Magnetic field of a wire, a loop and an inductor

This week's homework assignment is pretty difficult,  so I thought it might help to summarize a few essential things here. (These things are also discussed in the book (Ch's 26 and 27).)  [By the way, we have new constant: the Greek letter mu with a subscript zero. For this discussion, let's just call that mu; also let's use pi for the Greek pi (3.14159). That should make equations a little easier to read and write here. i think mu is 4 pi 10^{-7}.  what are its units?]

First, regarding magnetic fields:
The magnetic field of a straight wire is: B=mu I/(2 pi r).
The magnetic field in the center of a wire loop is:  B=mu I/(2 r) (larger by a factor of pi).
(r  is the distance from the wire; B is the magnitude of the magnetic field associated with the current flowing in the wire.)  These results come from the Biot-Savart equation and/or Ampere's relation.  Mostly we will just use these results and not get bogged down in the derivations.

There is a problem on the homework in which I believe you are asked for the magnitude of the magnetic field due to the current in a wire with a single loop in it. I think the answer involves combining the two results above, thus the total field is the contribution from the straight wire, B=mu I/(2 pi r), plus the contribution from the loop, B=mu I/(2 r).  (please post a comment if that doesn't work.)

A solenoid involves many loops of wire very close together with current flowing through all of them. The magnetic field everywhere inside a the solenoid is given by:

B=mu N/Z I = mu n I,

where I is the current, N is the number of loops in the solenoid, and Z is the length of the solenoid. ( Note that the units of B are the same as above because Z has units of length. Also, in truth, this i an approximation, in case you were wondering. It is not exactly correct, but it is a pretty good approximation in many cases of interest.)

Chapter 26 might be summarized by saying: current flowing in wires produces magnetic fields,  and we are particularly interested in the field produced inside a wire loop or a solenoid. Chapter 27 can be summarized as: changing magnetic fields generate electric fields.

In Chapter 27  the most important thing to learn and remember is that when a magnetic field changes, as a function of time, that creates an electric field.  That is what Faraday's law is all about.   Faraday's law is expressed in terms of a quantity called magnetic flux, or just flux, which is magnetic field strength multiplied by an area. For a solenoid, the flux is:
Phi = pi r^2  B
       = pi r^2 mu (N/Z) I

The reason that flux is important to us is that when it changes it produces an emf (a voltage-like quantity) given by:

emf = -d(Phi)/dt

i.e., an emf equal to the time derivative of the flux.

This relationship is the basis for the new circuit device known as an Inductor.  It means that when the current in a solenoid changes, that change will produce a voltage.   An inductor is a solenoid. The name inductor focuses on its function. The name solenoid focuses on its physical appearance. They both refer to the same thing.

The magnetic field inside the solenoid is B= mu N I/Z and the associated flux is pi r^2 times that. The emf produced by a changing current is then all those factors time dI(t)/dt, i.e.,

emf=-(pi r^2) mu N/Z dI/dt.

Actually, that is the emf per loop or turn. The total emf is N times that. From that, one obtains the inductance of a solenoid.

With this knowledge you should be able to do many of the homework problems. If not, please feel free to ask any questions here.

**Note added on Sunday:    Do you think that it might make sense to extend the due date  for some of the homework problems?  The reason I ask is that I believe that the homework problems can be divided into two categories: those that can be done using just the information provided above, and, those that can't. I would appreciate some feedback on this. Would some people go through the problems and analyze and explain which ones should be extended and why?   Your efforts will be much appreciated. (PS. It is especially helpful if you include a summary of  what the problem is asking as well as it's number.)

Reading

Read (from) chapters 26 & 27. In Ch 26 focus on the parts about the Biot Savart relations, Amperes Law, and using Amperes Law (453). These are all about magnetic fields and how they come from current flow in wires. We will be very interested in that!  (Magnetic field from current flow in wires.)

Also of primary interest to us is the observation that: changing magnetic fields lead to electric fields!  (as discussed in the post above and in Ch 27.) This relationship will be critical to everything we do from now on.

Please post your midterm poll comments here.

Midterm solution discussion

Problem 1: parts a), b) and c) were well understood, i think. When the two charges are the same, then the electric field points in or out. (For 2 minus charges, in; and for 2+ charges, out.) When the two charges are different, the field points up or down.  When the plus charge is on top, the field points down (away from the plus charge); when the plus charge is on the bottom then the field points up.  This will be important later.

Part d), the extra credit from problem one, was somewhat difficult and subtle. This is like a CO2 molecule, but with the carbon atom moved off-center. If the carbon atom were in the center, then the field would be nearly zero. However in this case the carbon has moved up (off-center) and thus the field points down. Understanding this will be useful over the next few weeks when we learn to understand how molecules create and absorb electrical waves (light and infrared radiation).

For problem 2, the key thing was to understand that the current flows through the whole system and is determined by the battery voltage and resistance of the system. The current from a to b depends on the voltage of the battery, and the resistors everywhere in the circuit. For example, when two parallel paths are available between b and c, you get twice as much current from a to b.

For problem 3, the voltage difference between a and b is always zero because there is no circuit element in that segment. The voltage decrease from b to c is exactly half the voltage decrease from b to d, which is the same as the voltage of the battery.

For problem 4, you drew arrows representing electric fields, calculated their magnitudes, and then divided them into components and added their components. The angle is 45°. For the plus charge the arrow points into the positive quadrant, i.e., both Ex and Ey are positive, while for the minus charge, the arrow points down, thus Ex is zero and Ey is negative. When you combine the fields from each charge, you get a positive Ex and a negative Ey (the sum of a large negative contribution and a much smaller positive one).

Problem 5 was the essay question. The gate closes at t = 0. What happens after that?
      There are many ways to tell the story. The plot centers on an exponentially decreasing current and an exponential buildup (increase) of charge in the capacitor. The characters, the most dramatic ones, are the resistor and capacitor. And, of course, the battery is lurking in the background, providing a voltage which drives the action.
      One might say (*though your story does not have to be this technical):
“When the gate closes current begins to flow through the circuit and into the capacitor. This leads to a build-up of charge in the capacitor. This buildup of charge in the capacitor causes current to decrease. My graphs of the exponentially decreasing current and exponentially increasing charge in the capacitor is a function of time are shown to the right. (not really shown, but you get the idea.) [aside: Current, after all, is measured in coulombs per second, so it makes sense that a flow of current leads to build-up of charge both intuitively and mathematically.]

      In the accompanying graphs [not actually shown], the behavior of the the current is shown as a function of time. The current starts at its maximum value, which occurs just after the gate closes, and decreases exponentially towards zero (as t approaches infinity), as shown.The reason for the decrease in the current is found in the increase of the charge in the capacitor. Initially the capacitor is  uncharged, therefore there is no voltage drop across the capacitor. At this moment all the voltage drop occurs across the resistor. Later on, when there is charge on the capacitor,  there is a voltage across the capacitor and this makes the voltage drop across the resistor smaller and hence the current flowing in the circuit decreases. The charge increases from zero at t= 0 to its asymptotic value as t goes to infinity as shown in the second graph. [Not actually shown]  This increase is also exponential and of the form (1-exp{-t/tau}). For both the exponential increase of the charge and the exponential decrease of the current the timescale is given by tau. This means, for example, that after tau seconds the current drops by a factor 1/e….

The critical thing in this story I guess, is the relationship between current flow and charge. In this case, with the battery and a capacitor which is initially uncharged, current flow leads to buildup of charge in the capacitor, which, in turn, leads to a decrease in the current. This “negative” feedback helps the system reach a stable equilibrium with a finite amount of charge in the capacitor and a current approaching zero.

Problem 6: Well please look at the accompanying solution and feel free to post your comments and questions here.

Midterm comments

Well, we had our midterm yesterday. What did you think of it?  Please post comments here.

PS. I will post the midterm problems, solutions and a poll tomorrow. You can suggest poll questions if you like.

midterm tips:

When calculating an electric field, first calculate the magnitude and draw a picture, then divide the field into components and use your picture to guide you as to which components are positive  and which are negative.

When calculating an electric field associated with two charges, do the above for each charge separately. Then you can get the total electric field by addition,  first for one component, then for the other.

Be ready to quickly sketch electric fields for uncomplicated charge distributions ( point charges)

When analyzing circuits, often you have to analyze the whole system to be able to understand one part. 

Be ready to describe, analyze and understand the behavior and influence of a capacitor as it charges or discharges in simple circuits.

Be prepared for an essay question that involves a circuit with a capacitor.

Poll on midterm content

Please feel free to vote on the midterm content poll --what you would like to see on the midterm-- and to comment here on that.

What you might want to have on your index card.

For the midterm you are allowed to bring  a 4 x 6 index card with some equations on it. Please don't get carried away or write too small.  The card should only have about 10 or 20 equations on it. That's all you need, probably not even that much really. Constants, like k and eps_0 are also again idea and also "e".

Some things you could include are:
  the equations that relate voltage, current and resistance.
  Equations that relate voltage, charge and capacitance.
  expressions for the energy stored in the capacitor in terms of charge and capacitance,  or voltage and  capacitance, or charge and voltage.
   An expression for the energy density (inside a capacitor) in terms of electric field
   An  expression for the electric field created by a point charge.
   An expression for the electric potential created by a point charge.
   values for k and eps_0

Is there anything else? That is all I can think of at the moment, but perhaps there are other things that would be useful also.

Special Sections this week

As you know, our midterm is on Wednesday, October 27 (in class). So we changed the times and rooms for the sections this week to:  (next week we will go back to our regular schedule)

Monday, October 25, 2 PM in room ISB235

Monday, October 25, 6 PM in room ISB235

Tuesday, October 26, 6 PM in room ISB235

Also, my office hours will be changed from Wednesday to Monday at 5 PM for this week only.

Homework 4 solutions

Please feel free to post comments which discuss and ask about the homework here:

Additional midterm practice problem ( midterm preparation post #3)

The enclosed drawing shows a circuit with a gate ( a "switch"), a resistor and a capacitor.
Initially,  the capacitor is charged with the charge of, lets say, 10 Coulombs,  and the gate is open.
Let's make the resistance 100 Ohms and the capacitance 0.1 Coulombs/volt*. (aside: how are these values relevant to making part the less difficult than it otherwise might be?)


a) In this configuration, with the capacitor charged and the gate open,  what is the voltage difference Va - Vb and why?  (The  “why” is just as important as the answer regarding Va - Vb.)

b) Now suppose you close the gate at t = 0.   Think about what happens and answer the following question: roughly (within about 10% accuracy) what is the amount of heat energy generated in the resistor in 10 ms immediately after the gate is closed? ( Let's assume the resistor is a pure heater and that all the energy dissipated or converted ( or whatever term you want to use)  in the resistor goes to heat.

c)  Explain your thought process and how you obtained your result  for part b).  Explain why you think it's accuracy is good enough.  Did you have to do any integration? Why or why not? (you can ask for a hint here. (via a comment))
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New problem about the same circuit:  With reference to the same circuit, in which the capacitor is initially charged and the gate is closed at t=0.

a)  graph Va-Vb  as a function of time.

b)  Graph Vb -Vc  as a function of time.

c)  What is the relationship between them?

* 0.1 Coulombs/Volt  is an unrealistically large value for an ordinary capacitor.  We use it here because otherwise the timescales would get very short, and having large negative exponents associated with the time can be distracting when one is first learning concepts.

PPS.  I will avoid using farads for capacitance.   One farad is the same thing as one coulomb per volt, so I don't think we need to introduce one more new unit.

Special Sections next week

As you know, our midterm is on Wednesday, October 27 (in class). So we changed the times and rooms for the sections next week to:

Monday, October 25, 2 PM in room ISB235

Monday, October 25, 6 PM in room ISB235

Tuesday, October 26, 6 PM in room ISB235

Also, my office hours will be changed from Wednesday to Monday at 5 PM.

Resistors and capacitors in circuits: series, parallel, etc.

When you have two resistors, one after another, in a circuit with a battery, the current through each resistor is the same, and that current is given by V/(R1+R2). (That is called series because the current goes through one element at a time, one after the other.)    To understand that, it might be useful to sketch a picture and to imagine the current, like a model train, going around the circuit first through one resistor, then through the other…

When you have two resistors on two different branches of a circuit,  current coming into the dividing point can either go into one branch or the other. Like a fork in the road, each car ( electron)  will either go on the left fork or the right fork.   If the two resistors are equal, the current will divide equally with half going through the left side of the circuit, and half taking the right fork and going through the right-hand side of the circuit. If resistor 1 has a higher resistance than resister 2, then more current will go through resistor 2.   The ratio, the way in which the current divides, is R1/R2.  So if R2  is twice as big as R1,  than the ratio is 2 to 1; which means that 1/3  of the current will go through R2 and 2/3 through R1.

To evaluate more complex circuits with resistors, one generally uses formulas which calculate effective resistances for resistors combined in complex ways. I won't ask you about that (on any test). No one remembers those formulas six months after the class anyway. We will focus on the concepts, and on simple circuits. Focusing on simple circuits takes us away from rote learning and application of formulas.

Simple circuits, however, are not necessarily easy, especially when there are capacitors involved. Capacitors tend to be more interesting than resistors, partly because they change with time. In a nutshell, resistors "dissipate" energy, and capacitors store energy.  To understand the nature of circuits which include capacitors, one ideally should have an understanding of current as a function of time in a circuit, and how the flow of current leads to the build-up of charge.

In all the problems we will consider that involve capacitors, there will be a gate (real or imaginary) and when you close the gate (turn on the switch), current will either flow in or out of the capacitor. (If it is charged to start with, current will flow out. If it is not charged to start with, current will flow in.)  Then, after a while, when the capacitor,  or really the circuit, reaches equilibrium.  In equilibrium no current ( zero)  will flow through the capacitor.  That is the key  to understanding capacitors in equilibrium. It is like they weren't there in the sense that no current flows through them.

In equilibrium,  or actually, steady state,  zero current flows through a capacitor. 

The time-dependent approach to that equilibrium state is generally exponential, just like we discussed in class. Sometimes in circuits that are a little complicated it is hard to know exactly what the exponential time dependence is, but it is always something like exp{-t/RC}, where R has units of resistance and C has units of capacitance.

Also, for an uncharged capacitor, in the beginning-- when the switch is first closed-- current will flow freely in and out of the capacitor. So, for a moment, the capacitor acts like a short-circuit––like a segment with no (zero) resistance. That is just for an instant. Then the current starts to decrease exponentially.  This concept, which seems less obvious to me, helps us understand the t=0 point on the graph of current versus time.

Notes on homework #4

Homework 4  has both some relatively easy and straightforward problems and some very difficult problems. From chapter 24, I think 30 is pretty easy; I noticed that in 14 and 16 you can actually just put the 1.6x10-19 in the denominator and MP will calculate the answer for you and give you full credit; in 37 something is open 20% of the time, ask about that if you want…

From chapter 25: 18 and 19 are pretty straightforward I think. 37 has to do with capacitors and resistors in equilibrium. 74 is related to what we did in class yesterday. V(t), I(t) and Q(t)  will all have that exponential form for their time dependence.

Problems 61 and 62 involve the dynamics of capacitors  and they are really difficult.   They both involve exponential time dependence very similar to what we derived  in class yesterday (Monday).   In  case you want to do those problems, and you don't have to,  I  enclose here some notes on those problems. They include more detail in the derivations then you really need to be able to do yourself, but I thought some people might be interested in seeing the time dependence derived in a fundamental way for these somewhat different situations.

Please feel free to post questions on any of the problems of homework 4 here.  also, please let me know if you think you might  benefit  from more time for homework 4.

Poll on iclickers

Lots of people, well physics professors really, told me that iclickers were really great, so I was a little surprised on our first day of class when they seemed rather unpopular. I would be very interested to understand more fully the your thoughts and feelings about iclickers (I believe that most of you have used them for two quarters).  So that is why I put up the poll on the right, which I hope asks relevant questions. Please feel free to comment here or suggest better poll questions that capture the essential pros and cons of iclickers.

Midterm preparation, post 2

(edited Sunday, Oct 24, 2 PM)
In response to the comments on the previous midterm preparation related post, I thought it would be a good idea to start a new post addressing what you should primarily focus on for the midterm. These are general guidelines. My approach to a midterm (and for the final) is to think of it as an opportunity for teaching as well as evaluation. With that in mind, I try to focus on the things that I think are most important for you to learn and remember.

With regard to circuits, you should be able to calculate and understand voltage and current in relatively simple (not complex) circuits involving batteries, resistors and capacitors. You should be able to calculate and understand the current through any resistor, or through any part of a circuit, and to understand how current divides at a junction and recombines at a junction. Our circuits will all be relatively simple. They will not involve complex combinations of resistors and junctions. They will be designed to illustrate basic principles and understanding.
   
The relationship between voltage and charge in a capacitor, as well as the manner in which a capacitor charges as a function of time are both important. Also important is an understanding of the energy stored in a capacitor–-how that is related to charge, voltage and capacitance–- and an understanding of the energy density of the electric field and how that relates to the total energy stored in a capacitor.

With regard to the electric field and electric potential, the two things we studied in the first few weeks of the class, you would want to understand those also. In particular, you should be able to sketch and calculate the electric field at a given point in space from a few discrete point charges. (By discrete we mean not continuous. You don't need to know how to calculate an electric field from continuous charge distribution.) You should also be able to calculate the potential from discrete point charges, though that may be less a point of emphasis. Additionally, it is advisable to understand the relationship between electrical potential and electric field. If you were asked about that it would only be in a fairly simple, quasi one-dimensional, context, like a (parallel plate) capacitor or a situation where the potential depends only on x.

One can see how the capacitor --in which the electric field between the plates played a central role-- connects the first part of the class (the one about fields and potentials) to the middle part (the one about circuits).  The energy carried by electric fields everywhere, not just in capacitors, is an important part of that.  Later in the quarter we will evolve back from talking about circuits to reconsideration of fields in space. We will find that circuits do not make sense unless one considers them in the context of the fields that surround them. So the connection between fields, energy, and circuits is an essential part of this course.

For the midterm,  bring an 4" x 6 " index card with a reasonable number (like 10 or 20) equations written on it. Please do not write too small or cram too many equations on your card. Unless you can remember everything without it, you will really need this card. You will not be given equations at the test.  (PS. put constants on it too, like k and epsilon_0 . also e...)

Homework 3 solutions

Revisiting today's class: what we learned about Q and E-fields (and energy)

In today's class, I think we may have learned something interesting  when we began to do problem 40. As you may recall in that problem we were given the energy density in the capacitor, and asked to find the charge, Q.  There was confusion, at first, when we realized that none of the equation that we had just summarized for the capacitor would  help us.  Those equations related V and Q and U and C,  but they did not involve E.  The equation that involved E, which related E to Q, was where we originally began our examination of the capacitor.  (From that we got V and all our other equations.)  For this problem we had to go back to that more fundamental  relation and equation.  Then we were able to solve it using the direct relation between E and Q,  and the relationship between energy density and E.

This illustrates a fairly fundamental and universal  aspect  of electrostatic problems. Very often one naturally starts with the charge distribution, one gets E from that, and then obtains V from E. A natural way to derive things is to go from q (or Q) to E to V.

Realizing that tells us how to do problem 80 (the " in impossible one").  First, you would need to figure out the electric field as a function of r for a given charge, Q;  then you can integrate that, from the inner radius to the outer radius, to obtain  the voltage difference between the inside and outside  conductors.  Capacitance is defined by the relationship, usually linear, between Q  and V.

Today's class, suggestions on how to prepare for class, capacitors and energy in electric fields

One of the keys to preparing for a class, in my opinion, is to review what came before. If you spend 15 minutes reviewing and thinking about the last class, that can enhance what you learn and remember each day.  Additionally, you can let that provide a basis for thinking about what might come next. Even if what you think of is not correct, it takes you out of a passive mode and into a more active, thinking mode, which will probably help your learning.

For example, on Friday we looked at capacitors. We began with the ansatz that the electric field between the plates of a capacitor does not depend on x, the position inside the capacitor.  We then showed that the potential, V(x),  does depend on x.  Finally we noticed that the difference between  the potential at one capacitor plate and the other capacitor plate is proportional to Q,  the magnitude of the charge on either one of the capacitor plates.  This potential difference is known as the voltage, V. ( We called it delta V in class.)  One can put that another way: the charge on the capacitor, Q,  is proportional to the voltage, V.  We call this proportionality the capacitance, C, and we thus write Q=CV.  (What does C depend on?)

Next class, today, we will look at the difference in energy between a charged capacitor and a uncharged capacitor (Q=0).  to put it another way, we will look at how much work one has to do to charge a capacitor.  That is also the energy stored in a capacitor. We will discuss the possibility of viewing that energy as being in the electric field. The idea that electric fields can contain energy plays a central role in the picture  and understanding we have of light and other forms of electromagnetic waves.

Derivation of the formula for the electric potential of a dipole

You do not really have to know this, but I heard that some people were interested in, and wondering about, the origin of the equation for the electric potential of an electric dipole.   The starting point to derive that equation is simply to write down the individual contributions to the potential (at a given point) from the positive charge and the negative charge and add them together (superposition).  That is the exact and correct expression for the potential. In order to derive the equation which has cos(theta)/r^2,  you need make an approximation and to use 2 things that are not easy or obvious.

The first one is a binomial expansion of the denominator in the expressions for the contribution to the potential from the individual charges.  To set up the binomial expansion you first factor out an r from the denominator,  and then expand, to linear order, in d/r ,  where d is  is the separation between positive and negative charge, and r is the distance from the center point between the two charges to the place at which you are evaluating the electric field.

The second thing is to know that x/r is equal to cos(theta).   An x emerges from the binomial expansion because we have chosen to orient the dipole along the x axis.  This substitution, of cos(theta) for x/r, is the last step in the derivation. [r and theta provide a 2nd way in which to specify points in 2 dimensions, the first being x,y --the "cartesian coordinates".  In the r, theta  coordinate system r ranges from 0 to infinity (it is never negative) and theta ranges from 0 to 360 degrees (2 pi).  Theta sweeps out from the x axis  in a counterclockwise direction and thereby one covers the entire x-y plane in a new way.  Typically these are called cylindrical coordinates.   When you sketch a picture,  you may be able to see why x/r is equal to cos(theta).]

The binomial  expansion  is valid in the limit d much  less than r,  and that limits the  range of validity of the cos(theta)/r^2 of formula.  Later I could scan a page which shows the derivation;  if you would like to try it yourself this outline tells you how to go about it.

Homework 2 solutions

Homework 2  solutions are posted here.  Please feel free to ask anything you like about these solutions, the tutorials, or anything else related to homework 2.

Homework 3 and reading for next week.

Homework 3 is  available for you to work on at the mastering physics website.  I  will add commentary here in response to your comments and questions. I look forward to seeing your comments, questions or other thoughts below.
       For  Monday,  please finish reading chapter 23. (I think that section 23.4 is particularly interesting and important as it contains the idea that there is always energy in electric fields.) Then, when you have time, I would suggest starting to read chapters 24 and 25, with your emphasis on chapter 25.  Our next homework, homework 4,  will involve both chapters 24 and 25; the emphasis will be on problems from chapter 25.
       In reference to homework or reading i  really do look forward to seeing comments and questions here. What points, either conceptual or computational, are you stuck on or would like to know more about?
Zack

PS. If you would like to see derivation of the formula for the electric potential of the dipole, then I could do a post on that. I heard that some people might be interested.
 If there is anything else you would like to see derived or explained, please feel free to ask.

Midterm preparation:

The midterm is scheduled for Wednesday, October 27. There are many ways to prepare.  What I recommend is to focus on working and redoing homework problems.   If you do homework problems in the style expected for the midterm, that is really good preparation; I think that will help you when you get to the midterm.   The following approach is useful for many problems:

1.  draw a picture
2.  define a coordinate system (sketch that on your picture)
3.  write down the relevant equations in terms of both given (known) and unknown quantities defined in a manner consistent with your picture and coordinate system 
4.  using the equations to solve for your unknown quantity or quantities (variables) in terms of known (given) quantities
5. graph your result, if requested

 All these parts are important.  True, on most HW problems they do not ask you to graph anything, but  graphing is important and will be present on the midterm to some degree.
        You will also be given problems in which you start with the graph (of some quantity as a function of a variable, like x)  and are then ask questions about that graph (including creating another graph from the graph you are given).  For example you may be given a graph of V(x) (Volts), as a function of x,  and then asked to graph the  x-component of the electric field (also as a function of x).

Here are some practice problems that you can use for preparation in addition to using some of the HW problems:

practice problem #1:
Suppose there are two charges which are .2 m apart. One has a charge, q,  of 1 C; the other has a charge of -1 C.

a)  sketch a picture of the two charges and mark the point halfway in between the two.
What is the electric field at that point? (What would the electric field at that point be if both charges were positive?)

b)  define and sketch a coordinate system and then calculate the electric field as a function of the distance from that center point, p,  along the direction perpendicular to the line connecting the two charges.  how many non-zero components does the electric field have?

c)   Graph each nonzero component of the electric field (as a function of the distance from that center point, p,  along the direction perpendicular to the line connecting the two charges).  put titles, such as x-component of electric field as a function of y,  labels, and scales on your graph and graph axes, respectively. Make it clear what your coordinate system is! Use picture and words! The correspondence between your variables, coordinate system and picture should be made very clear. You will get more credit if you make things clear; you will get less if you leaves things ambiguous. [The graders will grade by addition, from zero, not by subtraction for errors!]

practice problem #2:
Suppose there are two charges which are .2 m apart. One has a charge, q,  of 1 C; the other has a charge of -1 C. (This is similar, but asks about potential.) 

a)  sketch a picture of the two charges and mark the point halfway in between the two.
What is the electric potential at that point?

b)  define and sketch a coordinate system and then calculate the electric potential as a function of the distance from that center point, p,  along the direction perpendicular to the line connecting the two charges.  Is this easier or more difficult than the preceding problem? Why?

c)   Graph the electric potential (as a function of the distance from that center point, p,  along the direction perpendicular to the line connecting the two charges). Why is this more difficult/less difficult than problem 1?
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(Do the same two problems for 2 positive charges (+1 C each). How does that change your results and graphs.)

3. Suppose you are given an electric potential,: V(x)= Ax^2

b) graph V(x)

a) what are the units of A?

c) calculate and graph the x-component of the electric field as a function of x.

d) If a positively charged particle, the charge of +2 coulombs were placed at x = 2 meters, what force would it experience?

e) Does that force depend on x?  what would it be at x=4 m?

Reading

This week, I think you should read chapter 23.  Our next homework will be from chapter 23. I really like the treatment of electrical potential energy and capacitors in the beginning of this chapter. The chapter goes from p 384 to 393. I would suggest reading half of it before tomorrow's class and the rest by Friday.

Up to this point, my expectation is that you would have read chapters 20 and 22.  Chapter 20 contains the subject matter of the first homework assignment; the second homework assignment––the one you doing now––is based in Chapter 22.

We skipped chapter 21 because it involves  the concept of integrating an electric field as it pierces imaginary surfaces surrounding Gaussian volumes. While  Gauss's theorem is powerful and elegant, it presupposes, or requires, a familiarity with, and intuition for, 2-dimensional integrals over vector fields, which seems ambitious. I think it is often skipped in "modern times" as people have began to realize that it involves sophisticated mathematical and geometrical concepts,  which are very difficult to convey without glossing over crucial aspects.  Including this would take time away from the other material  and could lead to superficiality and rushing through other material.

Next week, I  would suggest that you read chapter 24. I will add more here later regarding points of emphasis.  Feel free to post any questions or comments here regarding reading.

Solutions: Homework # 1

Enclosed are the solutions for homework one. There are solutions for the problems, but not for the tutorials. Please feel free to post questions or comments about the solutions, and about the tutorials here.

Also, it is difficult to completely control the order of images at this blog, so please just be aware that the four pages above may be in a somewhat random or incorrect order.

Check out these pictures of electric fields, (equi)potentials and more cool stuff.

Check out these pictures of electric fields, (equi)potentials...
http://www.falstad.com/vector2de/

http://www.falstad.com/emstatic/

explore the options....

Problems with Mastering Physics?

Courtney, from Pearson/Mastering Physics, will come to our class on Wednesday. In the meantime you may contact her at:
Courtney.Austermehle@pearson.com
I have found her to be extremely helpful and capable.
-Zack

Homework 2, electrical potential (energy): feel free to post questions and comments here

Our second homework assignment, homework 2,  is available at the mastering physics website. The problems and tutorials are all from chapter 22.   The tutorial on electrical potential  and electrical potential energy  helps develop the analogy between gravitational and electrical potential energy. I hope you will find it helpful.   Regarding any of the problems, please feel free to post your questions and or comments here.

This homework deals with the subject of electric potential and electric potential energy. It will help you, hopefully, to understand these topics and to distinguish between potential and potential energy, a distinction which mostly has to do with units and meaning.  The units of electric potential energy are typically Volt-Coulombs,  which are  equivalent to Joules. the units of electric potential, on the other hand, are simply Volts.   So they differ by a multiplication by a charge of  1 Coulomb.

There is another common and "natural" unit used for electrical potential energy, which is the electron volt (eV).  That is what you get when you multiply volts times the charge of an electron. since an electron has a charge of  1.6 x10^-19 coulombs, 1 eV is equal to 1.6 x10^-19 Joules (That is, 1 eV is equal to 1.6 x10^-19 Volt-Coulombs.)

Electron volts are often  used to talk about atomic molecular systems, where  electron transitions and ionizations  typically involve energies of order 2 to 20 eV.

Photons of visible light have energies between about 2 and 4 eV.   UV photons have more energy, say 4 to 20 eV.   The  ability of UV photons to damage cells is associated with their  higher energy and the electronic processes that that allows.

 Back to the homework, I thought that problem 22.30  was/is an  interesting and informative problem,  but part C-- the way they express their answer-- seems cryptic. So I'll just tell you: the answer is 180. (You tell me what that means. I mean, it does mean something, once you see it you might know what it means, but it might be  difficult to realize in advance that that's what they want.
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additional commentary,  added  October 6:
for problem 54* ( Find the potential 16  from a dipole of moment 3.1  (a) on the dipole axis, (b) at 59 to the axis, and (c) on the perpendicular bisector. The dipole separation is much less than 16 .) (*the  numbers  maybe different for you):
I believe they expect you to use an approximate equation for the potential associated with a dipole, which is reasonable (valid) when the distance away from the dipole is much greater than the  separation between the two charges of the dipole. (Note how they say: “the dipole separation is much greater than…”.)   This equation involves cosine(theta),  and is valid even when theta is greater than 90°.  I'm not a big fan of using this equation in  an  intro course, because it engenders a plug and chug  mentality, so probably on tests you would be given the dipole separation and the magnitude of the individual charges and asked to find the potential, or field, at particular points in space  or, as a function of x or y  along a symmetry axis.

 However, I realize that later in this class we will need to understand cosine and sine  functions, including their behavior beyond the intuitive 0 to 90° range.  so let's review that here.

For theta in the range from 0 to 90°, the cosine can be thought of in terms of adjacent/ hypotenuse ratio with  reference to the standard picture of a triangle and an angle. ( sketch that now)  cosine starts at one, at theta = 0, and reaches zero at theta =  90°.    Beyond that it goes negative, it returns to zero at theta =  270°,  and returns to one at 360°.  Cosine has a minimum value of -1 at theta =   180°. (graph that now...).   The sine function is similar, and also different.  It starts at zero  and grows to a maximum of one at 90°. It is one when the cosine is zero,  and zero when the cosine is one. In that sense they are complementary.   Well,  I guess that's enough about sine and cosine  for now. Please feel free to ask any questions here.