1 = 1 m r 2=2 m What is the Total Force on q 2? 4) Finally, multiply by q 2 to get the force. . . (i. e. , the electric force at the location of q 2 points in the +x direction)

What happens to a conductor when you place it in an electric field and allow it to come to equilibrium? E conductor (Remember, charges are free to move around on the surface of conductors. )

1) no electric field exists inside the conductor. What if it did? Then an electrical force would be exerted on the charges present in the conductor. In a good conductor, charges are free to move around, and will when a force is exerted on them. If charges are moving around, we are not in equilibrium.

2) Excess charges on an isolated conductor are found entirely on its surface. The 1/r 2 nature of the electrostatic repulsive force is responsible for this one. The excess charges are trying to get as far away from one another as possible. It turns out, therefore, they all end up on the surface of the conductor.

3) The electric field just outside of a conductor must be perpendicular to the surface of the conductor. Again, what if this were not the case? Then a component of the electric field would exist along the conductor’s surface. This would yield an electrical force along the surface. As a good conductor, charges would move around in the 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!

What is ? To determine the direction of the , simply join point A to point B. The sign of their charges does not matter! B A Just connect the dots! points from the charge creating the field toward the charge of interest regardless of their signs!

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! 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!