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CurvedMirrors

Course: PHYS 131, Spring 2011
School: Cuyamaca College
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Word Count: 583

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Mirrors Almost Curved any operation that can be performed with a lens can also be performed with a curved mirror. Because light passes through the lens but bounces off the mirror, concave mirrors produce converging rays with positive focal lengths, and convex mirrors cause ray divergence and have negative focal lengths. Notice, too, that a positive focal length is now to the left of the lens and a negative f is to...

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Mirrors Almost Curved any operation that can be performed with a lens can also be performed with a curved mirror. Because light passes through the lens but bounces off the mirror, concave mirrors produce converging rays with positive focal lengths, and convex mirrors cause ray divergence and have negative focal lengths. Notice, too, that a positive focal length is now to the left of the lens and a negative f is to the right. Object and image distances must also be positive to the left because since the rays bounce off the mirror rather than passing through a lens. An ideal shape for focusing parallel rays would be a parabolic reflector, but there is no universally best shape for all situations, so spherical surfaces will be used again. Since there is now only one radius of curvature to keep track of and no index of refraction, the focal length is simply f = r/2 where the r is negative if the reflecting surface is convex. The rest of the equations are exactly as before: l = lf /(l f ), m = - l/l, and y =m y. Negative image distances are still called virtual, and so on. It is instructive to compare a few interesting situations that can occur for both mirrors and lenses. The first is when the object is placed one focal length in front of a converging mirror or lens. In each case, the thin lens equation predicts an image distance of infinity. This is just parallel-ray convergence run backwards: the outgoing rays are now parallel. The second case is when the object is placed two focal lengths in front of the same devices. For the lens, the image forms exactly two focal lengths from the side other of the lens, making the situation symmetrical. The image is inverted, but neither enlarged nor reduced. For the mirror, 2f is the radius of curvature. Every ray bounces right straight back to the object. The third case is when the radii of curvature are infinite, which makes the lens an ordinary pane of glass and the mirror flat, too. The equation becomes 1/l + 1/l = 0, which introduces a negative sign when you solve for the image distance and shows that mirrors turn object points far to the left into image points far to the right. This is called depth inversion and explains why a mirror image of your right hand when you wave at your own reflection seems to be a left hand; it has been inverted front-to-back, but it still has all the distinct features of your right hand. For the lens, meanwhile, we can get even more profound by noting that the thin lens equation should really be n/l + n/l = 1/f , where the indices in the object and image regions might not both be that of air. Now when f = , the equation describes the apparent depth of an object in a different medium, separated by a flat interface: l = - n' n l. The easiest way to remember how to use this equation is, Its deeper than it looks, referring to a pool of water. The apparent (image) distance to the bottom, l, as seen from directly above in air is the actual (object) distance l multiplied by the n for air and divided by n for water (with a minus sign for the fact that the bottom still appears to be below the surface. For pools of water, the apparent depth is only about three-fourths of the actual depth.
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Cuyamaca College - PHYS - 131
Decay SeriesOne very useful way to display all the known57isotopes is to create a graph with the number of56protons (Z) along the horizontal axis and thenumber of neutrons, which we will give the55symbol N, along the vertical axis. The stable54e
Cuyamaca College - PHYS - 131
DiffractionA kind of interference occurs whenever waves pass through very small openings in abarrier. On the far side, they spread out in expanding, round fronts that are capable ofinteracting with each other, resulting in an interference phenomenon ca
Cuyamaca College - PHYS - 131
DispersionChromatic dispersion occurs when the refractive index in a material is different fordifferent wavelengths of light. The reason this happens is that no known medium except acomplete vacuum is transparent to all electromagnetic waves. Glass, li
Cuyamaca College - PHYS - 131
Electric Force on an ElectronBecause electrons are extremely small and often escape atoms to move freely, they areusually treated as the idealization of a point-charge. Problems typically involve anelectron placed in a uniform electric field and allowe
Cuyamaca College - PHYS - 131
Superposition of Gausss LawElectric fields vectors from multiple sources simply add up at every location of interest.This is called superposition.As a first example, consider a pair of pointcharges placed as shown in the figure to the right.Using E=k
Cuyamaca College - PHYS - 131
Electromagnetic Intensity!!Previously, it has beenEBshown that when a linearelectric field varies in onlyone direction in space, aperpendicular magnetic fieldis created according todE/dx = dB/dt, and that theend result is a pair ofEM wavetra
Cuyamaca College - PHYS - 131
Electromagnetic SpectrumMaxwells theory predicts thatelectromagnetic waves travel ata speed of c = 300Mm/s anddiffer only in frequency andwavelength. The visiblespectrum of light is only a tinyportion of the totalelectromagnetic spectrum. Thefigu
Cuyamaca College - PHYS - 131
Electromagnetic WavesThe main reason Maxwells equations are named after himis that he incorporated them into a theory that predicted theexistence of electromagnetic waves. Suppose thatsomewhere out in space an electric field exists that isoriented pu
Cuyamaca College - PHYS - 131
AC Circuits!Z = R 2 + X 2 tanP = VIcos ()!R = R!0!X = ! 1 #90-1(X/R)&quot;C!X = ! L&quot;90c = 1/ LC! !V = IZAtrig ( t + ) ACircuitsI=dQ/dtR=d/AV=IRP = VIRRR' = 1 2R1 + R2C=A/dV=Q/CU=CVV=V e-t/V=V ( 1e-t/ )= R CV/V = N/NIN=INGausss
Cuyamaca College - PHYS - 131
Radiation ExposureWhen an atomic nucleus releases a radioactive particle, one radioactive count occurs.Non-radioactive substances dont ever release any counts, so the important measurementis how many counts per second a substance releases. This activit
Cuyamaca College - PHYS - 131
Eyesight OpticsModeled very simply, the human eye consists of a variable focallength convex lens that is intended to adjust itself so that,regardless of the object distance, the image distance remainslensabout 2.5cm in order to project a clear image
Cuyamaca College - PHYS - 131
Depicting Electrostatic Field Lines1. Field lines never cross over each other. If they did, it would mean that the electrostaticforce at the crossing point was in two directions at once.2. Field lines always emerge from positive charges and end onnega
Cuyamaca College - PHYS - 131
Cuyamaca College - PHYS - 131
Fission and FusionRadioactive decay is an example of fission, the splitting of a nucleus into smaller pieces.There is another form of this process: A heavy atom can be struck by a high-speedparticle, such as a neutron, contributing enough energy to cau
Cuyamaca College - PHYS - 131
Focal LengthIf parallel rays of light in a fast medium encounter ahemispherical interface to a slower medium, they willbe redirected to converge more or less on a singlefocal point. While the rays do not all perfectlyconverge on the same pointbecause
Cuyamaca College - PHYS - 131
Gausss LawThe equation for Gausss Law is E = Qin/A (where = 910), but using it is tricky.For each of the three major shapesplanes, cylinders (including lines), and spheresanimaginary Gaussian surface encloses them at a distance from the center r and th
Cuyamaca College - PHYS - 131
Half-LifeAn unstable nucleus can wait a relatively long time before it gets around to decaying.After it does so, it typically becomes a different element that may or may not beradioactive itself. Since the amount of time that passes before decay occurs
Cuyamaca College - PHYS - 131
The Hydrogen Electron CloudIn spherical coordinates for three-dimensional problems, Schrdingers equation becomes( r r )r / r + (sin ) / rsin + / rsin = -2m(E U) / , where everyincidence of a subscript r, , or indicates a partial derivative by that vari
Cuyamaca College - PHYS - 131
Induced CurrentThe magnitude of the induced current flow in a coil subjected to changing magnetic fluxcan be calculated from I = V/R where V = -N/t, but predicting the direction of thatcurrent flow can be tricky. As the current induced in the coil circ
Cuyamaca College - PHYS - 131
InductanceSuppose that a conductor moves through a uniform magnetic field asshown. The magnetic force on the electrons in it is F = qvBsin . + They pile up at one end until the force from the electric field theycreate (F = qE) is strong enough to bal
Cuyamaca College - PHYS - 131
InductorsNot only can a current be induced in a coil by an external magnetic flux changing withtime (V=-N/t), but a coil can induce a current in itself using the magnetic field itgenerates when a current is passing through it. This is by no mean perpet
Cuyamaca College - PHYS - 131
IsotopesSince it is known that a given element (identified by its atomic number Z) can have avarying number of neutrons altering the atomic mass A, a notation is required todistinguish these isotopes. Hydrogen, for example, has three main isotopes. All
Cuyamaca College - PHYS - 131
Kirchhoffs LawsKirchhoffs laws are two mathematical rules used in electrical circuit analysis. The first,known as Kirchhoffs Voltage Law (KVL) states that the sum of the voltages around anyloop in an electric circuit must be zero. Basically, this means
Cuyamaca College - PHYS - 131
Light Bulb ProblemsLight bulb problems have become extremely common on many standardized tests. Forthe most part, these problems can be solved like any other resistive circuit problem,treating the light bulbs as resistors. Usually, the problems involve
Cuyamaca College - PHYS - 131
Magnetic ForceAs with electric fields, magnetic fields can exert force on charged particles such as!!protons and electrons. The force is F = qv ! B where q is the charge on the particle in!!coulombs, v is the particles velocity vector in m/s, and B
Cuyamaca College - PHYS - 131
MagnetismWhenever electrical current flows, a measurable effect called magnetism is produced inthe vicinity of that current. For straight conductors, the magnetic field circles around thecurrent in a direction dictated by the Right-Hand Rule (RHR). The
Cuyamaca College - PHYS - 131
Maxwells EquationsMaxwells Equations arent really Maxwellswith the possible exception of the lastone. They are Gausss, Faradays, and Ampres, and you have mostly already beenintroduced to them: Gausss law says that the electric field through an enclosin
Cuyamaca College - PHYS - 131
Multiple-Lens SystemsIn most situations, one lens is not enough to build a useful optical device. Because of this,it is necessary to be able to make predictions about the final image location when rays oflight go through two or more thin lenses. The so
Cuyamaca College - PHYS - 131
Nuclear PhysicsAlpha decay is essentially limited to the radioactive elements more massive than lead. Itis as though they are trying to lose as much mass as possible to move downward inatomic number toward elements that have stable forms. We already kn
Cuyamaca College - PHYS - 131
The NucleusErnest Rutherfords proposal that the atom consisted of negatively charged electronsorbiting a positively charged nucleus triggered a great deal of curiosity about thestructure of that nucleus. It is important to understand how small it reall
Cuyamaca College - PHYS - 131
Physics 131 Practice Test 1fieldpoint1. Find the electric field vector at the point shown.Use k = 910.3my4mx-125nC2. What is the electric field 10cm from the central axis of a long, uniform metal bar uponwhich a total charge of 120C has been pl
Cuyamaca College - PHYS - 131
The Photoelectric EffectThe exploration of interference phenomena had indicated that light existed as wavesincontrast to gases, for example, which were clearly composed of discrete particles. Rightat the start of the twentieth century, however, experim
Cuyamaca College - PHYS - 131
PolarizationAlthough we know that the E-field, B-field, andBBEPoynting vector of an EM wave are all perpendicularto each other, that does not mean that any twoEparallel EM waves must have their electric fieldsEaligned in the same direction. The
Cuyamaca College - PHYS - 131
Practice Exam 21. Light with a wavelength of 600nm passes through the double-slitshown. If the distance to on the projection screen from the center to thefirst-order maximum is 0.1m, how far is the screen from the double-slit?60000nm-50mC2. Find the
Cuyamaca College - PHYS - 131
Practice Test 1 Answers1. (-36x -27y )N/C2. 0 (Charge all migrates to the surface of metal.)3. (9/0)r N/C4. 5mN5. 4F6. 50mW7. A,D,B,C or A,D,C,B8. 4e -t/0.6s V9. 290 A10. (-500y -1000z )q0 N
Cuyamaca College - PHYS - 131
Quantum NumbersSo far, we are aware of three integer quantum numbers for atoms: the principal (n &gt; 0),the angular (0 l &lt; n), and the magnetic (-l m l ). It was subsequently shown thatthere must be a fourth quantum number bearing a separate resonance of
Cuyamaca College - PHYS - 131
(Quiz 01 was simply attendance on the first day.)First find |E|=kQ/r. This requires finding r = (x + y) = (8 + 6) = 100 = 10.So |E| = -kQ/100. (Note that Q has been replaced with -Q as specified.Next, Ex = |E|(x/r) = (-kQ/100)(-8/10) = +8kQ/1000 = kQ/1
Cuyamaca College - PHYS - 131
(Quiz 01 was simply attendance on the first day.)First find |E|=kQ/r. This requires finding r = (x + y) = (8 + 6) = 100 = 10.So |E| = -kQ/100. (Note that Q has been replaced with -Q as specified.Next, Ex = |E|(x/r) = (-kQ/100)(-8/10) = +8kQ/1000 = kQ/1
Cuyamaca College - PHYS - 131
RC CircuitsA circuit consisting of one resistor and one capacitor can exhibit exponential behaviordue to the time it takes to charge the capacitor and to the resistors prevention ofunlimited current flow. The curves shown below are intended to remind y
Cuyamaca College - PHYS - 131
Rays and ReflectionA large number of electromagnetic waves all moving inthe same direction at the same time constitute what iscalled a wave front. Since each wave within it consists ofa small bundle of wavelets that were emitted together as apulse, i
Cuyamaca College - PHYS - 131
Refraction EffectsSuppose, for ease of calculation, that we have a sample of a material with an index ofrefraction of exactly 2, and we are observing the effects of shining a laser out of thematerial to the surrounding air (n = 1) at various angles. As
Cuyamaca College - PHYS - 131
Relativistic VelocityAt the core of physics, at least in principle, is the notion of logical consistency: A factisnt a fact if it contradicts itself. If we can overcome our natural human resistance toaccepting counterintuitive realities, there is a gre
Cuyamaca College - PHYS - 131
Special RelativityWe have seen that relativity requires a correction to the speed of an object as measuredfrom a different inertial framespecifically, this denominator: 1 vv/c. Notice,however, that for small values of v or v, this corrective factor is
Cuyamaca College - PHYS - 131
Resistive DC CircuitsDirect current (DC) means that the current flow in a circuit is steadyrather than varying in time. Batteries and plug-in chargers for variousdevices are two examples of DC. The usual symbol for a battery isshown at right, along wi
Cuyamaca College - PHYS - 131
ResistorsA resistor is an electrical circuit element made out of a material that conducts electricalcurrent, but not as well as a metal wire. Electrical current (I) is the flow of electricalcharges, and in direct current (DC) circuits, is pictured as g
Cuyamaca College - PHYS - 131
The Rydberg EquationUsing the quantum mechanical resonance model for the electron cloud in atoms, we canunderstand the absorption or emission of photons as resulting from the electron cloudexpanding out to a larger shell or contracting inward to a smal
Cuyamaca College - PHYS - 131
Simplifying Resistive and Capacitive NetworksAny network made purely of resistors or of capacitors can theoretically be simplifieddown to a single equivalent value by replacing each valid series or parallel pair with theappropriate equivalent. It is as
Cuyamaca College - PHYS - 131
Single-Aperture InterferenceWhen coherent light shines through a single small slit onto a screen, the multiple slitdiffraction equation (2l s i n = N ) fails to predict the maxima and minima. The firstproblem is that l was used to describe the distance
Cuyamaca College - PHYS - 131
Thin-Film InterferenceWhen any wave encounters an interface between mediums, there is always a reflectionbackward from the interface whether or not part of the wave continues into the newmedium. Consider a thin film of oil on the surface of a puddle of
Cuyamaca College - PHYS - 131
Thin LensesA lens wouldnt be much good if theonly function it had was to focus orspread parallel rays coming from anear-infinite distance. Fortunately,lenses perform interesting operationson rays under other conditions, too.For instance, a convergi
Cuyamaca College - PHYS - 131
Radiation ToxicityThe image at right depicts a molecule of an extremely toxic substance. Since itsinitial identification in 1917, it has been shown that a dose as little as 10 mg canresult in painful softening and swelling of the bones, yellowing and p
Cuyamaca College - PHYS - 131
TransformersA transformer is an electrical devicethat takes advantage of inductance inorder to convert one voltage/current toanother voltage/current. The basicIconstruction is shown in the figure toIthe right. Two coils of wire are wound+around
Cuyamaca College - PHYS - 131
The Uncertainty PrincipleIf the entire universe consists of waves when viewed on the smallest scales, what is aparticle? A particle is a wave packet. Consider the graph shown below where two sinewaves of slightly different frequency are superposed, cre
Cuyamaca College - PHYS - 131
Using RelativityHaving been shown the odd equations that must beemployed when relativistic speeds are involved, itis now time for you to see how to use them. Oneuseful device for glimpsing the peculiarities ofrelativity is a modified form of a space-
Cuyamaca College - PHYS - 131
Voltage (Electric Potential)If the path by which charged particles follow an electric field does not matter, such as forelectrons in an electrical circuit, it would be more convenient to have a non-vectorquantity that essentially indicates how much pot
Cuyamaca College - PHYS - 131
WaviclesIt has previously been shown to follow directly from Maxwells equations that for anelectromagnetic wave of constant average intensity, the photonic pressure P in emptyspace is given by P = P/Ac, where P is the power in the wave, A is the area,
Cuyamaca College - PHYS - 131
X-Ray SpectraWe have seen that hydrogen emission lines come in groups, and that the shortestwavelengths of light it can produce fall in the UV portion of the spectrum. Clearly, x-raywavelengths would have to come from other atoms. In fact, the Rydberg
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