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The IB Physics Compendium 2005: Electricity and magnetism
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5. ELECTRICITY AND MAGNETISM
5.1. Electric charge
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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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The IB Physics Compendium 2005: Electricity and magnetism
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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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The IB Physics Compendium 2005: Electricity and magnetism
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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 IB Physics Compendium 2005: Electricity and magnetism
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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.
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The IB Physics Compendium 2005: Electricity and magnetism
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5.2. Electric force and field
Coulomb's law for electric force
F = kq1q2/r2 where k = 1/40
[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 -value (electric permittivity). The kvalue and the permittivity in vacuum (or air) are given in the data booklet. The -value for other
materials is given when necessary. In vacuum or air 0 = 8.85 x 10-12 Fm-1 (F the unit 1 farad, not
explained here but a SI-unit). Some table list relative permittivity (r) values, where the actual
permittivity  = r0.
This can be compared to Newton's law of universal gravity F = Gm1m2/r2 but :
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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 q 1 and q2. The
directions of the forces are opposite (repelling or attracting) because of Newton's III law.
Note:
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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:
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The IB Physics Compendium 2005: Electricity and magnetism
E=F/q
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[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
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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
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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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The IB Physics Compendium 2005: Electricity and magnetism
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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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The IB Physics Compendium 2005: Electricity and magnetism
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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.
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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.
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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.
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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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The IB Physics Compendium 2005: Electricity and magnetism
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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 Gconstant 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
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W = Felectricx 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
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the gravitational potential V = Ep,gravitational / m
and this is here replaced by
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the electric potential V = Ep,electric / q
Remember that the gravitational potential V = Ep/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:
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the work = change in electric potential energy Ep = W = qEx
but since the electric potential is defined as V = Ep/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 = - V / x
[DB p.7]
Another way to write this is, now replacing x 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.
 Thomas Illman and Vasa övningsskola
The IB Physics Compendium 2005: Electricity and magnetism
GRAVIT.
ELECTR.
Homogenous Point,planet UNIT
F = mg
F= GMm/r2 N
g = F/m
g = GM/r2 Nkg-1=ms-2
Ep = mgh
Ep=-GMm/r J
V = Ep/m= gh V = -GM/r Jkg-1
Homogenous
F = qE
E = F/q
Ep = qEd
V = Ep/q =Ed
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Point, sphere
F = kQq/r2
E = kQr2
Ep = kQq/r
V = kQ/r
UNIT
N
NC-1=Vm-1
J
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 V between to
places in the uniform field or between the plates causing the field is
V = Ep / 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.
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The IB Physics Compendium 2005: Electricity and magnetism
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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 = 1/40
[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 :
VA - VB = V = qW
Electric potential from some charge systems
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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.
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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 :
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The IB Physics Compendium 2005: Electricity and magnetism
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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:
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isolated point charge: the equipotential lines are concentric circles or in 3 dimensions
spherical surfaces
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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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The IB Physics Compendium 2005: Electricity and magnetism
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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 =  q / t [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
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a source of "voltage" = potential difference, for example a battery
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The IB Physics Compendium 2005: Electricity and magnetism
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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 
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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  (kappa) or G = I / V with the unit 1 AV-1 = 1 -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 , while a
solution of very clean water or a covalent, non-polar compound has a low .]
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.
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if the conductor is ohmic, a graph of I as a function of V will give a straight line with the
gradient 1/R [ =  ], 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
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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
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The IB Physics Compendium 2005: Electricity and magnetism
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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:
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The IB Physics Compendium 2005: Electricity and magnetism
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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
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the current only has one possible way, and is the same through both resistors R 1 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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The IB Physics Compendium 2005: Electricity and magnetism
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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
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V/R = I = I1 + I2 and
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V/R = V1/R1 + V2/R2 but since V1 = V2 =V we get
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V/R = V/R1 + V/R2 and dividing both sides with V
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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
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The IB Physics Compendium 2005: Electricity and magnetism
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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 =  = rI + RI
where:
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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
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The IB Physics Compendium 2005: Electricity and magnetism
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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 = r0A/d
where 0 = the electric permittivity in vacuum (same as in F = 1/40)(q1q2/r2). 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/Ctot = 1/C1 + 1/C2 + ....
parallel Ctot = C1 + C2 + ...
(Note that is opposite to the formulas for resistors! There it was
series Rtot = R1 + R2 + ..., parallel 1/Rtot = 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,
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like magnetic poles repel, opposite poles attract
e06a
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.
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The IB Physics Compendium 2005: Electricity and magnetism
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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.
e06b
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 = 0I / 2r [DB p. 7]
where the direction of the field is given by the first right hand rule and 0 = the magnetic
permeability in vacuum = 4 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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e06c

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 = 0NI / l = 0nI
[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 0 with the larger iron.
e06d
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 
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[DB p. 7]
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where  = 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.
e07a
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 
[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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e07b






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 = 0I1 / 2r 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 = I2lB10I1 / 2r or :
F / l = 0I1I2 / 2r
[DB p.7]
and in the corresponding way, the same force but in the opposite direction will be acting on the I1wire (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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e07c
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.
Rvoltmeter = Vvoltmeter/Ivoltmeter => Vvoltmeter = RvoltmeterIvoltmeter
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
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only a small part of the current through the studied component will divert through the voltmeter.
Since
Runknown = Vunknown/Iunknown => Vunknown = RunknownIunknown
any change in Iunknown 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 90o angle then it will be affected by a
magnetic force perpendicular to the velocity and this force can act as a centripetal force:
e07d
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 = Ek = ½qvBr
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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:
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The IB Physics Compendium 2005: Electricity and magnetism
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the force acting on an electron is Fmagnetic = 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 = Felectric/q so the electrons are affected by Felectric in the opposite direction to
Fmagnetic
the more electrons gather in one end of the wire, the stronger E
eventually there will be an equilibrium where Fmagnetic = Felectric so
qvB = qE giving E = Bv
the potential difference or emf (or "voltage") V =  between the ends of the wire will then
follow the formula E = V / d where now V =  and d = l so
E = /l which gives /l = Bv and therefore
 = 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  = Blv can be written  = BlA / lt = BA / t
The quantity BA is called magnetic flux  with the unit 1 weber = 1 Wb = 1 Tm2

it follows from this that B =  / 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
 = BA cos 
[DB p.7]
where the  = the angle between B and the normal to A (not to the "surface" A).


If B is perpendicular to A then we have  = 0o giving cos  = 1 and  = BA
if B is parallel to the surface A then  = 90o and cos  = 0 so  = 0
Graphically, the flux  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.
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e08c
The  = BA / t would then become  =  / t or the induced emf would be the change in flux per
time. (can be written  =  / t)
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  = BA cos . 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  : 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
 = - N  / t
[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
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The IB Physics Compendium 2005: Electricity and magnetism
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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.
e09a
[Not required in the IB: Since the induced emf depends on the change of flux  and this flux in turn
is  = BAcos 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  = 2ft. If f = the frequency of rotation then:


(t) = BAcos(2ft) taking the time derivative to get
'(t) = -BAsin(2ft)2f
where
 the quantity -2fBA has the unit HzTm2
 using  = Blv => B = /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(2ft) and V(t) = V0sin(2ft)
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:
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The IB Physics Compendium 2005: Electricity and magnetism
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the power P =VI as a function of time P(t) = V(t)I(t) giving
P(t) = V0sin(2ft) I0sin(2ft) = V0I0sin2(2ft)
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 Prms = ½V0I0
we can use a formula Prms = VrmsIrms if we define :
e09b
Irms = I0 /2
Vrms = V0/2 [DB p.7]
resulting in Prms = VrmsIrms = (I0 /2)(V0/2) = ½V0I0 = Prms
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 
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 = Np (to which the AC source is
connected) and that in the secondary coil Ns we can from a given input voltage in Vp (which
usually would be an rms value, not a peak value) get different output voltages Vs in the secondary
coil :
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The IB Physics Compendium 2005: Electricity and magnetism
Vp / Vs = Np / Ns
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[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 Pp to output power Ps in which case
VpIp = VsIs
or
Vp/Vs = Is/Ip
[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 Np in a transformer with a very low
Ns, 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 Vs > Vp
a step-down transformer is one where Vs > Vp
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 90o. 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.]
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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:
εself = emfself = - LI/t
where L = inductance (induktans, induktanssi), unit 1 henry = 1 H = 1 VsA-1 = 1 s. For a solenoid
with the cross-section area A, the length l and the number of turns we have when 0 = the magnetic
permeability that
L = 0N2A/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 XC = 1/C where  = 2f 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 XL = L
The total impedance is (can be shown with complex number mathematics)
Z = (R2 + (XL - XC)2)
If there is no capacitor in the circuit Z = (R2 + XL2); inserting C = 0 would in principle give
infinitely high XC 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 = r0A/d  zero). For an
unbroken circuit with no capacitor in it, the term for capacitive reactance is just left out.
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If there is a resistor and a capacitor then L = 0 gives XL = 0 and Z = (R2 + (-XC)2) = (R2 + XC2)
and if there is only a resistor then Z = (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  in radians (2 means one whole "wavelength" of the sine curve ) is given
by
tan  = (XL - XC) / R
The AC circuit is in resonance when (XL - XC) = 0 so that Z = R which happens when XL = XC or
L = 1/C which gives  = 1/LC or
f = 1/2(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.
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