Part 1: Electric Forces and Fields.
INTRODUCTION: In ancient Greece amber became widely valued around
1600 BC. Greeks were fascinated by it. The ancient Greek word for amber
is "elektron", meaning - originating from the Sun. The Greeks were also
the first to describe the electrostatic properties of amber. Ancient Romans
loved amber as well. From the writings of Thales of Miletus it appears that
Westerners knew as long ago as 600 B.C. that amber becomes charged by
rubbing. There was little real progress until the English scientist William
Gilbert in 1600 described the electrification of many substances and coined
the term electricity from the Greek word for amber. As a result, Gilbert is
called the father of modern electricity.
One of nature's most spectacular display of electricity is the lightning
observed during a thunder storm. Benjamin Franklin (1706-1790) determined
that electricity originates from charges, positive or negative. We know now
that all material bodies possess electric charges. Electrons carry negative
charges while protons carry positive charges in the nucleus of an atom.
1. The electric force that stationary objects exert on each other is called the
electrostatic force. This force depends upon the distance between the two
point charges and the amount of charge on each. Experiments have
demonstrated that the greater the charge and the closer they are to each
other, the greater the force.
2. If charges have unlike signs, each charge is attracted to one other,
whereas like charges repel each other. These attractive forces and repulsive
forces act along the line between the charges, and are equal in magnitude
but opposite in direction (in accordance with Newton's 3rd law).
3. The French physicist Charles-Augustin Coulomb (1736-1806) experimented
with electric force between two point charges (the unit of charge is the
Coulomb, C). His work resulted in a law. Coulombs Law is defined: The
magnitude of the electrostatic force (F), exerted by one point charge on
another point charge is directly proportional to the magnitudes of the two
point charges, and inversely proportional to the square of the distance (r)
between the charges.
4. For a pair of charges q1 and q2, separated by a distance r, Coulomb's Law
may be stated as follows: F = k(q1q2/r2.
5. The constant of proportionality, k = 8.99x109 Nm2/C2. Such a force is
transmitted by the presence of an electric field. The electric field E due to a
point charge q is, E = k(q/r2.
6. Electric force and electric field are vectors. Hence, they have magnitudes
and directions. The electric force F and electric field E are related as follows:
F=qE, where the force is on charge q due to the presence of an electric field
at the position of q.
7. When an electric field is confined between two parallel metal plates, the
field is given by E = σ/εo, with σ, being the surface charge density, and εo
is the Permittivity of Free Space, or εo = 8.85x10-12 C2/Nm2.
8. The Principle of superposition also applies to the electric fields produced
by multiple charges. That is, the net electric field at a point due to several
charges is the vector sum of the electric fields due to individual charges.
9. For example, when more than two charges are present, the net force on
any one charge is equal to the vector sum of each of the forces produced by
10. In other words, the force on charge q1 due to the presence of charges q2
and q3, is the superposition of the forces exerted by q2 and q3. That is, the
net force F on charge q1 is, Fnet = F12 + F13.
where, F12 is the force on q1 due to the presence of charge q2 and F13 is the
force on q1 due to charge q3.
11. A capacitor is a device that stores charge. Capacitors are
formed by a pair of conductors (usually metal plates) separated by an
insulator. One of the many uses for capacitors is in computer memories.
A typical computer memory chip might contain 16,777,216 capacitors;
each capacitor is charged to approximately 5 volts to store the binary
digit 1, or 0 volts to store the binary digit 0. Another use of capacitors
is to store energy for relatively brief times; for example, the overhead
calculator that I use in class is powered by light energy instead of a
battery, and it has a capacitor to provide power during brief intervals
in which a shadow passes across its photocell. Additional applications
of capacitors include flash cameras, surge protectors, medical
defibrillators, touch pads, keyboards, car ignition systems, and radio
12. The electricity equations that we will have derived in class can also be
applied to capacitors since an Electric Field is maintained between its
plates. These are: E = F/q , E = kQ/d2 , W = qEd , W/q = Ed , V = Ed ,
and W = qV .
13. The type of capacitor we are most interested in will have a charge Q and
-Q on each conductor. There will also be a resultant potential difference
(voltage), V, between the two conductors.
14. This voltage is linearly dependent on the charge. If we triple the charge,
we triple the voltage. Because of this relationship, the ratio of Q / V is a
constant for that capacitor.
15. The value of Q / V for a given capacitor is known as its capacitance. This
gives the simple equation, C = Q / V . The unit of capacitance is the Farad,
named after Michael Faraday (1791-1867). It is equivalent to one coulomb
16. One Farad is an extremely large capacitance; most capacitors come in
units of micro (μ), nano (n), or pico (p) farads.
17. The capacitance of a capacitor is determined by two factors: (i) the
geometry of the capacitor, and (ii) the material between the conductors.
This material is known as a dielectric.
18. In a parallel plate capacitor, capacitance can be calculated by using the
equation, C = εoA / d , where C is capacitance, εo is the permittivity of free
space, A is the area of a plate, and d is the distance between the plates.
Part 2: Electric Circuits.
1. In electricity two fundamental concepts are current and voltage. For
any electrical element the voltage (V) across the element is the potential
difference between its two ends, while the current, I, through the element
is the rate at which electrical charges are flowing.
2. For many devices (but not all) the voltage and the current are
proportional to each other, and we can write V = I·R in which R is a
constant of proportionality known as the resistance.
3. The equation, V = I·R is known as Ohm's Law, and devices which obey
Ohm's Law are known as linear or ohmic devices.
4. Familiar examples are resistors which are found in radios, TV sets,
computers, and other electronic systems; the filaments of light bulbs; and
the heating elements of electrical ovens.
5. There are however other devices which do not obey Ohm's Law,
semiconductor devices such as transistors and diodes, and fluorescent
light bulbs. These are known as nonlinear devices.
6. Ohm's Law can be used to solve simple circuits. A complete circuit is one
which is a closed loop. It contains at least one source of voltage
(thus providing an increase in potential energy) and at least one potential
drop, i.e. a place where potential energy decreases.
7. If a potential difference (voltage) is maintained across a resistor, the
power, can be calculated with P = VI = I2R = V2/R.
8. Since energy is the ability to do work, and work is power x time, we also
have W = E = Pt = VIt = I2Rt = V2t/R. We usually measure this in kWh,
the kilo-Watt-hour with 1 kWh = 3.6x106 J. The cost is ~ $.12 from the
9. Because of the electrostatic force, which tries to move a positive charge
from a higher to a lower potential, there must be another 'force' to move
charge from a lower potential to a higher inside the battery.
10. This so-called force is called the electromotive force, or emf. The SI unit
for the emf is a volt (and thus this is not really a force, despite its name).
We will use a script E, to represent the emf.
11. A decrease of potential energy can occur by various means. For example,
heat lost in a circuit due to some electrical resistance could be one source of
12. For resistors in series, use simple addition: REQ = R1 + R2 + … + Rn .
For resistors in parallel, use reciprocals: 1/REQ = 1/R1 + 1/R2 + … + 1/Rn .
Part 3: Some Early History of the Theories of Magnetism.
900 BC - Magnus, a Greek shepherd, walks across a field of black stones
which pull the iron nails out of his sandals and the iron tip from his
shepherd's staff (authenticity not guaranteed). This region becomes
known as Magnesia.
600 BC - Thales of Miletos rubs amber (elektron in Greek) with cat fur and
picks up bits of feathers.
1269 - Petrus Peregrinus of Picardy, Italy, discovers that natural spherical
magnets (lodestones) align needles with lines of longitude pointing between
two pole positions on the stone.
1600 - The man who began the science of magnetism in earnest was William
Gilbert (1540 - 1603) whose book "De Magnete" was published in 1600.
Gilbert studied at St. John’s College, Cambridge, and became England’s
leading doctor, President of the Royal College of Physicians, and Queen
Elizabeth’s personal physician. At the same time, he worked on magnetism,
and after seeing his book Galileo pronounced Gilbert "great to a degree that
is enviable", not the sort of thing Galileo said too often.
1644 - Rene Descartes theorizes that the magnetic poles are on the central
axis of a spinning vortex of one of his fluids. This vortex theory remains
popular for a long time, enabling Leonhard Euler and two of the Bernoulli's to
share a prize of the French Academy as late as 1743.
1750 - John Michell discovers that the two poles of a magnet are equal in
strength and that the force law for individual poles is inverse square.
1774 – Anton Mesmer became interested in the effects of magnets on the body
and believed that he had discovered an entirely new principle of healing
involving "animal magnetism". This "animal magnetism" that he used was
different from physical magnetism in that he believed that he could "magnetise"
paper, glass, dogs and all manner of other substances.
1820 - Jean-Baptiste Biot and Felix Savart show that the magnetic force exerted
on a magnetic pole by a wire falls off like 1/r and is oriented perpendicular to
1825 - Ampere publishes his collected results on magnetism. His expression for
the magnetic field produced by a small segment of current is different from that
which follows naturally from the Biot-Savart law by an additive term which
integrates to zero around closed circuit. It is unfortunate that electrodynamics
and relativity decide in favor of Biot and Savart rather than for the much more
sophisticated Ampere, whose memoir contains both mathematical analysis and
experimentation, artfully blended together.
1846 - Faraday, inspired by his discovery of the magnetic rotation of light,
writes a short paper speculating that light might electro-magnetic in nature.
He thinks it might be transverse vibrations of his beloved field lines. He
also discovers diamagnetism. He sees the effect in heavy glass, bismuth,
and other materials.
1847 - Weber proposes that diamagnetism is just Faraday's law acting on
molecular circuits. In answering the objection that this would mean that
everything should be diamagnetic he correctly guesses that diamagnetism is
masked in paramagnetic and ferromagnetic materials because they have
relatively strong permanent molecular currents. This work rids the world of
1850 - William Thomson (Lord Kelvin) invents the idea of magnetic
permeability and susceptibility.
1895 - Pierre Curie experimentally discovers Curie's law for paramagnetism
and also shows that there is no temperature effect for diamagnetism.
1911 - Kamerlingh Onnes makes a momentous discovery of the phenomenon
of superconductivity in pure metals such as mercury, tin and lead at very low
temperatures, and following from this the observation of persisting currents.
Part 4: Magnetic Forces, Fields, and Electromagnetic Induction.
1. The force F produced by a magnetic field on a single charge depends upon
the speed v of the charge, the strength B of the field, and the magnitude of
the charge q, with F = qvBsinθ. θ is the smaller angle between v and B.
2. To find the direction of the force, use the First Right-Hand Rule with your
fingers in the direction of B, and your thumb in the direction of v. The force
will come out of the palm of your hand
3. If the charged particle moves parallel to the field lines (θ = 0), then the
magnetic force on the particle is zero. If a charged particle is moving
perpendicular to a uniform magnetic field, the path of the charged particle is
an arc (or circle).
4. The strength of the magnetic field depends on the current I in the wire and
r, the distance from the wire. The equation is B=μoI∕(2πr) , with the constant
μo, "mu naught", given as μo = 4π x 10-7 Tm/A .
5. The constant is the permeability of free space. The reason it does not
appear as an arbitrary number is that the units of charge and current
(coulombs and amps) were chosen to give a simple form for this constant.
6. The magnetic force is the source of the centripetal force on the charged
particle. This relationship can be used to find the radius of the arc when we
set the equations equal to one another, mv2/r = qvB , and solve for r.
7. Since the magnetic force is perpendicular to the velocity of the charged
particle, the force does not cause the speed of the particle to change, only
its direction. Thus, no work is done by the magnetic force on the charged
8. In regards to forces due to magnetic fields, Ampere found that a force is
exerted on a current-carrying wire in a magnetic field, F = BILsinθ, where
B is the magnetic field in N/Am, I is the current, L is the length of wire
in meters, and θ is the angle.
9. If the direction of the current is perpendicular to the field (θ = 90), then
the force is given by F = BIL.
10. If there is also a magnetic field between two charged plates in
addition to the electric field, and the fields are crisscrossed, that allows the
charge to pass through undeflected, qE=F, and F=qvB , yields v = E/B.
11. Again, for regions where both electric and magnetic fields exist:
V=Ed, qE=F, and F=qvB. Manipulating these formulas allows you to write an
expression for the accelerating voltage in terms of v, B, and d.
12. When a conductor of length, L, and velocity, v, moves across a magnetic
field, B, an Electromotive Force (Emf), ε, is induced in the conductor. This is
given by ε = BLv.
13. The current in the conductor is now given by I = ε / R, which is now Ohm's
Law for current from induced Emf.
14. The total magnetic flux through a plane area, A, placed in a uniform magnetic
field depends on the angle between the direction of the magnetic field and the
direction perpendicular to the surface area. The equation is Φ = BAcos(θ) .
15. Faraday discovered that when the magnetic flux, given by the Greek letter Phi,
Φ, changes with time, an electromotive force, or Emf, is produced. Or we can say,
ε = -N∙ΔΦ/Δt , with as the number of turns in the coil.
16. Since the magnetic flux is the product of the magnetic field, B, the area, A,
and the cos of the angle between the magnetic field and the normal to the
surface, there are three possible ways the flux can change with time; the field,
B, or the area, A, or the angle theta.
17. Lenz's Law: The polarity of the induced Emf is such that it tends to produce a
current that will create a magnetic flux to oppose the change in flux through the
circuit, ε = -ΔΦ/Δt .
18. Remember that a generator changes mechanical energy to electrical energy.
But a motor does the opposite. It changes electrical energy to mechanical.
19. In many cases voltage must either be "stepped-up" or "stepped-down"
depending on the application. These processes rely on transformer equations,
which are PP = PS , which means that the power of the primary circuit equals the
power generated in the secondary, if ideal.
20. Therefore, since P = VI , we have VP∙IP= VS∙IS . Physically this is
accomplished by the number of turns, N, in each coil. Now we have the
equation, VP/VS= NP/NS .
21. And to get full credit for your homework make sure you are following
(i) read the problem and identify the given variables
(ii) determine what you are asked to solve for
(iii) find the correct formula to use
(iv) use algebra to isolate the unknown
(v) substitute-in the given information and simplify.
For C-Code AND B-Code Problem Set #1 click HERE.
For C-Code AND B-Code Problem Set #2 click HERE.
For C-Code AND B-Code Problem Set #3 click HERE.
For C-Code AND B-Code Problem Set #4 click HERE.
For C-Code AND B-Code Problem Set #5 click HERE.