Newton’s Law

DP1 Physics

For Your notes on Newton's Laws Visit

http://sehsibphysicsi.edublogs.org/

Group 4 – Physics Project

 

Group 4 Project.

Theme Alternative energy sources and waste management for sustainable AKA, M

Research question: Exploring the potential viability of meeting the school power demand through harnessing solar energy within the possible areas in AKA,M .

Type of research explorative, investigative, grounded

Methodology: Quantitative

 

Thought process (Guiding Questions)

 

What kind of data do we need?

1. Electric consumption bills for the past couple of years
   
2. Trends over the years
   
3. Use school blueprints to get the area of the roof space
   
4. Light intensity in the locality
   
5. Weather Pattern in coastal region – average monthly sunlight levels – Kilindini
   
6. The amount of light intensity to produce a Watt of energy
   
7. Conversion rates with a variety of solar panels
   

How to collect the data?

1. Electricity bills from school – ask finance department (documents) – Aziz/H.C
   
2. School plans (Abraham/H.C/Ms. Clough) – or measure
   
3. Light Intensity Experiment – cloudy day and sunny day/morning, noon, evening OR a Solar Card
   
4. Weather Pattern from Internet or Metrological Center????
   
5. Check specification of Solar Panel – from shop and internet
   

What instruments will we need?

1. Solar Card/Light Meters
   
2. Internal Assessment Experiment Instruments
   
3. Measuring Tape
   
4. Stationery, Calculator, Laptops, GDC
   
5. Mobile Phone WITH CREDIT!!
   
6. Uncensored Internet Connection – No Firewall to restrict downloads from sites for example email attachments
   

   
    

   
    

How will data be stored?

1. Using Laptops and Paper
   
2. Imaging Devices
   

How are we going to process/analyze the data?

1. Tabulate the electricity bills and draw a trend graph, and find the mean/moving average over the years.
   
2. Graph of potential kilowatts energy to provides in each area of school
   
3. Make a map of school divided into sectors
   
4. Get the average of the light intensities at different times of the day and different areas of the school which will be put on the map of the school.
   
5. Average weather conditions over a period of time - table
   

 

Terms of Reference

All data collected goes to Secretary – Shailen & the nerve center – Levi

 

Duties

1. Electric consumption bills for the past couple of years – LEVI (Thursday 29^th May 2008)
   
2. Use school blueprints to get the area of the roof space - LEVI (Thursday 29^th May 2008)
   
3. Light intensity in the locality – DIANA AND SHAILEN – (Friday 30^th May 2008) by 10:45am
   
4. Weather Pattern in coastal region – average monthly sunlight levels – Kilindini – AKSHAY (FRIDAY 11:00am)
   
5. The amount of light intensity to produce a Watt of energy – SHAILEN (FRIDAY 11:00am)
   
6. Conversion rates with a variety of solar panels - SHAILEN (FRIDAY 11:00am)
   

How to Write a Term Paper

- Some Guidelines

Introduction

Writing papers may well be the opportunity for you to learn more about the subject you are studying than any other aspect of a course. It is worth doing well. You not only learn more, you also think more deeply about a topic when you have to put words on paper. Finally good grades depend on good papers.

I Collecting Information

Opinion is a fine thing, but in a college paper your opinions are only worthwhile if they are backed up by facts and arguments. You must collect information, and, since many topics will be new to you, it is worthwhile looking at the work and opinions of more than one author. You should certainly look at your textbook but also at other authors. Your professors will always be willing to give suggestions.

As well as your textbook, you should learn to use the library as a source of information. Make it a top priority to learn how to find a book in the Library.

II Recording Information

It is no use to just read a book and then write. You must record what you read so that you can review it before and during the writing of the paper. There are a number of ways to do this:

- You can mark the book - only if it is your own copy or a photocopy - with pencils or highlighting pens. You cannot use this method on Library books and it is of limited use as it can be difficult to locate what is really important if you have marked up half a book. It also reduces the resale value of books.

- You can use 3"x5" index cards and note down one, or a series of connected facts, on a card. You then use the cards to organize the information in the way you want to use it in the paper. One problem is that you may get bogged down in detail. The other is that it can be difficult to review index cards at examination time. In general this is the method that is successful for most people. Make sure that you note down on each card the source of your information or you lose track of what each card means.

- Finally you can try to summarize a chapter on letter or legal paper. You can note down both facts and arguments at length. This system can be cumbersome if you take a lot of notes, but is very good for reviewing before exams.

III Thinking About the Topic

After you have read as much as you need, DO NOT just start to write. Think about what you have read, mull over it on a walk, or discuss it with friends. The professor already knows about what you are writing and is looking to see how well you have understood a topic. It is no use at all to just present your reading notes stuck between an introduction and a conclusion.

Thinking about it is the most important stage of writing a paper.

IV The Plan

Sketch out on paper several ways of presenting your topic and your thoughts. You might think of doing this as a connected argument, or as a series of related headings organised in a way that makes sense of what you read. Another useful approach is to state, prove and defend a thesis.

You must always write out a plan. It will help you to be clearer both in papers and in tests. It is in fact another way of thinking about your topic.

V Writing and Editing

You cannot expect to just write out a paper and hand it in. Typo's alone will demand at least one re-type. So why not throw out the idea that what you write must be perfect first time?

It is a good writing technique to just WRITE down your thoughts as they come into your head (always keeping an eye on your paper plan). Do not stop to edit or correct spelling and grammatical mistakes. WRITING and EDITING are different skills. Even though you may think what you are writing is bad or plain stupid, once you have got it down on paper you can go back and look at what you have written. At that stage you can begin to knock it into shape, correct spelling and grammar and improve your style. Almost everybody thinks that what they are writing is bad at the time they write it: your aim is to find a way around this mental block.

You should note that in an exam, conditions force you to write and edit at the same time, however, the technique described here should help to improve your confidence in writing.

VI Finishing Touches

Before you hand a paper in make sure it looks good - use the Stylesheet handed out separately. Eliminate spelling and grammatical errors. Make sure all your references are noted. Add a booklist. Type the paper cleanly

Information

We did a test for the HL group on Thermodynamic and Waves.
Most of the members were complaining that they were not informed.

Please be informed this week we shall have a Test for both HL and SL group.

Best Wishes

SL option on offer NOW

The Following Option is on offer for SL students ONLY.

If you are interested, do attend the DP1 AHL sessions

OPTION A: SIGHT AND WAVE PHENOMENA 15

A1 The eye and sight 3

A2 Standing (stationary) waves 2

A3 Doppler effect 2

A4 Diffraction 1

A5 Resolution 4

A6 Polarization 3

Thermal Physics AHL

Shailen Explaining the reverse cycle of a Heat Engine

Schedule

Week 2

Tuesday Session to solve questions with both groups

Core Electric Currents

AHL Waves Properties

This Week no Labs

The double core and AHL will be used by the AHL group to catch up.

Long Format of Notes

5. ELECTRICITY AND MAGNETISM

 

5.1. Electric charge

 
 

        Atoms consist of heavy, positive protons and neutrons in the nucleus and light, negative electrons around it

        the two types of negative and positive electric charge are a fundamental property of materia, like mass

        the net charge is conserved, like mass (except that mass and energy can be converted to each other (relativity))

        masses always attract each other, but charges of the same type repel; different types attract

        the unit of charge is 1 coulomb = 1 C; the charge of one electron = e = - 1.6 x 10^-19 C (we can sometimes also use e = the elementary charge = 1.6 x 10^-19 C and then the charge of the proton is e, the charge of one electron is - e.)

        since the sign of the charge denotes its type ("positive" or "negative") but no direction, charge is a scalar quantity.

 
 

Conductors, semiconductors and insulators

 
 

A material which electrons can move easily through is a conductor, one where this is more difficult is an insulator. Metals are good conductors because metal atoms have a few electrons in the outer shell which are not very strongly attached to any particular nucleus. Semiconductors are materials where the possibility of conduction of charge depends strongly on some factor (direction, temperature, light, other).

 
 

 

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In a piece of metal the "unwanted" outer shell electrons are not connected to any particular metal nucleus and can easily be set in motion by any electric force acting on them. As a result of this, electrons may then be moving through the metal conductor at some drift velocity which may not be very high (compare to swithcing on the water in a garden hose - even if the water starts to move almost immediately, a water molecule does not immediately travel from the tap to the end of the hose).

 
 

When traveling through the metal the electrons will collide with the metal "cations" (positive ions) formed by the nuclei and the inner shell electrons. In these collisions they lose some of the kinetic energy they are given by the external battery or other causing the flow of electrons.

 

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Electrification by friction and contact

 
 

By rubbing materials against each other some electrons can be moved from one object to each other, which means one will have a positive and the other a negative net charge. This works best with insulators where the net charge on the surface of the material is not easily spread out through the whole object.

 
 

 
 

 
 

 

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If a charged object is brought to contact with a conductor with no net charge, this conductor will also be charged (but the net charge on the first object will decrease).

 

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Electrostatic induction

 
 

If an electrically charged object is place near another object where charges can move easily (a piece of metal), charges in this object will be attracted or repelled. If an object is allowed to touch another conductor or some charges are led to or from it from the earth, a conductor can be charged without touching it.

 

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The electroscope

 
 

A simple instrument to show the presence of electric charge is based on light pieces of conductors (metal) which all are in contact with each so that if the electroscope plate is touched by a charged object, the net charge is distributed over all inner parts of the instrument (but the outer parts are kept insulated).

 
 

Some of the inner metal parts are then easy to move by a repulsive force, which can be seen (gold leaves moving apart, or a metal needle turning in other types of electroscopes).

 
 

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If the conductor is hollow, the charge will be distributed on the outside of it, and the inside left uncharged (it will form a "Faraday's cage").

 

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This explains why it is relatively safe to sit in a car or an airplane in a thunderstorm, or why radios and cel phones may not work inside metal cages or buildings.

 
 

5.2. Electric force and field

 
 

Coulomb's law for electric force

 
 

F = kq₁q₂/r² where k = 1/4pe₀     [DB p.7]

 
 

where q₁ and q₂ are the charges, r the distance between them (or the distance between the centers of them if they are not very small "point charges").

 

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The Coulomb constant k = 8.99 x 10⁹ Nm²C^-2 in vacuum and approximately the same in air. In other materials a k-value can be calculated from the relevant e-value (electric permittivity). The k-value and the permittivity in vacuum (or air) are given in the data booklet. The e-value for other materials is given when necessary. In vacuum or air e₀ = 8.85 x 10^-12 Fm^-1 (F the unit 1 farad, not explained here but a SI-unit). Some table list relative permittivity (e_r) values, where the actual permittivity e = e_re₀.

 
 

This can be compared to Newton's law of universal gravity F = Gm₁m₂/r² but :

 
 

        we have charges instead of masses

        k is much greater than G, but mostly electrical forces are not noticed since ordinary materia consists of both positive and negative charges, and the Coulomb forces usually cancel out

        unlike the G-value, the k-value depends on the material (it is much different in water than in air or in oil).

 
 

The Coulomb formula gives the magnitude of the force on either of the charges q₁ and q₂. The directions of the forces are opposite (repelling or attracting) because of Newton's III law.

 
 

Note:

 
 

        if we have more than one charge present, we may have to split up the force(s) from some of them into components parallel or perpendicular to suitably chosen directions

 
 

Electric field

 
 

Coulomb's law gives the force acting on a charge q₁ caused by q₂. If we want to describe what force would act on an imagined small positive test charge q₁ here called just q, we can define the electric field strength as

 
 

E = F/q₁ which in the IB data booklet is given as:

 
 

E = F / q    [DB p.7]

 
 

a vector quantity with the unit 1 NC^-1

 
 

Using Coulomb's law for the field caused by a charge q₂ we get

 
 

E = F/q₁ = (kq₁q₂/r²) / q₁ = kq₂/r² which in the IB data booklet is given as:

 
 

E = kq / r²    [DB p.7]

 
 

Notice that like in Mechanics where m sometimes means the mass of a planet causing a gravitational field and sometimes the mass of a spacecraft in that field, here q also sometimes means a "big" charge causing a field, sometimes a small test charge in that field. If we further compare this to the force of gravity and remembering than mass is replacing charge we get

 
 

        F/m = g = the gravitational field strength (near earth the usual gravity constant 9.81 ms^-2 which is the same as 9.81 Nkg^-1 ; compare this to the unit 1 NC^-1 !!

 

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Note that since the imagined small test charge is positive the field is directed away from a positive charge, and towards a negative charge. The field of this type can be called a radial field.

 
 

The field lines drawn do not exist in reality (like the charge causing the field does), they are graphic descriptions of what would happen (what force would act) if the small positive test charge was placed in a certain place

 
 

        the closer the field lines are, the stronger is the field (nearer the charge; the further away, the weaker)

   
 

Electric field patterns for other situations

 
 

If we have two or more charges, the field in a certain point is the sum of the fields caused by the charges. Since the field E is a vector quantity, directions are relevant and it may be necessary to split the field vector into suitably chosen components.

 

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        two point charges of different type: on a line through the charges, the field is from the positive to the negative between them, away from the positive and into the negative on the far side of them. In other regions, the field lines are bent curves since at any point it is the resultant of a vector towards the negative and one away from the positive charge (remember that the field is defined from a hypothetical small positive test charge - if a negative charge is placed in the field, it will be affected by a force in the opposite direction to the field). Since the distance r to the charge appears in the E = kq/r² , the magnitudes of these vectors vary. The bent lines do not follow any known mathematical function (they are not parabolas, hypberbolas or other such curves) and have to be found by calculating the field in every point in the plane separately (in practice by computer).

 

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Note: If we place a small positive test charge at rest in the field, it will initially be affected by a force in the direction of the field in the point where it is placed, but its motion thereafter will not generally follow a field line - the electric force is parallel to the field, and the acceleration is parallel to the force (F = ma), but the new velocity v after a short time period t is v = u + at, where u and at are vectors, and generally not parallel.

 
 

        two point charges of same type: if they both are positive, they will "bend away" from a line where the distance to both is the same. If both are negative, the shape of the field lines is the same but the direction opposite.

 
 

        a charged metal sphere: outside the sphere, the field is the same as if all the net charge on the sphere was concentrated to its center; inside the sphere it is zero.

The field lines from a metal surface are always at a 90 degree angle to it (otherwise they would have a component parallel to it, and this component would result in a force parallel to the surface on any freely moving charges on it, and they would move until this is no longer the case).

=> if the hollow metal object has another shape, the E-field lines still have to be perpendicular to its surface. They will be closer together and the field stronger at sharp and "pointy" places.

 
 

        two oppositely charged parallel plates: between the plates, the field is the resultant of millions of field vectors each describing the effect of one small charge on either of the plates. The "sideways" components cancel out and the field lines are parallel, going from the positive to the negative plate. At the ends, outside the area between the plates, they are slightly bent.

 
 

A homogenous or uniform field is one which in some area has the same direction and magnitude. Can be produced by parallel metal plates.

 
 

5.3. Electric potential energy, potential and potential difference = "voltage"

 
 

Electric potential energy

 
 

The electric field between a positive and negative metal plate is homogenous and similar to the gravitational field near the surface of a planet (so near that the facts that the planet surface is not flat and the gravitational force and field get weaker far out in space can be disregarded).

 
 

 
 

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If a positive test charge q is "lifted up" from A to B or "falls down" from B to A, the change in its potential energy caused by electrical forces can be calculated. (There may be a force of gravity and gravitational potential energy involved also, but since the k-constant is much larger than the G-constant it can usually be disregarded. Also, since we assume the situation to be independent of any force of gravity, the plate pair can be turned any way we like; "up" just means towards the positive plate and "down" towards the negative.)

 
 

The work done by or against the E-field is then

 
 

        W = F_electricx but since E = F/q we get F = qE and then

        W = qEx = the change in potential energy

 
 

(compare this qEx to mgh where charge q corresponds to mass m, the electric field strength E to the gravitational field strength = the gravity constant g, and x or h symbolise how far "up" or "down" the field we have moved.)

 
 

Electric potential V

 
 

For the force of gravity we had

 
 

        the gravitational potential V = E_{p,gravitational} / m

 

and this is here replaced by

 
 

        the electric potential V = E_p,electric / q

 
 

Remember that the gravitational potential V = E_p/m = mgh/m = gh is rarely used since most applications of physics are placed near earth and the g-value always the same, so only the h-value is interesting, for example as in the height difference between to places. We now get:

 
 

        the work = change in electric potential energy E_p = W = qEx

        but since the electric potential is defined as V = E_p/q = W/q = qEx / q we get

        V = Ex, which using "deltas" and a negative sign to show that if we move against the field we gain potential energy and if we move with the field we lose potential energy:

 
 

E = - DV / Dx    [DB p.7]

 
 

Another way to write this is, now replacing Dx with d for the distance between two charged plates:

 
 

E = V / d    [DB p.7]

 
 

Comparing gravitational and electric quantities: A summary 

 
 

Here we will for clarity let the big central mass or charge be represented by M or Q, the hypothetical test- or other small mass or charge with m or q.

 
 

 

 

 

GRAVIT.			ELECTR.
Homogenous		Point,planet		UNIT	Homogenous	Point, sphere	UNIT
F = mg	F= GMm/r2	N	F = qE	F = kQq/r2	N
g = F/m	g = GM/r2	Nkg-1=ms-2	E = F/q	E = kQr2	NC-1=Vm-1
Ep = mgh		Ep=-GMm/r		J	Ep = qEd	Ep = kQq/r	J
V = Ep/m= gh		V = -GM/r		Jkg-1	V = Ep/q =Ed	V = kQ/r	JC-1 = V

 
 

Quantities corresponding to each other (gravitation - electricity), in addition to this the universal gravity constant G = 6.67 x 10^-11 Nm²kg^-2 is replaced by the Coulomb constant k = 8.99 x 10⁹ Nm²C^-2 .

 
 

F - F

g - E

Ep - Ep

V - V

M,m - Q,q

h - d

 

Potential difference = "voltage"

 
 

The potential difference V (if the potential in one point of comparison is zero) or DV between to places in the uniform field or between the plates causing the field is

 
 

V = E_p / q

 
 

so its unit is 1 JC^-1 which is called 1 volt = 1 V.

 
 

The potential difference between two points is what is commonly called the "voltage" between them.

 
 

It is extremely useful to remember this:

 
 

voltage = work or energy per charge

 

for later applications.

 

Since we have E = V/d we can write the unit for electric field strength E as 1 Vm^-1 in addition to the earlier presented unit 1 NC^-1 based on the definition E = F / q.

 
 

These units are the same : 1 Vm^-1 = 1 JC^-1m^-1 = 1 NmC^-1m^-1 = 1 NC^-1

 
 

The unit 1 electronvolt = 1 eV = an energy unit

 
 

If one electron with the charge q = e (or - e depending on which definition we follow) = 1.6 x 10^-19 C is accelerated through a potential difference of 1 volt, it will get an energy = the work done = qV = 1.6 x 10^-19 C x 1 JC^-1 = 1.6 x 10^-19 J = 1 eV.

 
 

A situation confusing enough to make angels cry is the fact that V is used both as the symbol and the unit for potential ( we can write V = 5.0 V ) and e both for the electron, the charge of an electron, and in the unit eV for the energy of an electron.

 
 

The unit 1 eV for energy is in atomic and nuclear physics also used for many other purposes than just electrons. The energy a charge - electron or other - gets when accelerated by a potential difference can be as kinetic energy, if air resistance and other forces are not considered:

 
 

qV = ½mv²

 
 

Electric potential from a point charge or charged sphere

 
 

 For the gravitational force, a different formula for potential energy had to be used in situations where an object was not staying near the surface of a planet but moving at significantly different distances to it (or rather its center), meaning that the force of gravity on it was not constant. The same can be found for electrical forces - and we can define electric potential V as:

 
 

V = kq/r where k = 1/4pe₀     [DB p.7]

 
 

The electric potential is a scalar, which is zero when r is infinitely large. If the potential difference between two points is calculated, this potential difference ("voltage") can be related to the energy or work W needed to transport a charge q against the field from one point to the other (or the energy released in the opposite case) as before :

 
 

V_A - V_B = DV = qW

 
 

Electric potential from some charge systems

 
 

        point charge : the potential positive near a positive charge (which would repel a small positive test charge - unlike gravity which is always attractive!) and negative near a negative charge. The value follows a hyperbolic curve, approching positive or negative infinity near the charge, and zero infinitely far from it.

 
 

        outside a hollow conducting sphere the potential follows a curve similar to that from a point charge at the center of the sphere; inside the sphere the value of the potential is constant at the value at its surface, since the field E inside it is zero, no resultant force would act on a test charge and no work would be needed or released inside the sphere.

 
 

 

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Equipotential lines or surfaces

 
 

A graphic way to illustrate electric potential are equipotential lines (or in a 3-dimensionsal situation surfaces) for which we have :

 
 

        they describe points where the potential has the same value

        they are always perpendicular to electric field lines

        the same work is needed/released when a charge is moved between two equipotential lines or surfaces

        no work is needed/released when a charge is moved along one

        they can be compared to altitude curves on a map for gravity (strictly, gravitational potential = altitude multiplied by the gravity constant)

 
 

Certain situations are commonlt studied:

 
 

        isolated point charge: the equipotential lines are concentric circles or in 3 dimensions spherical surfaces

 
 

        charged conducting sphere: outside the sphere, the equipotential lines/surfaces are the same as for the point charge

 
 

 

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        two point charges: near each of them they are approximately circles/spheres, between them is a straight line (in 3 dimensions, a planar surface). Note that they are always perpendicular to the field lines.

 
 

 
 

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        parallel oppositely charged plates: they are straigth lines parallel to the plates, or in 3-d parallel planar surfaces

 
 

 

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5.4. Electric circuits: current, resistance, power

 
 

Electric current

 
 

So far we have mainly studied electrostatics, the physics of electric charges at rest. Since the charges can be affected by forces, they may also move. We can then define electric current I as :

 
 

I = D q / Dt    [DB p.7]

 
 

or simpler I = q / t = the amount of electric charge transported per time.

Unit: 1 coulomb/second = 1 Cs^-1 = 1 ampere = 1 amp = 1 A. Since currents are easier to measure than charges, it is the ampere which is used as a fundamental unit in the SI-system, and 1 C = 1As

 
 

 

 

 

 

Electric circuit, conventional current and electron flow

 

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An electric circuit consists of

        a source of "voltage" = potential difference, for example a battery

        a resistor (or more complicated arrangements of components) where the energy/charge supplied by the battery is used

        connecting wires between the positive terminal of the battery and the resistor or other apparatus, and that and the negative terminal (for alternating currents the positive and negative terminal may switch many times per second). Two wires (or something else doing their job) are always needed to complete the circuit (unless the current is flowing to or from an enormous body like the earth)

 
 

The "conventional" current is from the positive to the negative terminal (the way a positive test charge would go), while the actual electron flow is in the opposite direction.

 
 

Electric resistance

 
 

For any circuit or component where the current I is caused by the potential difference V we define the electric resistance R as :

 
 

R = V / I    [DB p.7]

 
 

Unit: 1 VA^-1 = 1 ohm = 1 W

 
 

        The resistance describes how "hard" it is to move charges through the resistor - the higher R, the more "voltage" is needed to keep up a certain current. Good conductors have a low R, good isolators a very high R

 
 

[We could have defined the inverse quantity to describe how well a component conducts electricity: the electric conductivity k (kappa) or G = I / V with the unit 1 AV^-1 = 1 W^-1 = 1 siemens = 1 S, sometimes called 1 mho ("ohm" backwards!). In chemistry, the conductivity is related to the amount of ions in a solution; a solution of an ionic-bonded or polar compound has a high k, while a solution of very clean water or a covalent, non-polar compound has a low k.]

 

 

 
 

Ohm's law with V- and A-meters

 
 

The resistance R can be defined or measured for any component; but for metallic conductors at a constant temperature R is constant. This is Ohm's law.

 
 

        if the conductor is ohmic, a graph of I as a function of V will give a straight line with the gradient 1/R [ = k ], while a graph of V as a function of I will give one with the gradient R.

        if the conductor is non-ohmic, the graphs will be other curves

 
 

To experimentally produce this curve we need a circuit with

 
 

        an amperemeter = A-meter connected so the current flows through it ("in series" with the resistor). A good A-meter has a very low resistance which can be neglected.

        a voltmeter = V - meter connected "beside" the electron flow ("in parallel"). For a good V-meter, very little of the current flows through since it has a high resistance. (The A- and V-meters are based on magnetic phenomena studied later).

        a resistor

        a "voltage" source. Either the resistor or the the voltage source is variable.

        connecting wires, with a negligibly small resistance

 

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The filament lamp

 
 

This device which is an ordinary "light bulb", has a spiral metal wire which is heated by the flowing electrons colliding with the metal electrons until it glows brightly. The metal is tungsten with a high melting point. Since the temperature changes radically when a filament lamp is turned on, the R is not constant but increases with temperature. The result is that the slope of an I-V-curve decreases with higher V. There are other light sources and components with different characteristics.

 

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Circuit diagram symbols: see data booklet

 
 

Electric power

 
 

We recall that power P = energy or work per time. Now:

 
 

potential difference V = W/q and current I = q / t so we obtain VI = (W/q)(q/t) = W/t = P :

 
 

P = VI = I²R = V²/R    [DB p.7]

 
 

where the unit 1 watt = 1 W = 1 VA. Notice that since P = W / t we get W = Pt so 1 Ws = 1 VAs = 1 J is a unit of work or energy.

 
 

More common is the unit 1 kWh ("kilowatthour") = 1000 W x 3600s = 3.6 MWs = 3.6 MJ. The other relations are found by combining formulas:

 
 

        R = V/I so V = RI giving P = VI = RI²

        R = V/I so I = V/R giving P = VI = V²/R

 
 

The power is said to be "dissipated" meaning that this amount of energy per time is lost to heat.

 
 

Equivalent (effective) resistance for series and parallel circuits

 
 

[Kirchoff's laws, not required in the IB :

 
 

KI. The sum of currents flowing into a point = the sum of currents flowing out of it.

 
 

KII. The sum of all potential differences around any closed loop in an electric circuit is zero. "Voltage" sources usually counted positive and the "voltage drop" RI in resistors negative

 ]

 

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In a series circuit we have that

 
 

        the current only has one possible way, and is the same through both resistors R₁ and R₂, that is I₁ = I₂ = I (KI applied to arbitrary points in the circuit)

• KII gives V - R₁I -R₂I = 0 or
  
• V = R₁I + R₂I which using R₁ = V₁/I => R₁I = V₁ and R₂ = V₂/I => R₂I = V₂ gives:
  

 

 

        the sum of the voltage drops is the total voltage drop in the resistors or V = V₁ + V₂

        if we would like to replace the resistors R₁ and R₂ with only one with the same resistance R as they both have together, we get when dividing both sides in V = V₁ + V₂ with I:

        R = V/I = V₁/I + V₂/I = R₁ + R₂ so:

 
 

R = R₁ + R₂    [DB p.7]

 
 

For 3 or more resistors we have R = R₁ + R₂ + R₃ + ...

 
 

 

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In a parallel circuit the current can take different ways and splits up as I = I₁ + I₂ according to KI. We can apply KII in 3 different closed loops:

 

Loop 1 : through the battery V and R₁ but not R₂:

• V - R₁I₁ = 0 so V = R₁I₁ = V₁, the voltage over R₁
  

 

Loop 2 : through the battery V and R₂ but not R₁:

• V - R₂I₂ = 0 so V = R₂I₂ = V₂, voltage over R₂
  

 

Loop 3 : through R₁ and R₂ e.g. clockwise but not through the battery V:

• now the hypothetical loop passes through R₁ against the direction of the current; we are actually following the "current" -I₁ through it
  
• so R₁(-I₁) + R₂I₂ = 0 giving R₁I₁ = R₂I₂ or V₁ = V₂
  

 

All these lines of reasoning can then be followed by this:

        the potential drop is the same no matter which resistor we follow the current through: V = V₁ = V₂

        for the equivalent resistance R we then get R = V/I and then I = V/R so

        V/R = I = I₁ + I₂ and

        V/R = V₁/R₁ + V₂/R₂ but since V₁ = V₂ =V we get

        V/R = V/R₁ + V/R₂ and dividing both sides with V

        1/R = 1/R₁ + 1/R₂ , (with more similar terms for 3 or more resistors in parallel)

 
 

1/R = 1/R₁ + 1/R₂    [DB p.7]

 
 

 
 

Note : If we have both serial and parallel connections combined, we can stepwise replace them with effective resistances until the effective resistance of them all is arrived at.

 
 

5.5. Electromotive force (emf) and internal resistance

 
 

The source of electric potential difference is some device which gives a certain amount of energy/charge to the moving charges. This energy may come from chemical reactions in a battery. Since the electrons are moving around the circuit out from one terminal and eventually back into the other, they must also in some way move through the battery (possibly attached to ions moving in solutions in the battery or otherwise). Even if the resistance in the circuit outside the battery - the "outer circuit", with its "outer" or external resistance R (which may consist of a complicated set of serial and/or parallel connections which give this total effective resistance) there will always be some internal resistance r in the battery. This causes a potential drop (= loss of some energy per charge):

 
 

electromotive force = emf = e = rI + RI

 
 

where:

 
 

        emf is not at all a force, but a "voltage" = energy/charge; the one supplied by the original source of energy (for example chemical reactions). In Finnish "lähdejännite", in Swedish "källspänning".

        rI = the potential drop in the battery. Note that it depends on I - the higher current is drawn from the battery, the more loss inside it. (This is why in older cars the headlights would be dimmed when a lot of current was drawn to the starter engine).

        RI = the potential difference = "voltage" available in the external circuit connected to the terminals of the battery. Also called terminal voltage. (In Finland symbolised U = napajännite = polspänning). This is what earlier was symbolised V in R = V/I, the voltage over the external circuit, so we can write:

 
 

emf = rI + V

 
 

 

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5.6.* Capacitors

 

Capacitors (Sw. kondensator, Fi. kondensaattori) are devices where electric charge can be stored on plates or sheets of metal (often wrapped to a small cylinder) with a thin layer or air or insulating material in between. The amount of stored charge is much smaller than in a battery, but the advantage is that it can be taken out of there very quickly, in a small fraction of a second, which makes them useful in alternating current (AC) circuits where the direction of the current may change many times per second.

 
 

For the capacitor, the capacitance (kapacitans, kapasitanssi) is

 
 

C = Q / U

 
 

in the unit 1 farad = 1 F = 1CV^-1, if we use the symbol U instead of V for "voltage" = potential difference and as usual Q for the amount of electric charge stored in the capacitor (the charge is of opposite signs on opposing plates, but there is equally much positive and negative).

 
 

If the area of the plate (only one is counted, not both the positively and the negatively charged one) is A and the distance between the foils d then

 
 

C = e_re₀A/d

 
 

where e₀ = the electric permittivity in vacuum (same as in F = 1/4pe₀)(q₁q₂/r²). e_r = the relative permittivity = a number without unit which gives the correction for the vacuum value. Found in MAOL's tables for various substances.

The energy stored in a capacitor (in the unit J) is

E = ½QU

 
 

(which using C =Q/U can be written E = Q²/2C or E = ½CU² ) where the voltage U is in volts and the charge Q in coulombs.

 

Capacitors in series and parallel

 
 

Capacitors can be connected in series or in parallel like resistors. Then we have:

 
 

series => same charge, parallel = > same voltage

 
 

Compare to resistors:

 
 

series => same current, parallel => same voltage

 
 

For the total capacitance we have the formulas

 
 

series 1/C_tot = 1/C₁ + 1/C₂ + ....     parallel C_tot = C₁ + C₂ + ...

 
 

(Note that is opposite to the formulas for resistors! There it was

 
 

series R_tot = R₁ + R₂ + ..., parallel 1/R_tot = 1/R₁ + 1/R₂ + ...

 
 

 
 

5.7. Magnets and magnetic fields

 
 

Magnetic poles

 
 

Magnetism is the long-known phenomenon that pieces of certain materials (like an iron-rich ore, magnetite) turn towards north? The ends of magnetic materials can be called North and South poles, and like electric charges,

 
 

like magnetic poles repel, opposite poles attract

 
 

 

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Magnetic field lines

 
 

Electric fields do not materially exist, but describe what would happen to (in what direction a force would act on) a small positive test charge placed in a certain point.

 
 

Magnetic field lines similarly do not exist but

 
 

the magnetic field B describes in what direction the north end of a small test compass would point if placed in a certain point

 
 

        the unit of the magnetic field B (a vector quantity) is 1 tesla = 1 T, which will be explained later, as will why the quantity also can be called flux density.

 
 

A problem here is that we do not have isolated N or S poles - a magnetic piece of material always has both poles, and if it is sawed in two pieces these will also have both N and S poles.

 
 

We can also note that the test compass is not accelerated by a force along a field line, only turned into the correct direction (as if by a torque rather than a force).

 
 

Magnetic field lines from permanent magnets

 
 

        a bar magnet : the field lines go out from the N pole and into the S end and are otherwise shaped like the electric field lines around a + and - charge.

 
 

        Earth : the field lines are shaped as if a bar magnet was placed inside the Earth with a "magnetic north pole" near the geographic N pole, although physically this is a S pole, since it attracts the N poles of compass needles.

 
 

Magnetic fields caused by currents (all!)

 
 

In the 1800s it was discovered that compass needles also react to wires carrying electric current. Later it has been revealed that all magnetic fields are caused by currents.

 
 

        bar magnets : in the atom, electrons orbit the nucleus and this is like a current around it. In most materials the magnetic fields from the atoms are cancelled out since they are in random directions; some materials can be magnetised = have the small fields in more or less the same direction.

        Earth : flows of molten rock under high pressure in a plasma state act as convection currents inside the earth, powered by the heat from nuclear reactions inside Earth. These cause the "permanent" magnetic field, which every few million years or so changes direction, possible as a result of chaotic processes.

 
 

 

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More common examples of current-caused magnetic fields are:

 
 

        straight wires : when looking in the conventional direction of the current, the magnetic field lines are in concentric circles clockwise around the wire. We may also use

 
 

For a long thin straight wire carrying the current I the magnetic field B at a distance r from the wire is

 
 

B = m₀I / 2pr    [DB p. 7]

 
 

where the direction of the field is given by the first right hand rule and m₀ = the magnetic permeability in vacuum = 4p x 10^-7 TmA^-1. In air the same value can approximately be used, for other materials the value is different.

 
 

 
 

Right hand rule 1 : grip the wire with your right hand, the thumb in the direction of the current: the bent fingers will indicate the magnetic field

 
 

In this context it may be convenient to introduce a way to show in a graph

 
 

the direction of any vector quantity (current, field, ...) as a CROSS if perpendicularly into the page, a DOT if perpendicularly out of the page.

 
 

 
 

 

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        flat circular coil : if the straight wire is bent to a circle, the field will be in one direction inside it and in the opposite outside it.

 
 

        solenoid = long coil with several loops bunched together so that the fields inside the solenoid point in one direction, and outside it are like those around a bar magnet

 
 

For a solenoid of length l with the number of turns of wire N the magnetic field inside the solenoid is given by

 
 

B = m₀NI / l = m₀nI    [DB p.7]

 
 

where it can be noted that the B-value can be increased by inserting some materials like an iron core into the solenoid, replacing m₀ with the larger m_iron.

 
 

 

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Right hand rule nr 2 : grip the solenoid with your right hand, use the four bent fingers to follow the current in the solenoid: the thumb will indicate the North pole of this electromagnet.

 
 

5.8. Magnetic forces

 
 

Magnetic force on moving charges

 
 

If a straight wire with the length l carrying the current I is placed in a homogenous magnetic field B, the force F acting on it will be

 
 

F = I l B sin q    [DB p. 7]

 
 

where q = the angle between the direction of the current (opposite to the direction where negative electrons move!) and the direction of the magnetic field. If the angle = 90 degrees, then F = I l B.

 
 

 

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The direction of the magnetic force is given by

 
 

Right hand rule nr 3 : thumb in the direction of I, four fingers in the direction of B gives the direction of the force F in a third dimension out of the palm of the hand = in the direction where the four fingers can easily be bent.

 
 

        This makes it possible to express the unit of the magnetic field B in other units. Solving for B in a case with the angle = 90 degrees gives B = F/Il and therefore 1 tesla = 1 T = 1 N/Am = 1 NA^-1m^-1.

        Compare this to the electric field E = F / q giving the unit (which has no separate name) 1 N/C = 1 NC^-1.

        In both cases, the unit of the field is the unit of force divided by the unit of what will be affected by a force if placed in the field - in the electric case a charge, in the magnetic case moving charges as a piece of current-carrying wire.

 
 

 
 

In the formula for the magnetic force the factors Il can be replaced as:

       I l = (q / t) l but since the distance l that a charge moves through the wire in the chosen time t = its velocity v we get

       I l = q v so we can write

 
 

F = q v B sin q    [DB p. 7]

 
 

or F = q v B for the 90 degree case. This would describe the unit

1 T = 1 N/(Cms^-1)^-1 indicating that magnetic forces act on moving charges

 
 

Magnetic force on two parallel wires

 
 

If two parallel wires, assumedly very long and and thin, are near each other then they will act on each other with a forces that following Newtons's III law are of the same but opposite directions.

 

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        We study a length l of the parallel wires with currents in the same direction

        The wire carrying I₁ causes the field B₁ at the location of I₂ a distance r away.

        This field B₁ = m₀I₁ / 2pr is directed "downwards" (right hand rule 1)

        The force acting on the wire carrying I₂ will then be F = I₂lB₁

        It is directed towards the wire carrying I₁ (right hand rule 3)

        combining gives F = I₂lB₁m₀I₁ / 2pr or :

 
 

F / l = m₀I₁I₂ / 2pr    [DB p.7]

 
 

and in the corresponding way, the same force but in the opposite direction will be acting on the I₁-wire (as Newton's III law tells us).

 
 

Note : We may use the rule that currents in the same direction attract, in opposite directions repel - if we are not confused by this being contrary to the usual "opposite attract, same repel" rule which is valid for positive (+) and negative (-) charges as well as for North and South magnetic poles.

 
 

Defining the unit 1 amp

 
 

The unit for current, 1 ampere = 1 amp is defined from a situation like this: two infinitely long and thin parallel wires 1 m apart in vacuum carrying the current I which by definition is 1 amp if the force between them is 2 x 10^-7 N per meter of the wire pair.

 

 

The simple DC-motor (and A- and V-meters)

 
 

 

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For a rectangular loop of wire (or several loops) in a magnetic field carrying a current a force will be acting on the opposite sides, in opposite directions but producing a torque in the same clockwise/anticlockwise direction. The magnetic field can be made homogenous almost for all angles by having magnet poles shaped like "half-pipes" towards each other. The direction of the current needs to change every half turn of the loop, which is achieved by the commutator and brush contact arrangement in the picture. (An alternating current, one which periodically changes directions could have been used directly - see the chapter about alternating currents).

 
 

If instead of this a needle is attached to the loop(s) and a spiral spring which counteracts the torque by the magnetic force with one directly proportional to the angle turned is attached to this, then the loop with the needle will not rotate but turn an angle proportional to the current through it. This can be used as an ammeter (amperemeter). The ammeter is connected in series with the studied component so that the same current passes them; ammeters have a low resistance so as to decrease this current as little as possible.

 

A simple ammeter without a spiral spring opposing the magnetic torque is very sensitive and this instrument, called a galvanometer, can be used to detect the presence of very small currents.

 

If an ammeter is connected in series to a large known resistor then the needle reading will be directly proportional to the voltage over this instrument, a voltmeter.

 

R_voltmeter = V_voltmeter/I_voltmeter => V_voltmeter = R_voltmeterI_voltmeter

 

The scale of the instrument can be marked to directly show the voltage value, and different resistors give different maximum readings. The voltmeter is connected in parallel with the studied component so that the voltage over them is the same. Since the resistance of the voltmeter is high only a small part of the current through the studied component will divert through the voltmeter. Since

 

R_unknown = V_unknown/I_unknown => V_unknown = R_unknownI_unknown

 

any change in I_unknown will affect the value we wish to measure.

 
 

The magnetic force as centripetal force

 
 

If a moving charge enters a homogenous magnetic field at a 90^o angle then it will be affected by a magnetic force perpendicular to the velocity and this force can act as a centripetal force: 

 

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We will then have

 
 

        qvB = mv²/r and cancelling a factor v

        qB = mv/r which can be used to solve for r:

        r = mv/qB = p/qB with p = mv = momentum

        or p = qBr

        or if the factor v was not cancelled qvB = mv²/r giving

        mv² = qvBr and then ½mv² = E_k = ½qvBr

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.9. Induction

 
 

Previously we have studied

        magnetic fields causing forces on moving charges (=> motor)

 
 

Now we will focus on the opposite:

        moving magnets causing moving charges = currents (=> generator)

 

Induced emf in straight wire

 
 

If we move a straight piece of wire quickly between the poles of a U-shaped magnet a small current will be shown on a digital microammeter (or a galvanometer = sensitive ammeter).

 
 

 

e08a

 
 

To analyse this we focus on a straight wire of length l moving with the velocity v in a homogenous magnetic field B directed into the plane of the page.

 
 

 

e08b [switch signs, minus to left and plus to right!]

 
 

In a metal wire electrons with the charge q = e (or -e) can move. We will get:

 
 

        the force acting on an electron is F_magnetic = qvB

        this force will make electrons drift towards one end of the wire

        they will then cause an electric field parallel to the wire

        this field E = F_electric/q so the electrons are affected by F_electric in the opposite direction to F_magnetic

        the more electrons gather in one end of the wire, the stronger E

        eventually there will be an equilibrium where F_magnetic = F_electric so

        qvB = qE giving E = Bv

        the potential difference or emf (or "voltage") V = e between the ends of the wire will then follow the formula E = V / d where now V = e and d = l so

        E = e/l which gives e/l = Bv and therefore

 
 

e = Blv        [DB p.7]

 
 

If the wire is moving in a homogenous field it will in the time t sweep a rectangular area A with

 

        one side = the length of the wire = l

        another side = the distance traveled by the wire = vt

        from this we get A = lvt giving v = A / lt

        then e = Blv can be written e = BlA / lt = BA / t

 
 

The quantity BA is called magnetic flux F with the unit 1 weber = 1 Wb = 1 Tm²

 
 

        it follows from this that B = F / A wherefore the magnetic field intensity B also can be called the magnetic flux density ; normally "density" would mean mass/volume but "density" can be used in a more general sense as something per length, area or volume.

 
 

If the B is not perpendicular to the area A, then we can use

 
 

F = BA cos q    [DB p.7]

 
 

where the Q = the angle between B and the normal to A (not to the "surface" A).

 
 

        If B is perpendicular to A then we have Q = 0^o giving cos Q = 1 and F = BA

        if B is parallel to the surface A then Q = 90^o and cos Q = 0 so F = 0

 
 

Graphically, the flux F is represented by the number of field lines (crosses or dots if into or out of page) and B by the number of crosses or dots per given area

 
 

One way to use this emf to cause a current in an electric circuit would be to let the moving wire be in (assumedly frictionless) contact with rails which are connected by a stationary wire parallel to the moving one.

 

e08c

 
 

The e = BA / t would then become e = F / t or the induced emf would be the change in flux per time. (can be written e = DF / Dt)

 
 

It is more practical to have a closed circuit - which may be rectangular, circular or other and change the flux in it to cause an induced emf.

 
 

Changing the flux in a circuit: Faraday's and Lenz' laws

 
 

The change in flux can be achieved by changin any of the variables in F = BA cos q. The effects are easier to detect if a solenoid with N turns of wire is used instead of a single loop, and stronger the faster the change is done.

 
 

1. Changing B : the magnetic field near the pole of a bar magnet is stronger the closer to the magnet we are. By moving the magnet and the circuit relative to each other (moving either the magnet or the loop/ solenoid or both) we can affect the B-value.

2. Changing A
: for a single loop of wire placed between the parts of a strong U-shaped magnet a small current pulse may be detected on a digital microammeter if the loop is very quickly made smaller or larger.

3. Changing Q
: this can be achieved by rotating the loop or solenoid in a magnetic field and is most common in technical applications (generators).

 
 

For this (sometimes called magnetic flux-linkage) we have Faraday's law

 
 

e = - N DF / Dt        [DB p. 7]

 
 

The induced emf will cause an current in the wire which will cause a magnetic field around the wire, and affect the flux through the area A. We have two possibilities:

 
 

        either the induced current causes an additional change in flux which causes more emf to cause more current .... and so on making it possible to get an infinitely high current (or lots of free energy) by starting the process with a small input work. This is not happening in our universe, and would be against the law of conservation of energy

 
 

        or - which is the case - the induced current causes a change of flux opposed to the initial change which caused it. Therefore the minus in the formula = LENZ' LAW

 
 

5.10. Alternating currents

 
 

AC generators

 
 

A rotating loop of wire (rectangular or other) in a homogenous magnetic field works as an AC generator.

 

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[Not required in the IB: Since the induced emf depends on the change of flux F and this flux in turn is F = BAcosQ rotating the loop or coil in a magnetic field will produce an emf which follows a sine function, since the rate of change in (the derivative of ) the cosine function is a negative sine function, where the angle rotated Q = 2pft. If f = the frequency of rotation then:

 

• F(t) = BAcos(2pft) taking the time derivative to get
  
• F'(t) = -BAsin(2pft)2pf
  

 

where

• the quantity -2pfBA has the unit HzTm²
  
• using e = Blv => B = e/lv we get the unit 1 T = 1 V/(m²s^-1) so
  
• the unit HzTm² = s^-1V/(m²s^-1) = V = 1 volt
  

 

This is the voltage V₀ referred to below. For a constant resistance R this leads to a similar function for the current, with the constant V₀/R = I₀ in front of the sine function.

 

Peak and rms values

 
 

The alternating voltage and current as a function of time both follow a sine function:

 
 

I(t) = I₀sin(2pft) and V(t) = V₀sin(2pft)

 
 

where the I₀ and V₀ are the "peak" values of the quantities and f = the frequency of the alternating current; in Europe 50 Hz, in the US 60 Hz.. The effective value of them - sometimes called the rms value referring to a statistical "root mean square" concept, but we can think of it as an average - can be found using:

 
 

        the power P =VI as a function of time P(t) = V(t)I(t) giving

        P(t) = V₀sin(2pft) I₀sin(2pft) = V₀I₀sin²(2pft)

        if we plot the function y = sin²x we find geometrically that an average for y = one which gives the same area under the curve as under a horizontal line at the average is ½y (cut off peaks and use them to fill the 'troughs' - all on the positive side above the x-axis)

        analogous to that the average or effective power P_rms = ½V₀I₀

        we can use a formula P_rms = V_rmsI_rms if we define :

 
 

 

e09b

 
 

I_rms = I₀ /Ö2        V_rms = V₀/Ö2    [DB p.7]

 
 

resulting in P_rms = V_rmsI_rms = (I₀ /Ö2)(V₀/Ö2) = ½V₀I₀ = P_rms

 
 

Note : It is the rms value of the voltage, not the peak value, which is indicated for the ordinary household electricity, e.g. 230 V in Europe or 110 V in the US.

 
 

5.11. Transformers

 
 

We have earlier noticed that it is possible to induce a current in a loop of wire or a solenoid = bunch of loops by one of these:

 
 

        changing the magnetic field B

        changing the area A

        changing the angle F

 
 

If the field B is caused by a permanent bar magnet, changing it means bringing it closer to or further away from the loop or solenoid. But if the field is caused by another solenoid nearby, this field can be varied by varying the current in the other solenoid - which is exactly what happens in a solenoid connected to an AC source.

 
 

Without supplying a strict proof we, can notice that the number of loops N affects the induced emf = voltage. By varying the number of loops in the primary coil = N_p (to which the AC source is connected) and that in the secondary coil N_s we can from a given input voltage in V_p (which usually would be an rms value, not a peak value) get different output voltages V_s in the secondary coil :

 
 

V_p / V_s = N_p / N_s        [DB p.7]

 
 

or : where there are many turns of wire, there is a high voltage

 
 

An ideal transformer would convey all the input power P_p to output power P_s in which case

 
 

V_pI_p = V_sI_s

or

V_p/V_s = I_s/I_p        [not in DB]

 
 

that is, when the voltage is increased, the current is decreased and vice versa. High currents can be achieved by connect an AC source to a primary coil with high N_p in a transformer with a very low N_s, giving a high current which dissipates a lot of power P = RI², sometimes demonstrated by melting a nail.

 
 

In practice the efficiency of any transformer is less than 100% but can be rather high if it is equipped with an iron core:

 
 

 

e10a

 
 

        a step-up transformer is one where V_s > V_p

        a step-down transformer is one where V_s > V_p

 
 

Electricity from power plants is transformed up to high voltages where the current is lower and the power loss P = RI² minimized; then transformed down to the voltage delivered to the consumer (which may be further transformed down to e.g. 12 V and possibly converted to DC for some devices).

 
 

[Note : A simple transformer can be converted to a primitive metal detector by removing the iron core and turning one coil 90^o. It will then not work as a transformer since the field lines from the primary are mainly parallel to the loop area in the secondary, so no emf is induced there. But is a (not too small) metal object is place nearby, eddy currents will be induced in it, and these will induce some small emf in the secondary coil.]

 
 

5.12.* AC circuits

 

Self-inductance

 
 

Change in flux through a loop or a solenoid gives induced emf. If we connect or disconnect a solenoid to a voltage source, the current I through it will increase or decrease quickly, causing a "self-inductance" which opposes the change. For this:

 
 

ε_self = emf_self = - LDI/Dt

 
 

where L = inductance (induktans, induktanssi), unit 1 henry = 1 H = 1 VsA^-1 = 1 Ws. For a solenoid with the cross-section area A, the length l and the number of turns we have when m₀ = the magnetic permeability that

L = m₀N²A/l

 
 

The energy stored in the magnetic field around a coil or solenoid when the current I runs through it is:    

E = ½LI²

 
 

AC circuits and impedance = "AC resistance"

 
 

When a circuit is connected to an AC power source the total resistance is affected not only by resistors but also by capacitors and solenoids. The total

resistance for AC is called

 
 

        impedance Z = U/I (compare R = U/I), unit 1 ohm

 
 

Capacitors cause

 
 

        capacitive reactance
X_C = 1/wC where w = 2pf and f = the frequency in Hz.

 
 

This "reactance" has the same unit as resistance and impedance and only means the resistance caused by the capacitors in the circuit. (Note: Direct current can not go through a capacitor since the metal foils in it are separated by a layer of insulating material. High frequency AC can, since the charges that cannot go through the capacitor are stored on one of the plates or foils, but soon the current changes direction and they move back through the circuit and are stored on the opposite foil etc.)

 
 

        Solenoids cause inductive reactance
X_L = wL

 
 

The total impedance is (can be shown with complex number mathematics)

 
 

Z = Ö(R² + (X_L - X_C)²)

 
 

If there is no capacitor in the circuit Z = Ö(R² + X_L²); inserting C = 0 would in principle give infinitely high X_C and therefore also Z but that would represent a circuit which is broken (there is a place with insulating material that the current would have to go through) but where this break in the circuit is an infinitely bad capacitor, with so small area or high d that C = e_re₀A/d » zero). For an unbroken circuit with no capacitor in it, the term for capacitive reactance is just left out.

 
 

If there is a resistor and a capacitor then L = 0 gives X_L = 0 and Z = Ö(R² + (-X_C)²) = Ö(R² + X_C²) and if there is only a resistor then Z = Ö(R²) = R. (But L is never quite = 0 since any closed circuit contains at least one "loop" of wire).

 
 

Phase shift

 
 

When there is L and C in the circuit the current I and voltage U as a function of time will not be in phase; the phase shift j in radians (2p means one whole "wavelength" of the sine curve ) is given by

 
 

tan j = (X_L - X_C) / R

 
 

The AC circuit is in resonance when (X_L - X_C) = 0 so that Z = R which happens when X_L = X_C or wL = 1/wC which gives w = 1/ÖLC or

 
 

f = 1/2pÖ(LC)

 
 

This can be used radio tuning where the C-value of a capacitor is changed either by turning plates so that the area opposing a plate with opposite charge is changed, or by changing the distance between plates); thereby the resonance frequency is changed and difference stations can be tuned in.