Magnetism questions to review

1. In general, what causes magnetism?

2. What is electromagnetism?

3. What is the peculiarity involving magnetic north?

4. How could you find true (geographic) north?

5. What is a motor and how does it basically work?

6. What is electromagnetic induction?

7. What is a generator?

8. How do compasses respond to magnetic fields?

Final exam topics

The final exam is Thursday, December 10 at 7:30.  Note the later starting time.

PHONES AND CALCULATORS ARE NOT PERMITTED (OR NEEDED) ON THIS TEST.

The sheet of notes is still permitted.

The topics for the final exam are:

electrical charge
proton, neutron, electron, quark - particles
atomic number and elements
charging things - what happens
voltage
current
resistance
units of V, I and R
series circuit
parallel circuit
basics of circuits
bulb brightness predictions - it's related to current
V = I R
basic electrical schematics (and symbols - battery, resistor, wire)
magnetism
electromagnetism
electromagnetic induction
motors
compasses
finding north
generators (vs. motors)

You should definitely review demonstrations involving charge, light bulb brightness and removal, compasses, motors, etc.

Magnetism

Magnetism!

Some ideas from the Magnetism classes:

Similar to the case of charge, magnetic poles are divided into North and South poles.

A North magnetic pole is one that points toward the Earth's magnetic north pole.  This means that the Earth's magnetic north is ACTUALLY A SOUTH POLE (magnetically speaking).

Also:

- Like poles repel
- Opposite poles attract
- Each magnet must have at least one North and one South pole (though they may have more than one of each).  There is NO such thing as a magnetic monopole.
- Magnetic fields are real, but the lines are imaginary - Field lines indicate the direction that a compass needle would take in the vicinity of the magnetic field.
- There are naturally occurring magnetic minerals - a very common one is called magnetite (Fe3 O4)

Magnetic north on the Earth is near Ellesmere Island in Northern Canada, several hundred miles from true (geographic) North (the North Pole).  It is moving toward Russia at several miles per year.

For gory detail:

http://en.wikipedia.org/wiki/North_Magnetic_Pole

To find True/Geographic north, it is easiest to find Polaris (the current north star).  Polaris is actually not all that bright, though in the top 50 brightest stars in the night sky.  You need to find the Big Dipper (asterism at the rear end of Ursa Major).  Follow the “pointer stars” at the end of the dipper.  These visually lead you to Polaris.  [If you were to follow the “arc” of the handle, you’d come to a bright star, Arcturus – “Follow the arc to Arcturus.”]

FYI:

http://skymaps.com/

How do we get magnetism?

Magnetic fields are related to electrons spins.  Electrons act like tiny magnetic  spinning tops.  There is a tiny magnetic element associated with each electron spin.  If the spins align, more or less, the object is said to be somewhat magnetic.  More spin alignments (domains) means more magnetism.  Materials that do this easily are generally said to be ferromagnetic.  

As it happens, metals do this best (free electrons).  In the core of the Earth, molten metal convects (rises and falls), giving the Earth a good magnetic field – measurable from the surface and beyond.  Several planets have magnetic fields.

In general, the motion of charges leads to magnetic fields.  If you have charge traveling through a wire, electrons can be thought of as moving together – this causes a magnetic field, also known as electromagnetism.  The magnetic field caused by a current passing through a wire is often small, but if you coil the wire upon itself, the magnetic fields “add up”.  Several hundred turns of wire (with current running through it) can produced quite a strong electromagnet. 

A coil with current running through it can naturally react to a permanent magnet – if this is engineered well, we have a motor.  See illustrations and demos in class.

Electromagnetic Induction

Current causes magnetism – something shown in the early 19^th century by Hans Oersted.  As it happens, the reverse is also true – magnetism can cause current, but there must be some relative CHANGE in the magnetic field or location of conductor.   There must be relative change – either coil or magnet must move, relative to the other.

This phenomenon, wherein a change in magnetic field relative to a conductor, generates electric current is called “electromagnetic induction.”  It is the secret to understanding generators.  If something, say moving water from Niagara Falls, can cause a coil of wire (in a turbine) to spin, current is generated.  More spins of wire means more current.

It’s all about moving conductors in magnetic fields

In conclusion:

Electromagnetism:

Current (moving charges) à  Magnetic Field

Electromagnetic Induction:

Change in magnetic field (through conductor), or vice versa à electric current

>

Some images related to the above:

Below:  basic motors

Circuit problems

1.  Describe the difference between voltage, current, and resistance.  Give the proper units, too.

2.  What is the resistance of a light bulb that allows 2 A of current through it when connected to a 12-V battery?  (6 ohms)

3. A 5-ohm resistor is connected to a 10-volt battery. What current passes through the resistor?  (2 amps)

4. Two 100-ohm resistors are in series. What is their total resistance?  (200 ohms)

5.  In general, what is the difference between resistors in series and in parallel?  Recall the light bulb examples and how the brightnesses compare.

6.  Which has more resistance, 2 identical bulbs in series or the same 2 identical bulbs in parallel?

7.  For question 7, which set-up (series or parallel) would "kill" the battery quicker?

8.  You have 2 bulbs in series - remove one (unscrew it) and what happens?

9.  You have 2 bulbs in parallel - remove one (unscrew) and what happens?

10.  Draw the symbols for battery, resistance and wire.  Draw a schematic for 2 resistors in series.  Draw a schematic for 2 resistors in parallel.

11.  What are the main differences between static electricity and what happens in circuits?

Be sure to recall the various light bulb demonstrations.

Circuits - II

OK, so about regular circuits:

The images represent SERIES CIRCUITS and PARALLEL CIRCUITS.

In a series circuit, the current is constant and is set by the total resistance of the circuit (the sum of the resistors). If you remove one resistor (or light bulb, as in the first image), the current stops. If the resistors were identical bulbs, having more bulbs would result in dimmer bulbs, since the battery voltage is distributed among them.  Note that the sum of the voltages "over" the bulbs is equal to the total voltage provided by the battery (give or take some minor losses).  Identical bulbs (or resistors) have identical voltages "over" them - 3 identical bulbs connected to a 9-V battery would have roughly 3-V each over them.

In parallel circuits, current has multiple paths to take, so the total resistance of the circuit is actually LESS than if the resistors were alone or in series with other resistors - see details below. Since the bulbs are connected equally to the battery, they experience the same as the battery voltage - they are, therefore, of equal brightness (and the same brightness they would have if there were only ONE bulb connected). Of course, bulbs in parallel draw more current and thus cause a battery to die sooner.  You could have 10 bulbs or resistors connected in parallel to a battery - each will be as bright as if only 1 were connected to the battery (same voltage over each), though 10 bulbs will kill the battery 10 times faster.

Does this have anything to do with holiday lights?

What I've written above is primarily geared toward identical bulbs. In series, add up the resistances to get the total resistance. In parallel, it is more complicated. There is a formula one can use (1/Rp = 1/R1 + 1/R2 + ...), but we will only concern ourselves with the case of identical resistors in parallel. In that case, divide the value of the resistor by the number of resistors to get the total effective resistance. For example, two identical 50-ohm resistors in parallel is the same as one 25-ohm resistor. This seems strange, but it's a little like toll booths - when one toll booth is open, it can get crowded (the current is small). With multiple toll booths open, the resistance is effectively less, so the current can be greater.

In the first image below, the graphic represents the schematic view of a parallel circuits, with 2 resistors.  Note that 2 possible paths are available for current to take - current runs through EACH path, though there will be more current where there is less resistance.  The total current from the battery is equal to the sum of the currents through the 2 resistors.  It follows V = I R, though the V over each R is the same.  The I through each will therefore be V/R.

The second image illustrates the series circuit concept:  identical resistors in series will effectively give MORE resistance (the sum of the resistances, actually) to the battery, so the current will be LESS (and exactly the same in each resistor or bulb).  It also easily follows V = I R, with more R yielding less I (when V is constant).  Think of V = I R this way:  I = V/R.  More R, less I.

Bulbs in Series - same current through each, but the voltage from the battery "splits"

Bulbs in Parallel - same voltage over each, but the current from the battery "splits"

Introduction to Circuits

Thus far, we have only discussed "static" (stationary) charges.  Static charges alone are useful, but not nearly as much as charges in motion.  As you recall, electrons are the most easily moved particles.  However, for sake of ease in sign convention (positive vs. negative), we define the following:

Current (I) - the rate at which positive charge "flows"

I = Q/t

The unit is the coulomb per second, defined as an ampere (A).  One ampere (or amp) is a tremendous amount of current - more than enough to kill a person.  In fact, you can feel as little as 0.01 A.  Typical currents in a circuit are on the order of mA (milliamperes).

Essentially, current is how quickly charge travels (or charge per time, q/t).  The unit (a coulomb per second) is called the ampere (or amp, A). 

We need to define other new quantities in electricity:  voltage, resistance, power.

Voltage (V) - the amount of available energy per coulomb of charge.  The unit is the joule per coulomb, called a volt (V, in honor of Allesandro Volta, inventer of the battery).

V = E/Q

Batteries and other sources (such as wall sockets) "provide" voltage, which is really a difference between TWO points (marked + and - on a battery).  A wall outlet is a bit more complex - there are 2 prongs, but often also a third prong (the "ground", for safety purposes, through which excess charge can travel back to the Earth).

Resistance (R) - the ratio of voltage applied to an electrical device to the current that results through the device.  Alternately:  the amount by which the voltage is "dropped" per ampere of current.

R = V/I

You can also think of resistance as that which "resists" current.  Typically, resistors are made of things that are semi-conductive (they conduct current, but less well than conductors and better than insulators).  Resistors are often made of carbon, but can also be made of silicon and other materials.  The unit is the volt per ampere, defined as an ohm (Greek symbol omega)

A convenient way to relate all of the variables is embodied in an expression often called Ohm's Law:

V = I R

So, what exactly IS a circuit?

An electrical circuit can be thought of as a complete "loop" through which charge can travel.  Therefore, it actually has to be physically complete - there can be no openings.  That is, the current actually has to have a complete path to take.  I will demonstrate this in class with bulbs and wires; for now, see the image above.

What about power?

Also consider electrical power (P).  Power is the rate at which energy is used or expended:  energy per time.  Symbolically:  P = E / t.  The unit is the joule per second, called a watt (W).  In electricity, power is also given by:

P = I V

P = I^2 R

Power allows us to express the brightness of a bulb.  Consider that a 100-W bulb is brighter than a 60-W bulb.

Some folks like analogies.  Consider a water analogy.  Voltage is like a tank of water (how much water).  Resistance is provided by a drain or faucet.  The rate at which water comes out is the current.  It's only an analogy, but it gets the gist of circuit terminology ok.

https://phet.colorado.edu/en/simulation/circuit-construction-kit-dc

Charge questions

Things to think about:

1.  What exactly *is* charge?  How do we think of it?  How does this relate to protons and electrons, etc.?

2.  Explain the electrostatics demonstrations from class.

3.  Why is it that electrons are the easiest particles to manipulate?

4.  What does atomic number (Hydrogen = 1, Helium = 2, etc.) mean?

5.  What is the most common element, and why?

6.  What are quarks?

Electricity I - Charge!

Charge

- as fundamental to electricity & magnetism as mass is to mechanics

Charge is a concept used to quantatively related "particles" to other particles, in terms of how they affect each other - do they attract or repel?  If so, with what force?

Charge is represented by letter Q.

The basic idea - likes charges repel (- and -, or + and +) and opposite charges attract (+ and -).

Charge is measured in units called coulombs (C).  A coulomb is a huge amount of charge, but a typical particle has a tiny amount of charge:

- the charge of a proton is 1.6 x 10^-19 C.  Similarly, the charge of an electron is the same number, but negative, by definition (-1.6 x 10^-19 C).  The negative sign distinguishes particles from each other, in terms of whether or not they will attract or repel.  The actual sign is arbitrarily chosen.

The charge of a neutron is 0 C, or neutral.


But what IS charge?


Charge is difficult to define.  It is property of particles that describes how particles interact with other particles. 

In general, the terms are negative and positive, with differing amounts of each, quantified as some multiple of the fundamental charge value (e):

e = 1.6 x 10^-19 C

That's hard to visualize, since a coulomb (c) is a huge amount of charge.  One coulomb, for example, is the charge due to:

1 coulomb = charge due to 6.3 x 10^18 protons

A typical cloud prior to lightning may have a few hundred coulombs of charge - that's an enormous amount of excess charge.

If the charge is negative (-), the excess charge is electrons.

If the charge is positive (+), the excess charge is protons - however, we can NOT easily move protons.  That usually takes a particle accelerator.  Typically, things are charged positively by REMOVING electrons, leaving a net charge of positive.

Other things to remember:

Neutral matter contains an equal number of protons and electrons.

The nucleus of any atom contains protons and (usually) neutrons (which carry no charge).  The number of protons in the nucleus is called the atomic number, and it defines the element (H = 1, He = 2, Li = 3).

Electrons "travel" around the nucleus in "orbitals."  See chemistry for details.  The bulk of the atom is empty space.

Like types of charge repel.  Opposite types of charge attract.

The proton is around 2000 times the mass of the electron and makes up (with the neutrons) the bulk of the atom.  This mass difference also explains why the electron orbits the proton, and not the other way around.

Protons in the nucleus of an atom should, one would imagine, repel each other greatly.  As it happens, the nucleus of an atom is held together by the strong nuclear force (particles which are spring-like, called gluons, keep it together).  This also provides what chemists called binding energy, which can be released in nuclear reactions.


COULOMB'S LAW


How particles interact with each other is governed by a physical relationship called Coulomb's Law:

F = k Q1 Q2 / d^2

Or, the force (of attraction or repulsion) is given by a physical constant times the product of the charges, divided by their distance of separation squared.  The proportionality constant (k) is used to make the units work out to measurable amounts.

Note that this is an inverse square relationship, just like gravity.

The "big 3" particles you've heard of are:

proton
neutron
electron

However, only 1 of these (the electron) is "fundamental".  The others are made of fundamental particles called "quarks""

proton = 2 "up quarks" + 1 "down quark"
neutron = 2 "down quarks" + 1 "up quark"

There are actually 6 types of quarks:  up, down, charm, strange, top, & bottom.  The names mean nothing.

Many particles exist, but few are fundamental - incapable of being broken up further (so far as we know).

In addition, "force-carrying" particles called "bosons" exist -- photons, gluons, W and Z particles.

The Standard Model of Particles and Interactions:

http://www.pha.jhu.edu/~dfehling/particle.gif

Optics things to consider pre-test

Be sure to understand these concepts from class:

1.  Focal length/point

2.  How convex lenses and concave lenses redirect light rays

3.  How convex mirrors and concave mirrors redirect light rays

4.  The difference between real and virtual images

5.  The difference between nearsightedness and farsightedness

6.  How light behaves when it hits various types of optics

7.  How a flat/plane mirror works (bathroom mirror)

Exam 2 topics

Exam 2 is primarily a conceptual test.  There is not much math - see * below.


I.  Post-test, pre-wave content

how things fly & the Bernoulli Principle

energy

II.  Waves / Sound (mostly)

waves

- wavelength

- frequency

- speed

-amplitude

- crests and troughs

* wave speed = frequency x wavelength

(same equation as above, but with c for speed, when waves are electromagnetic)

mechanical vs. electromagnetic waves

EM Spectrum - recall the chart and the 7 major types of EM waves

harmonics:  how wavelength changes (* 2L/n), how frequency changes (* it increases linearly as n goes up), and how the speed (v) remains constant

* music - octaves, equal-tempered scale (1.059)

Doppler effect

- red shift, blue shift


III.  Waves / Light

light reflection

light refraction

index of refraction

lenses and mirrors (convex and concave)

real and virtual images

focal length

predicting light paths

electromagnetic spectrum again

What about curved mirrors?

Recall that light reflects from mirrors, according to the law of reflection.  However, it the mirrors is curved, light still obeys this rule - it just looks a bit different.  You have to visualize the curved mirror as a series of little flat mirrors.

A convex mirror (top) acts just like a concave lens - only virtual images are formed.  Think of convenience store mirrors.

 A concave mirror (bottom) acts just like a convex lens.  Think of makeup/shaving mirrors.

How do lenses form images?

Lenses

As shown and discussed in class, light refracts TOWARD a normal line (dotted line on the left image, perpendicular to surface of lens) when entering a more dense medium.

Note in this convex lens that this direction of bend changes from down (with the top ray) to up with the bottom ray. This is due to the geometry of the lens. Look at the picture to make sure that this makes sense.  As a result, the rays will intersect after leaving the lens.  An image can form!


The FOCAL LENGTH (f) of a lens (or curved mirror) where the light rays would intersect, but ONLY IF THEY WERE INITIALLY PARALLEL to each other. Otherwise, they intersect at some other point, or maybe not at all (if the object is too close to be focused on)!

Note that your (human) eye lenses are convex - slightly thicker in the middle.  Thus, your eyes form "real" images on the retina - upside-down!  Unless, of course, the object is too close.

If an image is projected onto a screen, the image is REAL. Convex lenses (fatter in the middle) CAN create real images - the only cases where there are no images for convex lenses are when the object distance (between object and lens) is equal to the f, or when do < f. In the first case, there is NO image at all. In the second case, there is a magnified upright virtual image "inside" the lens.

Concave lenses (thinner in the middle) NEVER create real images and ONLY/ALWAYS create virtual images.

Top image depicts parallel light rays hitting a convex lens and meeting at the "focal point."  A real image forms at the focal length of a convex lens, WHEN THE RAYS ARE INITIALLY PARALLEL.  People who are farsighted wear convex lenses.

The bottom image depicts parallel light rays hitting a concave lens and diverging.  In this case, under all circumstances (regardless of where the object is), only virtual images are formed.  These can not be projected onto a screen - rather, they appear to reside "inside" the lens.  People who are nearsighted wear concave lenses.




However, unless the light rays are exactly parallel (or the object is so far away, like the Sun, so that they are approximately parallel), the light rays do not behave exactly like this.  Rather, they form at a different location.

Extension to curved mirrors:

Convex lenses (which are defined to have a positive focal length) are similar to concave mirrors.

Concave lenses (which are defined to have a negative focal length) are similar to convex mirrors.


Summary

The key thing to note is that whether or not an image forms, and what characteristics that image has, depends on:

- type of lens or mirror
- how far from the lens or mirror the object is

In general, convex lenses (and concave mirrors) CAN form "real" images.  In fact, they always form real images (images that can be projected onto screens) if the object is further away from the lens/mirror than the focal length.   Think of using a magnifying glass to burn leaves - a real image of the Sun is forming on the leaves.

If the object is AT the focal point, NO image will form.

If the object is WITHIN the focal point (less than the focal point), only virtual images (larger ones) will form "inside" the mirror or lens.

Concave lenses and convex mirrors ONLY form virtual images; they NEVER form real images.  Think of convenience store mirrors and glasses for people who are nearsighted.

Extra info, FYI:

The location of images can be predicted by a powerful equation:

1/f = 1/di + 1/do

In this equation, f is the theoretical focal length (determined by the geometry of the lens or mirror), do is the distance between the object and lens (or mirror) and di is the distance from lens (or mirror) to the formed image.

We find several things to be true when experimenting with lenses. If the object distance (do) is:

greater than 2f -- the image is smaller
equal to 2f -- the image is the same size as the object (and is located at a di equal to 2f)
between f and 2f -- the images is larger
at f -- there is NO image
within f -- the image is VIRTUAL (meaning that it can not be projected onto a screen) and it appears to be within the lens (or mirror) itself

Optics HW - 1

1.  Review the concept of reflection, particularly the law of reflection.  Draw what happens when a light ray hits a mirror at various angles.

2.  Review the concept of refraction:  what it is, what causes it, what happens during it, under what circumstances does light bend, etc.  Draw what happens when a light ray hits a block of transparent plastic at various angles.  

3.  Show how to calculate the wavelength of WTMD's signal (89.7 MHz).

4.  Some questions related to how light is affected by optics.

Refraction animation

http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Waves/Refraction/Refraction.html

http://www.animations.physics.unsw.edu.au/jw/light/Snells_law_and_refraction.htm

http://www.freezeray.com/flashFiles/Refraction2.htm

Playing with Light 1 - Reflection and Refraction

Reflection - light "bouncing" off a reflective surface. This obeys a simple law, the law of reflection!

The incident (incoming) angle equals the reflected angle. Angles are generally measured with respect to a "normal" line (line perpendicular to the surface).

Note that this works for curved mirrors as well, though we must think of a the surface as a series of flat surfaces - in this way, we can see that the light can reflect in a different direction, depending on where it hits the surface of the curved mirror. More to come here.

Refraction:

Refraction is much different. In refraction, light enters a NEW medium. In the new medium, the speed changes. We define the extent to which this new medium changes the speed by a simple ratio, the index of refraction:

n = c/v

In this equation, n is the index of refraction (a number always 1 or greater), c is the speed of light (in a vacuum) and v is the speed of light in the new medium.

The index of refraction for some familiar substances:

vacuum, defined as 1

air, approximately 1

water, 1.33

glass, 1.5

polycarbonate ("high index" lenses), 1.67

diamond, 2.2

The index of refraction is a way of expressing how optically dense a medium is. The actual index of refraction (other than in a vacuum) depends on the incoming wavelength. Different wavelengths have slightly different speeds in (non-vacuum) mediums. For example, red slows down by a certain amount, but violet slows down by a slightly lower amount - meaning that red light goes through a material (glass, for example) a bit faster than violet light. Red light exits first.

In addition, different wavelengths of light are "bent" by slightly different amounts. This is trickier to see. We will explore it soon.

Refraction, in gross gory detail

Consider a wave hitting a new medium - one in which is travels more slowly. This would be like light going from air into water. The light has a certain frequency (which is unchangeable, since its set by whatever atomic process causes it to be emitted). The wavelength has a certain amount set by the equation, c = f l, where l is the wavelength (Greek symbol, lambda).

When the wave enters the new medium it is slowed - the speed becomes lower, but the frequency is fixed. Therefore, the wavelength becomes smaller (in a more dense medium).

Note also that the wave becomes "bent." Look at the image above: in order for the wave front to stay together, part of the wave front is slowed before the remaining part of it hits the surface. This necessarily results in a bend.

The general rule - if a wave is going from a lower density medium to one of higher density, the wave is refracted TOWARD the normal (perpendicular to surface) line. See picture above.

Introduction to Light

Recall that waves can be categorized into two major divisions:

Mechanical waves, which require a medium. These include sound, water and waves on a (guitar, etc.) string

Electromagnetic waves, which travel best where there is NO medium (vacuum), though they can typically travel through a medium as well. All electromagnetic waves can be represented on a chart, usually going from low frequency (radio waves) to high frequency (gamma rays). This translates to: long wavelength to short wavelength.

All of these EM waves travel at the same speed in a vacuum: the speed of light (c). Thus, the standard wave velocity equation becomes:

c = f l

where c is the speed of light (3 x 10^8 m/s), f is frequency (in Hz) and l (which should actually be the Greek letter, lambda) is wavelength (in m).

General breakdown of e/m waves from low frequency (and long wavelength) to high frequency (and short wavelength):

Radio

Microwave

IR (infrared)

Visible (ROYGBV)

UV (ultraviolet)

X-rays

Gamma rays

In detail, particularly the last image:

http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html

http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html

http://www.unihedron.com/projects/spectrum/downloads/full_spectrum.jpg

Don't forget - electromagnetic waves should be distinguished from mechanical waves (sound, water, earthquakes, strings on a guitar/piano/etc.). 

ALL E/M waves (in a vacuum) travel at the SPEED OF LIGHT (c).

The Doppler Effect

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

http://falstad.com/mathphysics.html

Run the Ripple tank applet -

http://falstad.com/ripple/

The key in the Doppler effect is that motion makes the "detected" or "perceived" frequencies higher or lower.

If the source is moving toward you, you detect/measure a higher frequency - this is called a BLUE SHIFT.

If the source is moving away from you, you detect/measure a lower frequency - this is called a RED SHIFT. Distant galaxies in the universe are moving away from us, as determined by their red shifts. This indicates that the universe is indeed expanding (first shown by E. Hubble). The 2011 Nobel Prize in Physics went to local physicist Adam Riess (and 2 others) for the discovery of the accelerating expansion of the universe. Awesome stuff!

http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/

It's worth noting that the effect also works in reverse. If you (the detector) move toward a sound-emitter, you'll detect a higher frequency. If you move away from a detector move away from a sound-emitter, you'll detect a lower frequency.

Mind you, these Doppler effects only happen WHILE there is relative motion between source and detector (you).

And they also work for light. In fact, the terms red shift and blue shift refer mainly to light (or other electromagnetic) phenomena.

Wave questions I and II

Wave questions I

1.  Differentiate between mechanical and electromagnetic waves.  Give examples.

2.  Draw a wave and identify the primary parts (wavelength, crest, trough, amplitude).

3.  Find the speed of a 500 Hz wave with a wavelength of 0.4 m.

4.  What is the frequency of a wave that travels at 24 m/s, if 3 full waves fit in a 12-m space?  (Hint:  find the wavelength first.)

5.  Approximately how much greater is the speed of light than the speed of sound?

6.  Harmonics

a.  Draw the first 3 harmonics for a wave on a string.
b.  If the length of the string is 1-m, find the wavelengths of these harmonics.
c.  If the frequency of the first harmonic (n = 1) is 10 Hz, find the frequencies of the next 2 harmonics.
d.  Find the speeds of the 3 harmonics.  Notice a trend?

7.  Show how to compute the wavelength of WTMD's signal (89.7 MHz).  Note that MHz means 'million Hz."

8.  A C-note vibrates at 262 Hz (approximately).  Find the frequencies of the next 2 C's (1 and 2 octaves above this one).

>

(answers)

1, 2.  See notes.

3.  200 m/s

4.  wavelength is 4 m.  Frequency is 6 Hz.

5.  3,000,000 / 340 --- that's around a million to one ratio

6.

a.  see notes

b.  wavelengths are:  2 m, 1 m, and 2/3 m

c.  frequencies are 10, 20 and 30 Hz, respectively, for n = 1, 2 and 3

d.  speeds are all constant:  20 m/s

7.  speed of light divided by 89.7 MHz.  That is 300,000,000 / 89,700,000, which works out to around 3.3 m.

8.  524 Hz and 1048 Hz

>

Wave questions II

Consider the musical note G, 392 Hz.  Find the following:

1.  The frequencies of the next two G's, one and two octaves above.

2.  The frequency of the G one octave lower than 392 Hz.

3.  The frequency of G#, one semi-tone (piano key or guitar fret) above this G.

4.  The frequency of A#, 3 semi-tones above G.

5.  The wavelength of the 392 Hz sound wave, assuming that the speed of sound is 340 m/s.

6.  What are the differences between longitudinal and transverse waves?  Gives examples of each.  What type of wave is sound?


Also for your consideration.  Understand the following concepts:

a.  harmonics on a string

b.  how waves form in a tube - what actually happens with the air inside?

c.  Here's a thought question for you - why does breathing in helium make your voice higher?

answers:

1.  392 x 2; 392 x 4

2.  392/2

3.  392 x 1.0594

4.  392 x 1.0594 x 1.0594 x 1.0594  (or 392 x 1.0594^3)

5.  340/392

6.  See notes.

Music

In western music, we use an "equal tempered (or well tempered) scale."  It has a few noteworthy characteristics;

The octave is defined as a doubling (or halving) of a frequency.

You may have seen a keyboard before.  The notes are, beginning with C (the note immediately before the pair of black keys):

C

C#

D

D#

E

F

F#

G

G#

A

A#

B

C

(Yes, I could also say D-flat instead of C#, but I don't have a flat symbol on the keyboard.  And I don't want to split hairs over sharps and flats - it's not that important at the moment.)

There are 13 notes here, but only 12 "jumps" to go from C to the next C above it (one octave higher).  Here's the problem.  If there are 12 jumps to get to a factor of 2 (in frequency), making an octave, how do you get from one note to the next note on the piano?  (This is called a "half-step" or "semi-tone".)

The well-tempered scale says that each note has a frequency equal to a particular number multiplied by the frequency that comes before it.  In other words, to go from C to C#, multiply the frequency of the C by a particular number.

So, what is this number?  Well, it's the number that, when multiplied by itself 12 times, will give 2.  In other words, it's the 12th root of 2 - or 2 to the 1/12 power.  That is around 1.0594.

So to go from one note to the next note on the piano or fretboard, multiply the first note by 1.0594.  To go TWO semi-tones up, multiply by 1.0594 again - or multiply the first note by 1.0594^2.  Got it?