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Electricity and magnetism

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Electricity and magnetism

The IB Physics Compendium 2005: Electricity and magnetism 1

    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 -19 the unit of charge is 1 coulomb = 1 C; the charge of one electron = e = - 1.6 x 10 C (we -19can sometimes also use e = the elementary charge = 1.6 x 10 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. ? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 2

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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).

    ? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 3

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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.

? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 4

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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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? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 5

    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

    2q/r where k = 1/4;, [DB p.7] F = kq120

where q and q are the charges, r the distance between them (or the distance between the centers of 12

    them if they are not very small "point charges").

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     92-2The Coulomb constant k = 8.99 x 10 NmC in vacuum and approximately the same in air. In

    other materials a k-value can be calculated from the relevant -value (electric permittivity). The k-

    value and the permittivity in vacuum (or air) are given in the data booklet. The -value for other -12-1materials is given when necessary. In vacuum or air = 8.85 x 10 Fm (F the unit 1 farad, not 0

    explained here but a SI-unit). Some table list relative permittivity () values, where the actual r

    permittivity = . r0

     2This can be compared to Newton's law of universal gravity F = Gmm/r but : 12

     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 and q. The 12

    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 q caused by q. If we want to describe what force 12

    would act on an imagined small positive test charge q here called just q, we can define the 1

    electric field strength as

    ? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 6

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

    E = F / q [DB p.7]

     -1a vector quantity with the unit 1 NC

Using Coulomb's law for the field caused by a charge q we get 2

     22E = F/q = (kqq/r) / q = kq/r which in the IB data booklet is given as: 11212

     2E = kq / r [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

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

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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.

    ? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 7

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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). 2 , the magnitudes of these vectors Since the distance r to the charge appears in the E = kq/r

    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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    ? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 8

    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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? Thomas Illman and Vasa övningsskola

The IB Physics Compendium 2005: Electricity and magnetism 9

    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

    x but since E = F/q we get F = qE and then W = Felectric

     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 / m p,gravitational

and this is here replaced by

     the electric potential V = E / q p,electric

Remember that the gravitational potential V = E/m = mgh/m = gh is rarely used since most p

    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= W = qEx p

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

     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 10

     GRAVIT. ELECTR.

    UNIT UNIT Homogenous Point,planet Homogenous Point, sphere

    22N N F = mg F= GMm/r F = qE F = kQq/r -1-2-1-122Nkg=ms NC=Vm g = F/m g = GM/r E = F/q E = kQr

    J J E = mgh E=-GMm/r E = qEd E = kQq/r pppp-1-1Jkg JC = V V = E/m= gh V = -GM/r V = E/q =Ed V = kQ/r pp

Quantities corresponding to each other (gravitation - electricity), in addition to this the universal -112-29gravity constant G = 6.67 x 10 Nmkg is replaced by the Coulomb constant k = 8.99 x 10 2-2NmC .

F - F g - E E - E V - V M,m - Q,q h - d pp

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 = E / q p

     -1so its unit is 1 JC 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:

for later applications.

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

     -1-1-1-1-1-1These units are the same : 1 Vm = 1 JCm = 1 NmCm = 1 NC

The unit 1 electronvolt = 1 eV = an energy unit

     -19If one electron with the charge q = e (or - e depending on which definition we follow) = 1.6 x 10

    C is accelerated through a potential difference of 1 volt, it will get an energy = the work done = qV -19-1-19= 1.6 x 10 C x 1 JC = 1.6 x 10 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.

? Thomas Illman and Vasa övningsskola

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