# Physics 122 Lecture Summaries, Spring 2006

## Week 1

### Thursday January 26, 2006

We went over the syllabus in class. A really brief review of some basic ideas covered last semester (vectors, forces, Newton's laws, conservations of energy, conservation of momentum). The crucial role of experiment in determining the validity of our scientific models. The four fundamental forces of nature: the Gravitational, Electromagnetic, strong, and weak forces. Covered gravitation last semester, most of this semester will deal with electromagnetic interactions, maybe mention the strong and weak forces at the end of the semester. We estimated the total number of gallons of gas used in the US by people driving cars to be ~100 billion gallons!! The actual number is something like 150 billion.

## Week 2

### Tuesday January 30, 2006

Vibrations and waves. Spent the entire class discussing the oscillating mass/spring system. We tried to understand the origin of Hooke's Law (F=-kx) by looking at the demo of a mass hanging vertically from a spring. When the spring is stretched or compressed, the force is proportional to the distance the spring was stretched or compressed but in the opposite direction. This is an example of a restoring force. In this case, there is an equilibrium point, and when the mass is moved away from equilibrium, the force is such that it pulls the mass back in the direction of the equilibrium. This results in a bounded motion, with the mass oscillating back and forth between two points each an equal distance from the equilibrium.We compared this to the case of unbounded motion where the mass, once moving away from equilibrium, continues moving away indefinitely. The type of motion described here is Simple Harmonic Motion (SHM). This is an extremely important concept in physics, with many problems in physics often being reduced to SHM. SHM is characterized by the amplitude (A), period (T) or frequency (f). The amplitude corresponds to the maximum displacement of the mass, the period is given by the length of time required to complete 1 cycle of the oscillation (i.e. to return to the starting point of the oscillation). The frequency is the inverse of the period (f = 1/T), and corresponds to the number of cycles per second.
We derived the expression for the potential energy stored in a spring (PE = 1/2 kx^2) and worked out an example illustrating the conservation of energy and how to do calculations involving energy losses due to friction.
Finally, by making a comparison with uniform circular motion, we derived the period of the mass/spring system. We found that T = 2*pi*sqrt(m/k). This tells us that if we increase the mass, the period will increase, while for a larger spring constant, the period is smaller. One important point to note here is that the period does not depend on the amplitude of the oscillation.

### Thursday February 2, 2006

More simple harmonic motion. Returned to the analogy between the motion of the mass on a spring to derive an expression for the position of the mass as a function of time x(t) = A cos (wt). Where w = 2*pi*f and has units of radians/sec, f is the frequency (number of cycles per second), has units of 1/s or Hz, and is the inverse of the period (time to complete one cycle). We also derived expressions for the velocity and acceleration. v(t) = wAsin(wt) and a(t) = -w^2Acos(wt). Note that a(t) ~ - x(t). Also tried to understand this by looking at the relation between the displacement and acceleration. x = -m/k a = -m/k delta(v)/delta(t) = -m/k delta(delta(x)/delta(t))/delta(t). Or in words, the displacement is proportional to the rate of change of the rate of change of the displacement. We then graphically verified that the expression x(t) = Acos(wt) satisfies this condition.
Another example of simple harmonic motion: the pendulum. By considering the free body diagram of a mass hanging from a string of length l at an angle theta with respect to the equilibrium position we found that the force along the direction of motion of the mass is F = -mg sin(theta). This does not look like the force law for simple harmonic motion. However if we look more closely at the sin function we find that sin(theta) ~ theta for small theta (theta less than 15-20 deg). For small angles then, F = -mg*theta, or in terms of the arc length, F = -(mg/l)*s. If we make the analogy that "k" = mg/l and "x" = s, all the results that we obtained for the mass and spring system can be applied to this case. In the mass and spring case, we found that the period T = 2*pi*sqrt(m/k). Inserting our effective k for the pendulum we find that T = 2*pi*sqrt(l/g). So the period of the is independent of the mass and the amplitude (at least for small amplitudes where our approximation is still valid), and longer length strings give a longer period.
Next we considered the graph of the potential energy vs position and used this to examine in more detail the relation between the potential energy, kinetic energy, and total energy. How the shape of the potential energy graph relates to equilibrium, simple harmonic motion, and bounded motion. Considered different potential energy curves and discussed what they had in common with simple harmonic motion. A few examples were sketeched to give an indication of the usefulness of simple harmonic motion in physics (motion of planets, vibrations of ionic molecules). The point of this part of the lecture was more to give you some idea of the connection to more useful things that we might want to calculate. The oscillatory motion of a mass connected to a spring might seem to have little connection to natural phenomena that we would like to understand, however the ideas developed here can be applied to many more interesting systems.
Damping. Importance of understanding what are our assumptions. We see that the oscillations of the pendulum eventually stop. This is due to forces such as friction and air-resistance. The simplest model for this is to change the force law to F = -kx -bv, or a = -k/m*x - b/m*v. There are three solutions to this depending on the relative size of the parameters in the equation. Underdamped (least amount of damping), critcally damped (quickest return to equilibrium without overshooting it once), and overdamped (longest time to return to equilibrium).
Finally, we began our discussion of waves in general. A wave is a propagation of a disturbance typically through some medium (except in the case of electro-magnetic waves). Two types of waves; transverse waves: propagation perpendicular to the direction of motion of the medium, longitudinal waves: propagation parallel to the direction of motion of the medium. Continuous waves are characterized by the wavelength (lambda), amplitude, and period. We looked at transverse waves on a stretched spring, and showed how each point on the spring undergoes simple harmonic motion. Lastly, there are 2 speeds to consider when discussing waves. The propagation speed of the disturbance v = lambda*f which in terms of the properties of the medium is v = sqrt(F,mu), where F is the tension in the string, or spring, and mu on the linear mass density mu = m/L. This is unrelated to the speed at which the medium itself moves.

## Week 3

### Tuesday February 7, 2006

Wrapped up our discussion of simple harmonic motion with a brief examination of the motion of a pendulum. Carried out a calculation of the period and wrote the expression for the displacement as a function of time for a specific pendulum (initial angle theta = 15 deg, l = 1m, m=0.15kg, note the mass is irrelevant in the calculations). Used this as a chance to talk about making approximations in calculations. We had T = 2*pi*sqrt(1m/9.8m/s^2) ~ 2*3*sqrt(1/10) ~ 2*3*sqrt(1/9) = 2*3/3 = 2s. To get the amplitude we used s = l*theta = 1m*15deg*pi/180 (pi rad/180 deg is to convert to radians) ~ 15*3/180 = 15/60 = 0.25m. And so s(t) = 0.25*cos(2*pi*t/2) = 0.25*cos(pi*t).
Waves. Waves are a kind of travelling disturbance. Recapped what was said about waves in the previous lecture. The wave speed or propagation speed is the speed at which the disturbance travels. This is as opposed to the speed at which the medium moves. In the case of transverse waves travelling in the horizontal direction, the propagation speed is in the horizontal direction while the motion of the medium is in the vertical direction. A continuous, sinusoidal wave is described by it's wavelength (lambda), frequency (f), and amplitude (A). The wave speed is giving by v = f*lambda = number of cycles per second * the distance per cycle. I tried to give some motivation for how this is related to the properties of the medium. If we consider at experiment where we have a string connected at one end to something that we can make vibrate at a specific frequency and at the other end connected via a pully to a mass which provides tension in the string we can see a few things (for a picture of this setup, see p.447 in the text).
Assume in the original setup, the mass m gives a tension T to the string. The string is then vibrated at a frequency f, with wavelength lambda, and has a wave speed v =f*lambda.
Case 1:
We quadruple the mass providing the tension (T). Since the tension is equal to mg, increasing the mass by a factor of 4 increase the tension by a factor of 4. In this case we would see that the wavelength increases by a factor of 2, and since the frequency is fixed in our setup, the wave speed has increased by a factor of 2. This suggests that v ~ sqrt(T).
Case 2:
We change the string to a thinner (lighter string). In this case the mass of the string has decreased and we observe that the wavelength becomes longer. Since the frequency is the same, this means the wave speed has increased.
Case 3:
If now we increase the length of the string, while keeping everything else the same, we see the wavelength is unchanged. Since we increased the length of string we have increased the mass as well. What is the difference in these last two cases? While in this case we have changed the total mass of string, we have not changed the mass per unit length. While in case 2, we did change the mass per unit length (less mass in 1 meter of the thin string than in 1 meter of the thick string).
It can then be shown that if the mass per unit length is mu = m/L, then v = sqrt(T/mu) = f*lambda.
Mathematical Description of a travelling wave. In general, if y(x,t=0) = F(x), and the wave speed for the medium is v, then y(x,t) = F(x-vt). For example, if y(x,t=0) = Ae^-(x/b)^2, y(x,t) = Ae^-((x-vt)/b)^2.

### Thursday February 9, 2006

Went through the mathematical description of a travelling wave. If at time t = 0, the wave is described by a function y(x,t=0) = F(x), then at any time t, y(x,t) = F(x-vt). See the example at the end of Tuesday's notes and we also considered y(x,t=0) = Asin(2*pi*x/lambda), a travelling sine wave. For this case y(x,t) = Asin(2*pi*(x-vt)/lambda) = Asin(2*pi*(x/lambda-t/T)).
We considered the case of a symmetric and an assymetric travelling wave and the resulting motion of the medium as these waves travel past a particular point. Note that in the case of the asymmetric pulse, the resulting motion of the medium is the mirror image of the travelling wave.
Principle of Superposition: When two waves meet at the same place, we add the amplitudes to get the resulting wave. We say that the two waves are in phase if their maxima and mimima coincide, and out of phase if when one wave is at a maximum, the other is at a minimum. Constructive interference occurs when we add 2 waves that are in phase, destructive interference occurs when we add 2 waves that are out of phase. We also considered reflections of travelling waves when one end of a spring is held fixed. Here when the incident wave (the one travelling into the fixed point) hits the fixed point, it exerts an upward force on the fixed point. By Newton's 3rd Law we know that the fixed point (the wall, the clamp, whatever is holding the spring in place) exerts an equal and opposite force on the spring. This results in a reflected wave that is inverted relative to the incident wave.
A new type of wave called a standing wave was discussed. If we shake a spring which was fixed at one end at just the right frequency, the incident and reflected waves will combine in such a way that the wave appears to stand still. Depending on the frequency at which the spring is vibrated, more wavelengths can be fit between the two ends. The progression of wavelengths is refered to as harmonics. The 1st harmonic consists of 1/2 wavelength, the 2nd harmonic consists of 1 full wavelength, the 3rd of 1.5 wavelengths, and so on. So in each case we add 1/2 wavelength. For a spring of length L, with wave speed v, the frequency of the nth harmonic is fn = nv/2L. From this we can calculate the allowable frequencies of such a system.
From our discussion of damping, we know that real systems are not frictionless, and the simple harmonic motion will die out over time, losing energy to the environment. If we add energy back into the system, the oscillations are able to continue. However, the amplitude of the oscillations depend on the frequency at which energy is put back into the system. Think of a swing set. If you are given one initial push, you will swing back and forth, but eventually come to a stop unless someone is there to push you again (assuming you don't swing your legs back and forth). However if the person trying to push at some random frequency, sometimes they push at the right time, sometimes they will push while you are still on your way back, actually slowing you down instead. When they are pushing at the right frequency, your amplitude becomes the highest. This frequency is called the resonant frequency, and the system is said to be in resonance when driven at the resonant frequency.
Finally we talked about sound. All sounds comes from a vibration of some kind (vocal chords, a string, a speaker membrane, etc) which is transfered to vibrations of the molecules in the air. This disturbance travels as a longitudinal wave which eventually reaches your ear. Just like all waves this disturbance travels with a speed v, for sound waves v ~ 340 m/s, although this depends on temperature.
Lastly we looked at the Doppler effect. This describes the phenomenon that we notice when a fire truck drives past us, we hear the frequency increase as it approaches, and then decrease as it passes us. We derived the expression for the change in frequency for different cases (stationary observer and moving source, moving observer and stationary source, and the general case where both move). See the text for an alternate derivation from the one considered in class.

## Week 4

### Tuesday February 14, 2006

We first reconsidered forced vibrations and resonance. A glass beaker was placed in front of a small speaker. Into the speaker we placed a single with a variable frequency and amplitude. This caused standing waves to be set up in the beaker which were monitored with a microphone and seen through a camera. As the frequency that was generated we could see that the amplitude of the standing waves reached a maximum at a particular frequency (the resonant frequency). If we then turn up the amplitude of the driving force, the amplitude of the standing waves increase as well, until eventually the breaker shatters.
Began our discussion of electricity. We considered the existence of electrical forces. Sometimes these are attractive, sometimes repulsive, sometimes simply nonexistent. We explain this by assuming the existence of positive and negative charges and that like charges repel each other, while opposite charges attract each other. We then summarized a number of observations that can be made about charges and electrical forces:
1) Most objects have equal amounts of positive and negative charges.
2) Opposite charges attract, like charges repel.
3) Charge is conserved.
4) Charge can be transfered.
5) Electric forces descrease with distance.
6) Some materials allow charges to move freely (conductors), others don't (insulators).
We then talked about different ways of transfering charge. Conduction involves bring a charged object in contact with a neutral conductor. If the charged object is negatively charged, the charges in the conductor will rearrange themselves such that the positive charges are close to the negatively charged object, while the negative charges on the conductor move as far away as possible. When the charged object makes contact, the negative charges move onto the conductor. This neutralizes some of the positive charge that was built up on that side, leaving the conductor with a net negative charge.
Induction involves bringing a charged object near a conducting sphere as above, but without making contact. Instead when the charge has separated due to the presence of the charged object, a ground connection is made, allowing the excess charge that had built up on one side of the conductor to be drained off. The ground is then disconnected and the charge object removed, leaving the conducting sphere with a net charge. Grounding refers to making a connection to an object that basically has an infinite resevoir of charge such as a very large metal object.
Lastly, polarization involves bringing a charged object near an insulator. While the charges in an insulator cannot move freely, they are able to reorient themselves in such a way that the opposite sign charge becomes closer to the charged object (e.g. if a negatively charged object is brought near an insulator, the positive charges in the insulator will move slightly closer to the negatively charged object). This is what happens when you charge up a ballon by friction, and then see that it will stick to a wall, which is an uncharged insulator.
We ended by discussing what is going on physically. As you know from chemistry, all matter is made of atoms. Atoms in turn consist of a nucleus (made up of positively charged protons and neutral neutrons) surrounded by negatively charged electrons. The charge on the proton is e = 1.6x10^-19 C, that on the electron is -e = -1.6x10^-19C, where C is Coulombs and is the SI unit of charge. The mass of the proton is about 1000 times more than the mass of the electron, and so for a given force, the electrons will experience a greater acceleration. Therefore it is the electrons that are typically the particle involved when there is a transfer of charge. On top this, the electrons occupy the outer regions of atoms, making it easier for them to move around.

### Thursday February 16, 2006

After considering a demo illustrating the electric force, we examined the mathematical expression of this force. The electric force between 2 chaged objects is given by Coulomb's law: F = k*|q1|*|q2|/r^2, where |q1| is the magnitude of the charge of object 1, |q2| the magnitude of the charge of object 2, r is the distance between the two objects, and k is Coulomb's constant with k ~ 9 x 10^9 N m^2/C^2. To determine the direction of the force, remember that opposite charges attract and like charges repel. We calculated the magnitude of the electric force between the electron and proton in the hydrogen atom, and compared this to the magnitude of the gravitational force between them. The electric force Fe = 8.2x10^-8 N, while the gravitational force Fg = 3.6x10^-47 N. Or Fe/Fg = 2.3x10^39. The electric force is much much stronger here. In general electrical forces are much stronger than gravitational forces until you get to large objects like planets. Another way to think about this is in terms of the demonstration we looked at in class with the rubber rod and the glass beakers. It was pretty easy to give enough charge to these objects so that the electrical force between them was easily visible. However, the gravitational force between these objects is so small that we can completely ignore it. We then carried out a calculation of the force exerted on a charge(A) by 2 other point charges (B and C). The main point to remember from the calculation is that we add the vector components of each of the forces between the particles to get the net force.
When we carried out this calculation for the force exerted on object A by objects B and C, we noted that F = (charge on object A) x (stuff which depends on all the other charges). If we change the charge of object A, instead of recalculating everything, we would just multiply the same "stuff" by the new charge to get the force. Mathematically then we can write F = qE, where E is what we call the electric field. The electric field is created by all the other charges and is independent of the charge for which we are calculating the force. More explicitly we can write F (on q due to all other charges) = q * E(due to all other charges). So in the absence of the charge q we can calculate the electric field due to the charges in the environment.
The remaining topics which were covered were graphical representations of the electric field, proporties of conductors in electrostatic equilibrium, and finally Gauss's law. Please look over the text if any of these were unclear or if you missed class.

## Week 5

### Tuesday February 21, 2006

We began by reviewing some of the new ideas introduced last week. Specifically Coulomb's law and Electric fields. Outline of the problem solving procedure for calculating forces on charges. For three charges A, B, and C to calculate the force on charge C we:
1) Draw free body diagram.
2) Determine the magnitudes Fca and Fcb from Coulomb's law.
3) Find the x and y components of Fca and Fcb and note if they are in the positive or negative x and y directions.
4) Add the components taking into account the signs:
(Fc)x = (Fca)x + (Fcb)x
(Fc)y = (Fca)y + (Fcb)y
5) Determine the magnitude and angle of the resultant force:
|Fc| = sqrt((Fc)x^2+(Fc)y^2)
tan(theta) = (Fc)y/(Fc)x
This same process can be followed for calculating the electric field.
We finished by making an analogy between electric force/fields and gravitational force/fields. Just as we write F = qE for electrical interactions, we can write F = mg, for gravitational interactions. The gravitational field can be taken as uniform and constant near the surface of the earth. The analogous situation for electric fields is the uniform and constant field between 2 parallel, equal and oppositely charged conducting plates. Then we considered this analogy for the case of the work done by these fields in moving a particle from point A to point B. After noting how one should be careful with the signs when calculating the work done by the field, we wrote the general expression: Wab = qE(yb-ya).

### Thursday February 23, 2006

We worked through a calculation together in class. The example was of an electron injected into the region of uniform constant electric field between 2 parallel, equal and oppositely charged plates separated by 10 cm. The electric field was 1000 N/C in the negative x direction, so the work done by the electric field in moving the electron from one side of the plates to the other is W = qE*delta(x) = (1.6*10^-19 C)*(-1000 N/C)*(0.1 m) = 1.6*10^-17 J. The change in potential energy is delta(PE) = - W = -1.6*10^-17 J. Given the initial velocity of the electron when it enters the region with the electric field, we can find its final velocity using conservation of energy: delta(PE) + delta(KE) = 0. We just determined delta(PE) above, and delta(KE) = 1/2*m*(vf^2-vi^2), where m = 9*10^31 kg is the electron mass, vi is the initial velocity (1000m/s in this example), and vf is the final velocity. Applying the principle of conservation of energy, we can now find the final velocity.
So far we have only spoken about the potential energy when we have a uniform and constant electric field. The potential energy associated with a pair of charges q1 and q2 is given by PE = k*q1*q2/r, where k is Coulomb's constant and r is the distance between q1 and q2. Note that if q1 and q2 have the same sign charge, the potential energy increases as r decreases, while if they have opposite signs, the potential energy will increase as r decreases. Note also that we need to define a point at which the potential energy is zero since we are only interested in differences in potential energy.
In the same way that we defined the electric field as E = F/q, we define the electric potential as V = PE/q. And again we are only interested in differences in potential between two points a and b delta(Vab) = Vb-Va = delta(PE)/q. The units of electric potential are J/C or volts (V). The potential difference tells you the work per unit charge that would be required to be supplied by some other force to move a charge from point a to point b in an electric field. For the case of a uniform, constant electric field, the potential difference between points a and b is delta(V) = delta(PE)/q = q*E*(xb-xa)/q = E*(xb-xa). In general, for a point charge q, the potential is V = kq/r. Using this, we considered the potential at some point due to two point charges, and then calculated the work done by the electric field in moving a charge in from infinity to the point P. The work done is W = -(delta(PE)) = -q*delta(V).
We then noted that if the potential difference between two points is zero (detla(V)=0), then the work done in moving a charged object from a to b is zero. We then used this to note that the surface of a conductor is all at the same potential. This is referred to as an equipotential surface; a surface on which all points are at the same potential. One property of equipotential surfaces is that they are always at right angles to the electric field lines. This allows to draw the equipotential surfaces if we know what the electric field lines look like.
We then began introducing some of the ideas which will allow us to talk about electrical circuits. A battery is an object which maintains a constant potential difference between 2 points (the battery's terminals). Through a chemical reaction inside of the battery there is a separation of positive and negative charge leading to a difference in potential. If a wire (a conductor) is connected the two terminals charge will flow from one terminal to the other.
Capacitors allow us to store charge. The capacitance tells us how much charge can be stored on the capacitor per volt of potential difference. C = Q/delta(V), and is measured in farads (F), where 1 F = 1C/V. The simplest example of a capacitor is the parallel plate capacitor. We derived the expression for the capacitance for this type of capacitor: C = (epsilon)o*A/d.
We then considered combinations of capacitors in parallel and series combinations. The equivalent capacitance of two capacitors in parallel is just the sum of the capacitances Ceq = C1+C2. For capacitors in series this is 1/Ceq = 1/C1+1/C2.
We ended with a discussion of the energy stored in a capacitor. The energy stored in the capacitor is E = 1/2*Q*V = 1/2*C*V^2 = 1/2*Q^2/C.

## Week 6

### Tuesday February 28, 2006

We started by reviewing some basic ideas about the electric potential and the relation to the change in potential that occurs when moving a charge between 2 points A and B in an electric field. One useful way to think about the change in potential energy is to consider how much kinetic energy the system could gain if the charges were released at the 2 points A and B. If we brought 2 like charges closer together in moving from A to B, the potential energy has increased since at the closer point B the charges will feel a stronger force than at A and so the acceleration away from each other will be greater. This greater acceleration will lead to a higher final velocity and greater kinetic energy.
We reviewed some ideas about capacitors and then went on to talk about dielectrics in capacitors. A dielectric is any insulating material such as rubber, paper, plastic, etc. A parallel plate capacitor is charged by a battery to have +Q and -Q on it plates and then the battery is removed. A dielectric material is then inserted between the plates of the capacitor. In the dielectric the electrons are more tightly bound to the nuclei, and are not free to move about the material. However in the presence of the electric field between the plates a slight separation of positive and negative charge will occur. This separation of the charge results in an induced electric field between the plates of the capacitor. This induced field (Eind) will be in the opposite direction of the original field between the plates (Eo), and it is assumed that it is both proportional to the original field and smaller in magnitude. That is: Eind = aEo, where a is a constant which is less than 1. Then the net field between the plates of the capacitor is Enet = Eo - Eind = Eo-aEo = Eo(1-a). We redefine the constant factor 1-a=1/k, where k here is called the dielectric constant. Since a<1, 0<(1-a)<1 and therefore k>1. Since Enet = Eo/k, Enet1, C>Co, the capacitance has increased. We also talked about dielectric breakdown. This occurs when the electric field through the insulator becomes too great and allows the insulator to act as a conductor. A demo in class was done to illustrate this effect. Lightning also an example of dielectric breakdown. The storm clouds build up a charge which then induces a charge on the ground. When the electric field between the earth and the ground exceeds a maximum field strength, the air becomes like a conductor and the cloud discharges in a bolt of lightning.
We then begin to discuss the flow of charge a little more carefully. We define the current as the rate at which charge flows perpendicularly through some cross-sectional area. In equation form the current is I = delta(Q)/delta(t), and has units of C/s which we define to be Amperes (amps, A). We talked about the motion of the electrons in a conductor in the presence of an electric field, and introduced the idea of the drift velocity.

### Thursday March 2, 2006

Continued our discussion of current. Derived the expression for the current in terms of the drift velocity: I = q*n*A*vd, were q is the charge of the charge carriers in the metal, n is the density of charge carriers, A is the cross-sectional area, and vd is the drift velocity. The typical drift velocity is rather slow, we considered an example which gave a drift velocity of 2.5*10^-4. This is very slow. For 1 m of wire, it would take more than 1 hour for an electron to travel from on end to the other. However we know that when we flip the light switch, the light turns on almost immediately. This led us to consider a few analogies for the flow of current through a conductor when a potential difference is applied across the conductor. One idea is that current is like the flow of water in a hose. A potential difference is like when we turn on the faucet that is connected to the hose. This applies a higher pressure to one end of the hose, and if there is already water in the hose, water begins to flow out the other end of the hose almost immediately, just as the light turns on almost immediately.
We also talked about connections of batteries in series, and defining a reference point of zero potential in circuits. Then we considered what happens when we connect a light bulb to a different number of batteries of equal voltage. Intuitively we expect that with a higher voltage the light bulb will shine more brightly. The reason for this is that as we increase the voltage, we also increase the current. The ratio of the potential difference to the current is called the resistance: R = V/I. The unit for resistance is V/A or Ohms (Omega). The relation between the current through some object (like a wire, a lightbulb, etc) and the voltage across that object could take many forms. When there is a linear relation, we call the object Ohmic. All other relations we call non-Ohmic. When there is a linear relation that holds over all values of V and I, then we can write V = I*R, where R is a constant. This is called Ohm's law. This is an empirical law, unlike Netwon's laws.
We then talked a bit about the internal structure of metals. Ideally, metals have a regular lattice structure with a regular spacing between the nuclei and the electrons free to move around them. If this were the case, the electrons could move through the metal with little resistance. In reality the nuclei are constantly in motion due to thermal vibrations. In addition to this there are impurities in the metal (other atoms which mess up the lattice structure). These are two sources which serve to increase the number of collisions the electrons will have with the nuclei, and thus increase the resistance.
We could do a number of experiments with wires of different diameter and different lengths and measure their resistances. Doing this we would find that R = pL/A, where p is the resistivity of the material, L is the length, and A is the cross-sectional area. We considered an analogy of water in a sink flowing through drains with different area, and different thickness of filters in the drain. We finished by considering the energy consumed in moving charge along a conductor. The change in potential energy is delta(PE) = delta(Q)*delta(V). The power is delta(PE)/delta(t) = delta(Q)*delta(V)/delta(t) = I*V. For Ohmic materials P = IV = I^2R = V^2/R. This is the energy consumed by a resistor. We considered the analogy of current flowing through a resistor with moving a piece of rope. One person hold the rope in one hand in a tight fist (the tightness being like the resistance). Another person is pulling the rope (acting as the battery). The rate at which the rope moves around is the current. As the rope moves through the person's hand, their hand will heat up. This corresponds to the energy consumed by the resistor.

Exam Review

Exam 1

## Week 8

### Tuesday March 14, 2006

Review of exam 1.

### Thursday March 16, 2006

Circuits. We talked about different circuit elements we have seen so far: batteries, resistors, and capacitors. In addition to these we have the "ideal wire" which is a wire that is used to connect these different elements and is usually assumed to have 0 resistance. We then discussed the difference between series vs parallel connections. For two resistors connected in series the current through each resistor is the same, but the potential difference across each resistor is different. For two resistors connected in parallel the potential difference is the same across the two resistors, but the current through each resistor is different. To understand this better we considered a demo involving two light bulbs connected either in series or in parallel. It was asked which would be brighter, the parallel connection or the series connection. Before turning on the lights to see which was the case we did a little experiment. We taped together 2 straws, one set in series another in parallel. Then each group had one person take a full breath of air and blew out through the straws. The length of time to empty your lungs was compared for the series and parallel case. Most people found that it took longer for the straws connected in series. To connect this to the case of the resistors and battery we take each straw to be like a resistor, and the pressure supplied by your lungs to be like the potential difference supplied by the battery. As was seen in the experiment, the two straws connected in series gave a greater resistance and it took longer to completely exhale. This is like having a greater equivalent resistance and lower current. With the straws, the current is amount of air moving through the straw per unit time. The experiment then tells us that the parallel connection has the higher current, and therefore the bulbs connected in parallel should be brighter. This is what we observed when we turned on the bulbs.
We then derived expressions for the equivalent resistance for resistors connected in series and parallel. For the series case the resistances just add: Req = R1 + R2+..... For the parallel case we have 1/Req = 1/R1 + 1/R2 + .... Note that this is the reverse of how capacitors add.
We then worked through a number of different circuit examples, calculating the current, potential drop, and power consummed by the resistors in the circuit.
We then discussed Kirchhoff's rules for more complex circuits. These are:
1. The sum of the potential drops around any closed path is 0.
2. The current going into a junction is equal to the sum of the currents leaving a junction.

For solving complex circuits in this way one should follow this general approach:
1. Draw circuit diagram and label everything.
2. Give a direction to all the currents in the circuit.
3. Apply the junction rule to any junctions. I1 = I2 + I3.
4. Calculate the change in potential around closed loops (pick the independent loops).
5. Solve simultaneous set of equations. With # unknowns = # equations.
6. Check resuls.

Spring Break.

## Week 10

### Tuesday March 28, 2006

Reviewed DC circuits. Solved an example in detail applying Kirchhoff's rules. We then moved on to discuss RC circuits. These are circuits which include capactors and resistors. In general when we connect a battery to a capacitor, it will take some amount of time for the full charge to be built up on the capacitor. Once it reaches it's full charge (specified by Q = CV), current no longer flows to the capacitor. The equation which describes the charge on the capacitor as a function of time is given by q(t) = Q(1-e^-t/RC). RC is called the time constant, and has units of seconds. When t=RC, q = Q*0.632, or in words: when a time equal to the time constant has passed, the capacitor will have reached 63.2% of its full charge. A large time constant (R*C is large) means that it will take a long time for the capacitor to become fully charged.
If instead we have a fully charged capacitor which is then discharged through a resistor, the charge on the capacitor as a function of time is given by: q(t) = Qe^-t/RC. Now in this case, when t = RC, only 36.8% of the charge remains on the capacitor. Now here if R*C is large, it will take a long time for the capacitor to discharge.

### Thursday March 30, 2006

Magnetism. Magnetic forces, a new kind of force. Mapping the magnetic field of a bar magnet with a compass. The magnetic field of the earth (geographic north/south vs magnetic north/south poles). Some observations about magnetic fields/forces:
1) Some objects are attracted to the bar magnet, others are not.
2) Magnetic poles always occur in pairs (dipoles). No magnetic monopoles have been observed (although some theories predict that they exist).
3) The magnetic field points in the direction of the force the would be felt by a north magnetic pole at each point in space.
4) Like magnetic poles repel, opposite poles attract.
We considered happens to a charged particle in a magnetic field. If the charged particle is stationary, it remains stationary (no magnetic force on the charged particle). If the charged particle moves in the same direction in which the magnetic field points, the charged particle continues in a straight line with the same velocity with which it started (no magnetic force on the charged particle). If the charged particle moves perpendicularly to the magnetic field, the particle feels a force which is perpendicular to both its velocity and the magnetic field. Summarizing this, we can write: Fmagnetic = qv(perp)B, where v(perp) is the component of the velocity that is perpendicular to the magnetic field and B is the magnetic field (or B-field for short), and is measured in units of Tesla (T). Note 1T = 1N/C*m/s. Another unit sometime used is the Gauss (G) 1G = 10^-4 T. Since v(perp) = vsin(theta), with theta being the angle between the velocity vector and the magnetic field, we can write: Fmagnetic = qvbsin(theta). This gives the magnitude of the force, to determine the direction we must use the right hand rule. If you put your right hand with your fingers in the direction of the velocity and the bend your fingers to make a 90 deg angle and rotate your hand so that your fingers now point in the direction of the B-field, your thumb will point in the direction of the magnetic force. Since current is just a collection of moving charges, currents in a B-field should feel a magnetic force as well. This is given by F = IlBsin(theta), where l is the length of the current carrying wire. We finished by considering the torque on a current loop in a B-field and the creation of a magnetic field by a long current carrying wire. The B-field a distance r from a wire with current I is: B = mu0*I/2*pi*r, where mu0 is a constant called the permeability of free space and is equal to 4*pi*10^-7 T*m/A.

## Week 11

### Tuesday April 4, 2006

Finished up magnetism. Summarized properties of magnetic fields and magnetic forces. The magnetic force is parallel to both the B-field and the velocity of the charged particle or current. Must apply the right hand rule to determine the direction of the force. Some applications: cosmic rays and the aurora, torque on a current carrying loop of wire and electric motors.
B-fields of a long current carrying wire. Analogy with the electric field Felectric = q * E, Fmagnetic = qvsin(theta) * B. Magnetic force between two current carrying wires. Currents in the same direction attract, currents in the opposite direction repel.
So far we have seen that the sources of E-fields are electric charges, and sources of B-fields are moving electric charges. We then did a demo that showed that a changing B-field could produce an E-field. In this demo a bar magnetic was quickly inserted through a loop of wire which was attached to a device which could measure the current through the wire. When the magnetic was inserted into the loop of wire we saw a current flowed momentarily through the circuit, indicating the presence of an E-field. We then considered the case of the B-field created by a current in a straight wire vs that created by a wire with a capacitor in the middle ( -------- vs ---||---- ). In order for there to not be a discontinuity in the B-field, there must be a B-field between the plates of the capacitor. As charge builds up on the plates of the capacitor, the E-field between the plates is changing with time. It was then suggested by Maxwell that this changing E-field could be a source of the B-field between the plates. This was later verified. So now we have the following:
Sources of B-fields:
moving electric charges and changing E-fields.
Sources of E-fields:
electric charges and changing B-fields.
This is the basic idea of electromagnetism, and leads to the idea that electromagnetic waves can propagate through space. If we go through the mathematics, we find that electromagnetic waves propagate at a speed v = 1/sqrt(epsilon0*mu0) = 3*10^8 m/s = c, the speed of light! Light is a kind of electromagnetic wave.

### Thursday April 6, 2006

Wrapped up out discussion of electromagnetism: sources of E and B fields, propagating electomagnetic waves, the electromagnetic spectrum (a few examples of different electromagnetic waves), light as an electromagnetic wave, the speed of light.
From the discussion on light we moved into optics. In order to talk about mirrors, lenses, reflection, etc we introduced the ray model:
1) Light rays travel in straight lines at a speed v = c/n, c is the speed of light, and n is the index of refraction of the medium in which light is travelling.
2) Light rays do not interact with each other.
3) A light ray continues forever unless otherwise indicated.
4) When light rays move from one medium to another, they can be reflected, refracted, or both.
5) An object is a source of light rays (either by refleecting other light rays or because it is a self-luminous object like a light bulb or the sun.
6) Your eye "sees" an object when bundles of diverging rays from each point on the object enter the pupil and are focused on the retina.
Given this model we first discussed reflection. We used the optical board to observe light rays reflected off a mirror. We could see that the angle of incidence was equal to the angle of reflection. These two angles are defined with respect to the line normal to the surface of the mirror.
When a light ray enters a medium which has a different index of refraction, its speed changes and the ray bends. If the ray moves to a medium with a larger index of refraction the ray slows down and bends toward the normal. The angle of incidence (theta1) and the angle of refraction (theta2) are related by: n1sin(theta1) = n2sin(theta2), where n1 and n2 are the indexes or refraction for the different media. This relation is known as Snell's Law because he figured it out. In general, the index of refraction depends on the wavelength of light (n=n(lambda)). This is called dispersion because if a beam of white light enters a different medium, the different wavelength components will spread out (or disperse).
Total internal reflection occurs when a light travels from a medium with a high index of refraction to a medium with a lower one. For angles of incidence beyond some critical angle (thetaC) all light is reflected back. This angle is reached when the refracted ray travels parallel to the surface where the two media meet. In this case the refracted angle is 90 deg. and so sin(thetaC) = n2/n1.

## Week 12

### Tuesday April 11, 2006

We considered refraction from another point of view: Fermat's Principle or the principle of least time. This states that when light travels from one point to another it follows the path that takes the least time. If light starts in one medium where the speed of light is higher and moves to one where it is lower, the light ray will follow a longer path in the medium where it travels faster. Next we discussed images. Images occur when light rays converge to a point. We then used this and the law of reflection to see where the image is formed in the case of a flat mirror. We found that the object distance (do) equals the image distance (di) and the object height (ho) equals the image height (hi). The image is virtual meaning that the image is formed behind the mirror at a point where the light rays do not actually cross, but where they appear to cross. The magnification of the image is defined as M = hi/ho and M = 1 in this case. We also pointed out that the mirror reverses front to back, not left to right, and that in order to see an object of height h in the mirror, we only need a mirror of height h/2 (independent of how far from the mirror the object is). Note that this is only in the case of a flat mirror. We then introduced spherical mirrors.

### Thursday April 13, 2006

Spherical mirrors. Finding images formed by spherical mirrors. Please refer to the text for all drawings. We derived the mirror equation: 1/do + 1/di = 2/R and the relation M = hi/ho = - di/do, where the minus sign accounts for inversion of the image. Given the object distance and the radius of curvature for the mirror we can find the image distance. We discussed the sign convention that must be used when applying this equation (again see the table in the text which outlines the sign convention).
If do is very large, the mirror equation becomes 1/di = 2/R. Also when do is very far the diverging rays appear to be parallel. This tells us that parallel rays all focus to the same point f = R/2. This distance is called the focal length and the point is called the focal point of the mirror.
The general procedure for finding the image with a spherical mirror includes drawing the following 3 rays:
1) A ray parallel to the principal axis, reflected through the focal point.
2) A ray through the center of curvature, reflected back upon itself.
3) A ray through the focal point, reflected parallel to the principal axis.
This image is at the point where these 3 rays intersect (or appear to originate from). When then went through a number of different ray diagrams (see text for examples).
We then moved on to discuss thin lenses. We looked at different examples of lenses and then we focused on biconvex and biconcave lenses. Lenses have to focal points, one on each side of the lense. The convex lens is a converging lens for which parallel rays entering on one side of the lens are refracted and meet at the focal point on the other side of the lens. The concave lens is diverging and parallel rays spread out and appear to originate from the focal point on the same side of the lens from which the rays arrive. We then went through the procedure for finding the images formed by lens which also use 3 rays:
1) A ray drawn parallel to the axis is refrated through (converging) or appears to come from (diverging) one of the focal points.
2) A ray drawn through the center of the lens which continues in a straight line. This is in the approximation that the lens is thin.
3) A ray drawn which is drawn through the focal point (converging) or would go through the focal point if it continued in a straight line (diverging) which is refracted parallel to the principal axis.
The image forms where the rays cross or appear to originate from.
We also used the result that 1/do + 1/di = 1/f applies to lens as well, but with a slightly different sign convention. Note that in the case of a diverging lens, the focal length is negative.

## Week 13

### Tuesday April 18, 2006

Summarized mirrors and lenses. Went through some calculations with converging and diverging lenses. We then discussed some basic ideas associated with cameras. Handed out study guide for exam 2 and discussed some of the points on the study guide.

Exam review

Exam 2

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