• `d or `delta:  D
• `lambda:   l
• `omega:    w
• `rho:  r
• `theta:  q
• `mu:      m
• `pi:       p
• `phi:       f
• `gamma: g
• `sigma:  s
• `Sigma:  S

Fluids:

• P = F / A (definition of pressure). Use if you know pressure and force, or force and area, or area and pressure.

• A1 * v1 = A2 * v2:  For a confined fluid, A1 * v1 = A2 * v2 (continuity equation for incompressible fluids—the amount flowing past one point is equal to the amount flowing past another). Ratio of velocities inverse to ratio of areas: v2 / v1 = A1 / A2. Remember area proportional to the square of the diameter or of the radius.

Thermal Energy Transfers and Materials:

• `dQ = m c `dT:  The specific heat c is typically given as number of Joules per kg, per Celsius degree. Thermal energy required to raise a sample is specific heat, multiplied by number of kg, multiplied by number of degrees temp change in Celsius. In equation form, `dQ = m c `dT.

• `dQ / `dt = k A ( `dT / `dx):  The rate of thermal energy transfer is proportional to temp gradient `dT / `dx and temp difference: `dQ / `dt = k A ( `dT / `dx). Note t is time and T is temp. `dx is typically thickness of container. `dQ / `dt is energy transfer in Joules / sec.

Kinetic Theory of Gases:

• If a single mass is divided into large number of particles and they still move parallel to the axis, there is still the same momentum change in same time interval so the above expression for pressure is not changed.

• P = 2/3 KE / V or PV = 2/3 KE
• PV = n R T so n R T = 2/3 KE so KE = 3/2 (nRT)

(n is the number of moles. To find KE per particle divide by the number of moles to get the KE of 1 mole, then by Avagadro's Number to get the KE of 1 particle.)

(note for General College and especially University Physics students:   the KE referred to here is the KE of a single particle at at the rms velocity)

Gas Laws:

• PV = n R T with R = 8.31 J / (particle Kelvin) so, for example:

V2 / V1 = T2 / T1:  If P and n are constant then V is proportional to T and you can use 'straight ratios' V2 / V1 = T2 / T1 to calculate desired temp or volume.

V2 / V1 = P2 / P1:  If V and n are constant then P is proportional to T and you can use 'straight ratios' V2 / V1 = P2 / P1 to calculate desired pressure or volume.

P2 / P1 = V1 / V2:  If T and n are constant then P is inversely proportional to V and you can use inverse ratios P2 / P1 = V1 / V2 (note that one of the ratios is 'upside down') to calculate desired temp or volume.

If you know three of the four quantities P, V, n and T you can find the fourth.

Thermodynamics:

• When a system converts thermal energy to mechanical work as it runs through a repeating cycle, energy `dQin is transferred to the system from the 'hot source', `dQout is transferred out to the 'cold sink' and the system does work `dW.
• `dQin = `dW + `dQout:   Energy conservation tells us that `dQin = `dW + `dQout.
• efficiency = `dW / `dQin:  The efficiency of such a system is

The maximum possible efficiency of any such system is (Th – Tc) / Th, where Th and Tc are the temperatures of the hot and cold sources.  This efficiency is theoretically achievable only by a Carnot Cycle, which consists of two adiabatic processes bounded by two isothermals.

Waves and Optics:

• v = f * `lambda: The frequency f tells us how many peak-to-peak cycles of a traveling wave past in a unit of time. The wavelength `lambda tells us the distance between consecutive peaks of the wave. Thus when we multiply the frequency by the wavelength we obtain the distance travel by the disturbance in a unit of time, which is the velocity at which the disturbance propagates.

T = 1 / f:  The period T is the time required for a peak-to-peak cycle, and is equal to the reciprocal of the frequency: T = 1 / f.

`omega = 2 `pi f:   The angular frequency `omega is the velocity of the point moving around the reference circle which models the simple harmonic motion of a single point as the wave passes. Since there are 2 `pi radians in a circle, `omega = 2 `pi f.

• y = A sin( (2 `pi f) (t - x / v) ): If the disturbance at the left-hand end of a wave traveling to the right has the form y = A sin ( (2 `pi f) t), then if the wave has propagation velocity v the disturbance will require a time delay `dt = x / v to reach position x. It follows that the disturbance at position x will satisfy the equation y = A sin( (2 `pi f) (t - x / v) ).

• v = `sqrt ( T / `mu ): In a uniform string whose mass per unit length is `mu = m / L, under uniform tension T, application of the impulse-momentum theorem shows that a transverse disturbance will travel at velocity v = `sqrt( T / `mu ).

  v = `sqrt( B / `rho):  In a liquid medium the velocity with which a disturbance is propagated is given by v = `sqrt( B / `rho), where B is the bulk modulus and `rho the density of the liquid.

  v = `sqrt( Y / `rho):  In a solid medium the velocity with which a disturbance is propagated is given by v = `sqrt( Y / `rho), where Y is the Young's modulus and `rho the density of the solid.

  v = `sqrt( `gamma * p / `rho):  In an ideal gas we have propagation velocity v = `sqrt( `gamma p / `rho), where `gamma is the ratio Cp / Cv of specific heat at constant pressure to specific heat at constant volume, p is the pressure of the gas and `rho its density. Since for an ideal gas p / `rho = RT / M, where R is the gas constant, T the temperature and M the molar mass, we can also write v = `sqrt( `gamma * R T / M).

• `lambda = L * (2 / n), n = 1, 2, 3, ...: For a standing wave in a linear medium, where the wave is constrained to have nodes separated by distance L, then the number of node-antinode distances spanning distance L is a multiple of 2. Since 4 node-antinode distances are required to span a wavelength, the possible wavelengths are 2 L (corresponding to 2 node-antinode spans in distance L), L (corresponding to 4 node-antinode spans in distance L), 2/3 * L (corresponding to 6 node-antinode spans in distance L), 1/2 * L (corresponding to 8 node-antinode spans in distance L), etc..

  The corresponding frequencies are the f = v / `lambda values v / ( 2 L), v / (L), v / (2/3 * L), etc., making up the series 1/2 * v / L, 1 * v / L, 3/2 * v / L, 4 * v / L, summarized by the formula f = (n / 2) * v / L.

  The same results apply for much the same reasons when antinodes occur with separation L.

• `lambda = L * (4 / (2n - 1) ), n = 1, 2, 3, ... For a standing wave in a linear medium, where the wave is constrained to have nodes separated from antinodes by distance L, then the number of node-antinode distances spanning distance L is 1 plus a multiple of 2--i.e., an odd number. Since 4 node-antinode distances are required to span a wavelength, the possible wavelengths are 4 L (corresponding to 1 node-antinode spans in distance L), 4/3 L (corresponding to 3 node-antinode spans in distance L), 4/5 * L (corresponding to 5 node-antinode spans in distance L), 4/7 * L (corresponding to 7 node-antinode spans in distance L), etc..

  The corresponding frequencies are the f = v / `lambda values v / ( 4 L), v / (4/3 * L), v / (4/5 * L), etc., making up the series 1/4 * v / L, 3/4 * v / L, 5/4 * v / L, 7/4 * v / L, summarized by the formula f = (2n-1)/4 * v / L.

• f ' = f / (1 - vSource/vSound). When a source of sound emitting pulses with frequency f is moving toward an observer with velocity vSource then if the speed of sound is vSound, the frequency detected by the observer is f ' = f / (1 - vSource/vSound). This is an instance of the Doppler Effect.

• f ' = f (1 + vObserver / vSound ): If an observer is moving toward a source which is emitting a sound with frequency f, then if the speed of sound is vSound the frequency detected by the observer will be f ' = f (1 + vObserver / vSound ).   This is another instance of the Doppler Effect.

• path difference = a sin(`theta). If waves are emitted in phase from two points separated by distance a, then if the waves are detected at a distance much greater than a, and at a position P such that a line from either source to P makes an angle `theta with the perpendicular bisector of the line segment connecting the sources, the distances traveled by the two waves will differ by distance a sin(`theta).

  If this path difference is equal to half of a wavelength the two waves will arrive at P exactly out of phase and the net disturbance at P will be 0. We call this destructive interference.

  If the path difference is equal to a whole wavelength or to any multiple of a whole wavelength the two waves will arrive at P exactly in phase and the net disturbance will be double that of either of the individual disturbances. We call this constructive interference. The condition for constructive interference is that path difference = n * `lambda for n = 0, 1, 2, ....

  If the path difference is equal to a whole wavelength plus a half wavelength we will again have destructive interference. The condition for destructive interference is that path difference = (n + 1/2) * `lambda for n = 0, 1, 2, ... .

• sin(`theta1) / sin(`theta2) = n2 / n1: Snell's Law states that if `theta1 and `theta2 are the angles of incidence and refraction for a beam of electromagnetic radiation which is directed from one material into another, then the sines of those angles are inversely proportional to the indices of refraction n1 and n2 of the two materials. The index of refraction for a material is the number n such that v = c / n, where v is the propagation velocity of the electromagnetic radiation in the material and c the propagation velocity of electromagnetic radiation in a vacuum. For most materials n varies with the wavelength of the electromagnetic radiation, which leads to the phenomenon known as dispersion.

• Critical Angle (angle of total internal reflection):  Total internal reflection occurs whenever the angle of incidence is greater than that for which the angle of refraction is 90 degrees. If `theta1 and `theta2 are the angles of incidence and refraction and n1 and n2 the corresponding indices of refraction, then sin(`theta2) = n2 / n1 * sin(`theta1) and if sin(`theta1) = n1 / n2, then sin(`theta2) = 1 and the angle of refraction will be 90 degrees. The value of `theta1 for which sin(`theta1) = n1 / n2 is called the critical angle.   For any angle of incidence greater that the critical angle we will have total internal reflection.

  Summary of Topics from Introductory Problems on Electricity and Magnetism

  ---

• Coulomb's Law and its Consequences:

  F = k * q1 * q2 / r^2, where k = 9 * 10^9 N m^2 / C^2, q1 and q2 are two point charges, r is the distance separating the charges and F is the force between the two charges. The force F is exerted on each charge by the other.  If q1 and q2 are of like sign the force will be one of attraction along the line between the two charges; otherwise the force will be one of repulsion in the direction of the line defined by the two charges. This law is known as Coulomb's Law.

• E = k q / r^2: The electric field E at a point P, due to a point charge q at another point Q, is the force experienced per Coulomb of test charge when the test charge is introduced at the point P. This field is directed along the line defined by P and Q, and is directed away from P if q is positive and toward P is q is negative.

• F = q * E: The force exerted on a charge q in electric field E is equal to the product q * E. If q is positive the force will be exerted in the direction of the field; if q is negative the force will be exerted in the direction opposite that of the field.

• Electrostatic Flux = 4 `pi k Q: The total electrostatic flux through a closed surface is 4 `pi k Q, where Q is the total charge enclosed by the surface.

E = k Q / r^2:  If the charge distribution is spherically symmetric, then the electric field due to that distribution will be spherically symmetric about the same central point. The field will be directed radially outward from the center and will hence be perpendicular to the surface of any concentric sphere. Thus for any such sphere we have uniform electric field E = 4 `pi k Q / ( 4 `pi r^2) = k Q / r^2, where Q is the total charge enclosed by the sphere. Note that this agrees with Coulomb's Law in the case of a point charge q.

E = 2 k `lambda / r:  If a charge distribution is cylindrically symmetric, then the electric field due to that distribution will be cylindrically symmetric about the central axis of the distribution. The field will be directed radially outward from the central axis and will hence be perpendicular to the curved surface of any coaxial cylinder. Thus if `lambda is the amount of charge per unit length enclosed by the cylinder, a cylinder of radius r and length L will enclose charge Q = `lambda * L. Since the curved surface of the cylinder will have area A = 2 `pi r * L, the field at the surface of the cylinder will be E = 4 `pi k Q / r^2 = 4 `pi k ( `lambda * L) / (2 `pi r * L) = 2 k `lambda / r.

E = 2 `pi k `sigma:  If charge is uniformly distributed over a large plane area, then symmetry arguments show that near that plane but not close to its edges the electric field due to the distribution is very nearly perpendicular to the plane. It follows that if the charge per unit area is `sigma, then a rectangular box with its central axis oriented perpendicular to the plane, with the box intersecting the plane and having cross-sectional area A perpendicular to the plane, will contain charge Q = `sigma * A. The electrostatic flux will exit this box through its two ends and not through its sides, so the flux will be E * ( 2 A). We therefore have E ( 2 A) = 4 `pi k ( `sigma * A) and E = 4 `pi k ( `sigma * A ) / ( 2 A) = 2 `pi k `sigma. Note that as long as we remain near the plane, in the sense already specified, the electric field remains constant.

Work, Energy, Potential Difference, Power:

• `dW(ON) = q * E * `ds: The work done ON a charge q which moves through a displacement `ds in the direction of a uniform electric field E is `dW = F * `ds = (q * E) * `ds.  The work done BY the charge against the field is the negative of this quantity.

• V = E * `ds. The potential difference V between two points is the work required per unit charge to move charge from the first point to the second. In a uniform electric field E the work to move a charge q through displacement `ds in the direction of the field is `dW(ON) = q * E * `ds, so the potential difference is V = `dW(ON) / q = q * E * `ds / q = E * `ds. The unit of this potential difference is the Joule / Coulomb, which is the Volt.

• F = q * `dV / `ds. If `dV is the potential difference corresponding to a displacement `ds in the direction of a constant electric field, then the potential gradient of the field is `dV / `ds. In this case the force on a charge q is the product of the charge and the potential gradient.

• `dKE = -`dW(BY). The change in the kinetic energy of a charge particle, in the absence of dissipative forces, is equal and opposite to the work done by the particle against an electrostatic field. Electrostatic fields are conservative, and the forces on charged subatomic particles within such fields usually exceed any dissipative forces to the extent that the dissipative forces are negligible. In this case energy conservation tells us that the kinetic energy change of such a particle will be equal to the work done on the system by the electrostatic field, or to the negative of the work done by the system against the electrostatic field.

• P = I * V: The power required to move charge at the rate I (standard unit Coulombs / sec or amps) through a potential difference V (standard unit Joules / Coulomb or volts) is the product I * V of the rate and potential difference (when we multiplied Coulombs/second by Joules/Coulomb we get Joules/second, or watts).

Conduction by Charge Carriers:

• (Rate of charge passage = N / L * vDrift).  If there are N charge carriers uniformly distributed over a length L, then the number of charge carriers per unit length is N / L. If those charge carriers, which typically have small mass and which therefore have large thermal velocity, have an average net drift velocity vDrift along the length, then the number of charges passing a given point per unit of time is given by N / L * vDrift.

• Rate of charge passage = charge density * A * vDrift.  The number of charges per unit of length in a uniform conducting wire depends on the volume density of charge carriers in the wire and on the cross-sectional area A of the wire. We have N / L = charge density * A, so that the number of charges passing a given point per unit of time is charge density * A * vDrift.

• The average drift velocity of the charge carriers in a substance is proportional to the strength of the electric field, which is equal to the potential gradient. For a uniform conducting wire with potential difference `dV between its ends, the strength of this electric field or potential gradient is `dV / L.

• (Rate of Charge Passage proportional to A * `dV / L).  The drift velocity is affected by various factors that vary with temperature and from one substance to another. For given substance at a given temperature the drift velocity is proportional to `dV / L, so that the number of charges passing a given point per unit time is proportional to charge density * A * `dV / L. Since charge density also depends on the substance and temperature, we can say that the number of charges passing a given point per unit of time for a given substance and temperature is proportional to A * `dV / L.

• (`dV / R proportional to A * `dV / L).  The number of charges passing a given point per unit of time is the current, designated I. Thus for given substance at a given temperature I is proportional to A * `dV / L. Ohm's Law tells us that I = `dV / R, where R is the resistance of the conductor to the flow of current (see below; for the present model `dV is the voltage V across the conductor). Thus `dV / R is proportional to A * `dV / L, so 1 / R is proportional to A / L and R is proportional to L / A.

• R = `rho * L / A.  We call the proportionality constant between R and L / A the resistivity of the substance, and we use the symbol `rho for the resistivity. Thus R = `rho * L / A. The resistance to the flow of current is proportional to the length of the conductor (greater L implies lower potential gradient, or electrical field strength, `dV / L, which results in lower drift velocity and therefore lower current) and inversely proportional to its cross-sectional area (greater A implies more charge carriers and hence more current flow for a give drift velocity).

Ohm's Law and Circuits:

• I = V / R. Ohm's Law expresses the fact that the current I in a conductor is proportional to the voltage across the conductor and inversely proportional to the quantity called resistance. Using V for the voltage and R for the resistance this is expressed as I = V / R.

• P = V^2 / R. If a voltage V is applied across a resistance R the resulting current is I = V / R. A current I passing through a potential difference V requires power P = I * V so P = (V / R) * V = V^2 / R.

• R = R1 + R2, series resistances. If a current I passes through two resistances R1 and R2 without branching, we say that the two resistances are in series. In this case voltages V1 = I * R1 and V2 = I * R2 are required across the resistances. Since the resistances, hence the voltages, are in series the total voltage across the combination is V1 + V2 = I * R1 + I * R2, or V = I * (R1 + R2). Thus I = V / ( R1 + R2) and we can say that the resistance of the series combination is R1 + R2.

• 1 / R = 1 / R1 + 1 / R2, parallel resistances: If a voltage V is maintained across a parallel combination of two resistances R1 and R2, then the entire voltage is experienced by both resistances. Hence current I1 = V / R1 flows through the first resistance and I2 = V / R2 flows through the second, giving a total current I = I1 + I 2 = V / R1 + V / R2. So I = V ( 1 / R1 + 1 / R2). Since I = V / R, we have 1 / R = 1 / R1 + 1 / R2.

Capacitors and Capacitance:

• C = Q / V. A capacitor, when a potential difference is maintained between its terminals, stores charge in proportion to that potential difference. Capacitance is defined as C = stored charge / voltage = Q / V.

• C = A / (4 `pi k Q) (parallel-plate capacitor). A parallel-plate capacitor consists of two flat plates each with area A, with a uniform separation d. If charge Q is taken from one initially uncharged plate and placed on the other, then the plates will have charges Q and -Q and charge densities Q / A and -Q / A. The resulting electric fields will therefore be E1 = 2 `pi k Q / A, directed away from the first plate, and E2 = 2 `pi k Q / A directed toward the second plate. Provided d is small compared to the dimensions of the plates, between the plates the field will therefore be 4 `pi k Q / A, directed from the first plate toward the second, and outside the plates field will be zero. The voltage between the plates will be V = E * d = 4 `pi k Q / A * d, and the capacitance, defined as Q / V, will be C = Q / (4 `pi k Q / A * d) = A / (4 `pi k * d).

• C = C1 + C2 (capacitors in parallel). A parallel combination of two capacitors with capacitance C1 and C2 will maintain the same potential difference across both. The total charge is therefore the sum of the two charges that would result from that potential difference, and as a result capacitance is the sum of the two capacitances.

• 1/C = 1/C1 + 1/C1 (capacitors in series). A series combination of two capacitors with capacitance C1 and C2 will result in equal charges Q on both capacitors. This results in series voltages V1 = Q / C1 and V2 = Q / C2 so that total voltage is V = Q ( 1 / C1 + 1 / C2) and we have V / Q = 1 / C1 + 1 / C2. Since the capacitance of the combination is C = Q / V, we see than 1 / C = 1 / C1 + 1 / C2.

• B = k ' I * `dL / r^2: Magnetic fields are caused by electrical currents. A current I flowing thru a short straight segment `dL (think for example of a short segment of a straight conductor) will contribute k ' I `dL / r^2 to the magnetic field B at a point P, lying at distance r, provided that a line from the segment to P is perpendicular to the segment. k ' = 10^-7 Tesla / (amp meter). The direction of the field contribution at P is perpendicular to both `dL and to the line from the segment to P, according to the right-hand rule.

• B = k ' I * `dL / r^2 * sin(`theta): A current I flowing thru a short straight segment `dL will contribute k ' I `dL / r^2 sin(`theta) to the magnetic field B at a point P, lying at distance r, if the line from the segment to P makes angle `theta with the segment. If `theta is a right angle then this rule is identical to the rule of the preceding paragraph. k ' = 10^-7 Tesla / (amp meter). The direction of the field contribution at P is perpendicular to both `dL and to the line from the segment to P, according to the right-hand rule.

• B = 2 k ' * `pi I / a (field at the center of a single circular loop of radius a). If we add up all the k ' I `dL / r^2 contributions from a single loop of wire of radius a, adding to obtain the field at the center of the loop, we find that if the loop is in a horizontal plane and the current goes around the loop in the counterclockwise direction the magnetic field contributed by each segment is directed vertically upward. Thus all the individual k ' I `dL / r^2 contributions are in the same direction and their magnitudes add to the total magnitude of the magnetic field. Since the total `dL is equal to the circumference 2 `pi a of the loop and r = a we can replace `dL by 2 `pi a and r by a to get B = k ' I ( 2 `pi a ) / a^2 = 2 k ' * I / a.

• `phi = F * A * cos(`theta): The flux, designated `phi, of any constant field F which penetrate a planar surface whose area is A is equal to the product of the component of F perpendicular to the surface, and the area of the surface. Here `theta is the angle of the field F with a line perpendicular to the surface. The component of F perpendicular to the surface is therefore F * cos(`theta), and the flux is the product of this perpendicular component and the area: flux = F * cos(`theta) * A = F * A * cos(`theta).

• Vave = `d`phi / `dt (ave voltage = ave rate of change of electrostatic flux): When the electrostatic flux `phi = E * A cos(`theta) due to an electrostatic field E through a loop of area A is changed by amount `d`phi in a time interval of duration `dt, a potential difference with average value Vave = `d`phi / `dt is created around the loop.

• | Vave | = 4 * A * E * f (ave magnitude of a voltage produced when a loop of area A is rotated at frequency f in the presence of an electric field E): When a loop of area A is rotated with frequency f about an axis perpendicular to a constant electric field E, then the time required for the loop to rotate from perpendicular to parallel with the field is 1/4 of the period of rotation. The period of rotation is the reciprocal T = 1 / f of the frequency. So in `dt = 1/ 4 T = 1 / (4 f ) we have a change in flux from `phi = A * E to `phi = 0, a change of magnitude | `d `phi | = A * E. This implies average voltage vAve = `d`phi / `dt = A * E / (1 / (4 f) ) = 4 * A * E * f.

• F = q v B (force on a moving charge in the presence of the magnetic field, velocity and field perpendicular): When a charge q moves with velocity v perpendicular to a magnetic field B, it experiences a force F = q v B directed perpendicular to both the velocity v and the magnetic field B, with the direction determined by the right-hand rule.

• m v^2 / r = q v B (force on a moving charge in the presence of a perpendicular magnetic field is equal to the centripetal force on the charge): When a charge q moves with velocity v perpendicular to a uniform magnetic field B, the force being perpendicular to the velocity constitutes a centripetal force m v^2 / r. The particle will move in a circular orbit of radius r, where r is the solution to the equation m v^2 / r = q v B.

• v = | E / B | (velocity of a charge to particle moving in a straight line through 'crossed' electric and magnetic field): When a particle with charge q passes through uniform electric and magnetic fields which make right angles with one another, and the velocity of the particle is perpendicular to both of these fields, then if the particle travels in a straight line this means that net force on the particle is zero. This can only happen if the electrostatic force q E is equal and opposite to the magnetic force q v B, so that | q E | = | q v B |. Solving for v, the magnitude of the velocity, we easily obtain v = | E / B |.

• F = I L B sin(`theta) (force on a straight current segment in the presence of a uniform magnetic field): When a current I flows in a straight conductor of length L, in the presence of a magnetic field B making angle `theta with the conductor, then the field exerts a total force F = I L B sin(`theta), with the direction of the force perpendicular to both B and I, the direction determined by the right-hand rule.

Quantization of Energy in Electromagnetic Radiation:

• c = `lambda * f: Light, and generally all electromagnetic radiation, is characterized by a uniform speed c of approximately 3 * 10^8 m/s in a vacuum, and by a spectrum of wavelengths `lambda, which can be measured using diffraction gratings and photographic plates. Frequencies are easily determined once wavelengths are measured.

• E = h * f: The photoelectric effect demonstrates that electromagnetic radiation with frequency f transfers energy in discrete amounts equal to the product of Planck's constant h = 6.63 * 10^-34 J s and the frequency of the radiation. In this sense electromagnetic radiation behaves as a stream of photons, particles which carry energy but which have no mass.

• p = E / c: A photon with energy E will have momentum p = E / c.

Wave Properties of Particles:

• `lambda = h / p: A beam of massive particles (i.e., particles with mass) each with momentum p will exhibit interference effects identical to those of a wave having wavelength `lambda = h / p, where h is Planck's constant.  This wavelength `lambda is called the deBroglie wavelength.

• `dx * `dp = h (uncertainty principle): The uncertainty in x, the position of an object, and p, the momentum of the object, are denoted by `dx and `dp. It is not possible to measure both position and momentum to an arbitrarily high degree of precision. The more precisely one is measured the less precisely is possible to determine the other. The product of the uncertainties is approximately equal to Planck's constant. i.e., `dx * `dp = h.

Quantization of Orbital Energies in Atoms:

• m v^2 / r = k q^2 / r^2 (condition relating velocity and radius of 'orbit' of electron around proton). If an electron 'orbits' a proton at a distance r, then the velocity must be such that the Coulomb attraction k q^2 / r^2 (q = charge of electron) between the proton and electron constitutes the centripetal force m v^2 / r required to maintain the circular path. This relationship can easily be solved for v in terms of r, obtaining r = k q^2 / (m v^2).

• m * v * r = n ( h / (2 `pi) ), n = 1, 2, 3, ... . (quantization condition for angular momentum): Angular momentum can occur only in whole-number multiples of h / ( 2 `pi ). Combining this quantization condition on the angular momentum m * v * r with m v^2 / r = k q^2 / r^2 we conclude that for an electron orbiting a proton, only certain orbital radii and their associated velocities and energies can occur. The resulting model explains to a high degree of precision the spectrum observed as a result of energy transitions within a hydrogen atom.

• r = n^2 h^2 / ( 4 `pi^2 k q^2 m) = n^2 * .509 *10^-10 m approx., n = 1, 2, 3, ... . (possible radii of electron orbit around proton) Combining m v r = n ( h / (2 `pi) ), the quantization condition on angular momentum, with m v^2 / r = k q^2 / r^2, which states that Coulomb Force = centripetal force, we obtain r = n^2 h^2 / ( 4 `pi^2 k q^2 m). Using h = 6.63 * 10^-34 J s, k = 9 * 10^9 N m^2 / C^2, q = 1.60 * 10^-19 Coulombs and m = 9.11 * 10^-31 kg (mass of electron) we find that h^2 / ( 4 `pi^2 k q^2 m) = .509 * 10^-10 m, approx., so we can write r = n^2 * .509 Angstroms (approx).  An Angstrom is 10^-10 meter.

• 2 `pi r = n * `lambda   (orbital radius contains an integer number of particle waves):  For an orbital radius satisfying the quantization condition, the circumference of the orbital is a whole number of deBroglie wavelengths, suggesting that the electron in forms allowed orbits by creating a standing wave in each.

Conversion between Mass and Energy and Nuclear Decay Modes:

• E = m c^2: As predicted by Special Relativity and as confirmed by experiment, mass can be converted to energy and vice versa. The energy equivalent to mass m is E = m c^2.

• Alpha decay consists of the emission of a helium nucleus (2 protons and 2 neutrons all bound together) from the nucleus of a larger atom (i.e., one containing in its nucleus more than 2 protons and 2 neutrons). This process reduces the number of protons and neutrons in the larger nucleus each by 2. The mass of the resulting nucleus plus the mass of the alpha particle is less than the mass of the original nucleus. The resulting mass defect appears in the form of kinetic energy as the alpha particle and the remains of the original nucleus fly apart in opposite directions. In the usual case, where the alpha particle is much less massive then the remains of the original nucleus, almost all of this kinetic energy resides in the alpha particle.

• Beta decay consists of the emission of an electron from the nucleus of an atom. This process results when a neutron decays into a proton and an electron (note that the total charge is conserved--the original neutron has zero charge, and a proton and an electron have a net charge of zero). This process thus increases the number of protons in the nucleus by 1 and reduces the number of neutrons by 1. The mass of the resulting nucleus plus the mass of the beta particle (the electron) is less than the mass of the original nucleus. The resulting mass defect appears in the form of kinetic energy as the electron and the new nucleus fly apart in opposite directions. Since the electron is very much less massive then the new nucleus, virtually all of this kinetic energy resides in the electron.

• Gamma decay is analogous to a decrease in the orbital energy of an electron in an atom, but Gamma decay takes place within the nucleus and involves nuclear particles. The energies within the nucleus are much higher than the energies of atomic electrons, and the energies of the Gamma photons are hence much greater than the energies emitted by atomic electrons. The nucleus after a Gamma decay is less massive than before, and the mass defect is carry off by the energy of the photon.

Special Relativity:

• `dt ' = `dt * 1 / (1 - v^2 / c^2) (time dilation): Einstein's Theory of Special Relativity is based on the assumption that the laws of physics are the same in all interial reference frames. One consequence of these laws is that the speed of light in a vacuum is c = 3 * 10^8 m/s, approx.. If we consider a photon bouncing back and forth in the vertical direction between two mirrors in a spaceship which passes at velocity v in the horizontal direction, we are led to the conclusion that the time between round trips as measured by an occupant of the spaceship will be different than the time between round trips as measured from our frame of referencet. Specifically if in our frame of reference we measure a time interval `dt ' between two events which occur at the same location as observed by the occupants of the passing spaceship while the occupants of the spaceship measure time interval `dt, our time interval `dt ' will be greater than `dt by factor 1 / (1 - v^2 /c^2).  The contraction of length and the increase of mass, as well as the relative nature of simultaneity, also follow (see below).

• `dx ' = `dx * `sqrt(1 - v^2 / c^2) (length contraction): If a spaceship passes us moving moving at velocity v relative to our frame of reference, and if as measured by the occupants of the ship its length in the direction of its velocity is `dx, then the most accurate measurements in our frame of reference will determine that the length of the ship is `dx ' , which is less than `dx by factor `sqrt(1 - v^2 / c^2).

• m ' = m * 1 / `sqrt( 1 - v^2 / c^2) (relativistic mass): If a spaceship passes us moving moving at velocity v relative to our frame reference has mass m, as measured by the occupants of the spaceship (who could, for example, measured the acceleration and the net force applied to the spaceship to determine its mass), then any measurement made in our frame of reference to determine the mass will indicated that the mass is m ' , which is greater than m by the factor 1 / `sqrt(1 - v^2 / c^2).

• Relative nature of simultaneity: If a spaceship passes us moving moving at velocity v relative to our frame reference, then two events that occur simultaneously but at different locations as measured by the occupants of the spaceship will not occur simultaneously in our reference frame.

• KE = m c^2 / `sqrt(1 - v^2 / c^2 ) - m c^2.  If a spaceship with mass m, as measured in the frame of reference of the spaceship, passes us moving moving at velocity v relative to our frame reference, then its kinetic energy will be m c^2 / `sqrt(1 - v^2 / c^2 ) - m c^2. For velocities v which are small compared to c, this expression reduces to .5 m v^2. The expression for kinetic energy can also be written by factoring the m c^2 out, obtaining m c^2 ( 1 / `sqrt(1-v^2/c^2) - 1). This form suggests a 'rest energy' of m c^2, to which kinetic energy is added as velocity increases.

• Correspondence Principle: The Correspondence Principle states that the results of relativistic mechanics reduce, at small relative velocities v, to the results of classical Newtonian mechanics. Stated another way, there must be a smooth transition from the everyday physics of low relative velocities to the relativistic physics of high relative velocities.

Brief Notes about Particle Physics:

• If a particle such as a proton or and electron is accelerated, usually by means of electromagnetic forces, to a high relativistic velocity then the possibility exists that the total energy of the particle can be converted to other particles. This usually happens when the particle interacts with other particles. The rest mass of the particles so created can exceed the rest mass of the original particles as some of the kinetic energy of the original particles converts to mass. This conversion satisfies the equation E = m c^2.

• When new particles are created a number of conservation laws apply. The total energy of the particles involved is the same before as after collision, provided we include the energy equivalent E = m c^2 of the rest mass of the particles as part of the energy. Momentum must be conserved (keeping in mind that we're talking about relativistic momentum m v * 1 / `sqrt(1-v^2/c^2). Electric charge must be conserved (for example the total charge after a collision involving a proton and a neutron must be the same, +1, as before the collision). Other more subtle conservation laws also apply. These conservation laws lead to a classification of the hundreds of different particles observed and of their properties. This classification leads to a very successful mathematical model which successfully predicts the nature of particles prior to their being observed. This model is called the Quark Theory.

• It is believed that protons, neutrons and hundreds of other particles are in fact combinations of more fundamental particles called Quarks.