Electricity and magnetism

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 material, like mass

· the net charge is conserved, like mass (except that mass and energy can be converted to each other (relativity) E = mc²

· masses always attract each other, but charges of the same type repel; different types attract (unlike charges attract and like charges will always repel)

· 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 in which electrons can move easily through is called a conductor. A material where this is more difficult is called 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 switching 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 travelling 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 energy source 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 in 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 placed 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 outward 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 is 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 cell 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 materials consist 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’s Law 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 that 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 a 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, hyperbolas, 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 at 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 at 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 any other 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

(we can now 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
E_p = mgh	E_p=-GMm/r	J	E_p = qEd	E_p = kQq/r	J
V = E_p/m= gh	V = -GM/r	Jkg^-1	V = E_p/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

E_p - E_p

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, approaching 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 commonly investigated:

· isolated point charge: the equipotential lines are concentric circles or in 3 dimensionsal 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 straight 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 primarily investigated 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 unit 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 insulators 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 ammeter = 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 investigated later).

· a resistor

· a "voltage" source, Either the resistor or the voltage source is variable.

· connecting wires, with a negligibly small resistance

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The filament lamp

This device is an ordinary "light bulb”, which has a spiral metal wire that 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