Force at a distance is explained by one of two types of models, Virtual Particle Models, and Geometric Models.

Transcription

1 Physics 06 Final Review Modified from College Physics, 8 th Ed., Serway and Vuille. You should know all the concepts and equations in this summary. The Nature of Forces Force at a distance is explained by one of two types of models, Virtual Particle Models, and Geometric Models. Our best understanding of electromagnetic forces uses a virtual particle model called QED. Our best understanding of gravitational forces uses a geometrical model called General Relativity. Section 5.: Properties of Electric Charges Electric charges have the following properties:. Unlike charges attract one another and like charges repel one another.. Electric charge is always conserved. 3. Charge comes in discrete packets that are integral multiples of the basic electric charge e. 4. The force between two charged particles is stronger when the charges are closer. When a charge is brought near an insulator, it can cause the charge to be polarized. Neutral objects can experience electrostatic force when charges are polarized. (This is why balloons stick to ceilings.) Section 5.: Insulators and Conductors Conductors are materials in which charges move freely in response to an electric field. If charges remain fixed on atoms, the materials are called insulators. Section 5.3: Coulomb's Law Coulomb's law states that the electric force between two stationary charged point particles separated by a distance r has the magnitude F q q k [5.] e r where q and q are the magnitudes of the charges on the particles in coulombs and k e is the Coulomb constant. The direction of the force is along the line of centers. It is attractive when the charges are opposite and repulsive when they are like. Coulomb s law also works for charged spheres. Section 5.4: The Electric Field An electric field E is a set of vectors defined at every point in space. The electric field is defined by the equation: F = qe [5.4]

2 The direction of the electric field at a point in space is the direction of the electric force that would be exerted on a small positive charge placed at that point. The magnitude of the electric field due to a point charge q at a distance r from the point charge is Section 5.5: Electric Field Lines q E k [5.6] e r Electric field lines are useful for visualizing the electric field in any region of space. The electric field vector E is tangent to the electric field lines at every point. Furthermore, the electric field is stronger where electric field lines are closer together. Section 5.6: Conductors in Electrostatic Equilibrium A conductor in electrostatic equilibrium has the following properties:. The electric field is zero everywhere inside the conducting material.. Any excess charge on an isolated conductor must reside entirely on its surface. 3. The electric field just outside a charged conductor is perpendicular to the conductor's surface. 4. On an irregularly shaped conductor, charge accumulates where the radius of curvature of the surface is smallest, at sharp points. 5. The electric potential (voltage) is constant throughout a static conductor (see Section 6.3). 6.: Potential Difference and Electric Potential The change in the electric potential energy of a system consisting of an object of charge q moving through a displacement Δx in a constant electric field pointing in the +x direction is given by U qe x x [6.] where E x is the component of the electric field in the x-direction and x electric potential between two points A and B is x f x. The difference in i U V VB VA [6.] q where ΔV or just V is the change in electrical potential energy as a charge q moves between A and B. The units of potential difference are volts. The electric potential difference between two points A and B in a uniform electric field pointing in the +x direction is V E x [6.3] x where x x f x is the displacement between A and B. i Section 6.: Electric Potential and Potential Energy Due to Point Charges The electric potential due to a point charge q at distance r from the point charge is

3 V q k [6.4] e r The electric potential energy of a pair of point charges separated by distance r is U qq k [6.5] e r These equations can be used in the solution of conservation of energy problems and in the work-energy theorem. Section 6.3: Potentials and Charged Conductors The electric potential is constant throughout a conductor in electrostatic equilibrium. Section 6.4: Equipotential Surfaces Every point on the surface of a charged conductor in electrostatic equilibrium is at the same potential. Further, the potential is constant everywhere inside the conductor and equals its value on the surface. The electron volt is defined as the energy that an electron (or proton) gains when accelerated through a potential difference of V. The conversion between electron volts and joules is ev = J (Don t Memorize, just plug in e in C) [6.7] Any surface on which the potential is the same at every point is called an equipotential surface. The electric field is always oriented perpendicular to an equipotential surface. Section 6.6: Capacitance A capacitor consists of two metal plates with charges that are equal in magnitude but opposite in sign. The capacitance C of any capacitor is the ratio of the magnitude of the charge Q on either plate to the magnitude of potential difference V between them: Q = CV [6.8] Capacitance has the units coulombs per volt, or farads; C/V = F. Section 6.7: The Parallel-Plate Capacitor The capacitance of two parallel metal plates of area A separated by distance d is where ε 0 is a constant called the permittivity of free space. A C [6.9] 0 d

4 Section 6.8: Combinations of Capacitors The equivalent capacitance of a parallel combination of capacitors is C eq C C [6.] C3 If two or more capacitors are connected in series, the equivalent capacitance of the series combination is [6.5] C eq C C C 3 Problems involving a combination of capacitors can be solved by applying Equations 6. and 6.5 repeatedly to a circuit diagram, simplifying it as much as possible. This is followed by working backwards to the original diagram, applying C=Q/V, the fact that parallel capacitors have the same voltage drop, and the fact that series capacitors have the same charge. Section 6.9: Energy Stored in a Charged Capacitor Three equivalent expressions for calculating the energy stored in a charged capacitor are Energy stored Section 6.0: Capacitors with Dielectrics CV [6.7] When a nonconducting material, called a dielectric, is placed between the plates of a capacitor, the capacitance is multiplied by the factor κ, which is called the dielectric constant, a property of the dielectric material. The capacitance of a parallel-plate capacitor filled with a dielectric is A C [6.9] 0 d Section 7.: Electric Current The electric current I in a conductor is defined as I Q [7.] t where ΔQ is the charge that passes through a cross section of the conductor in time Δt. The SI unit of current is the ampere (A); A = C/s. By convention, the direction of current is the direction of flow of positive charge. Section 7.: A Microscopic View: Current and Drift Speed The drift speed of electrons in a circuit is typically a fraction of a mm/s.

5 Section 7.4: Resistance, Resistivity, and Ohm's Law The resistance R of a conductor is defined as the ratio of the potential difference across the conductor to the current in it: V = IR [7.3] The SI units of resistance are volts per ampere, or ohms, Ω = V/A. If a conductor has length l and cross-sectional area A, its resistance is R [7.5] A where ρ, is an intrinsic property of the conductor called the electrical resistivity. Section 7.6: Electrical Energy and Power If a potential difference V is maintained across an electrical device, the power, or rate at which energy is supplied to the device, is P = IV [7.8] Section 8.: Resistors in Series The equivalent resistance of a set of resistors connected in series is R eq R R [8.4] R3 The current is the same through each resistor in series, and is the same as the current through the equivalent resistance. The voltages across the individual resistors add to give the voltage across the equivalent resistance. Section 8.3: Resistors in Parallel The equivalent resistance of a set of resistors connected in parallel is [8.6] R eq R R R 3 The potential difference across any two parallel resistors is the same. The currents through each resistor add to give the current through the equivalent resistor. Section 8.4: Kirchhoff's Rules and Complex DC Circuits Complex circuits can be analyzed using Kirchhoff's rules:. The sum of the currents entering any junction must equal the sum of the currents leaving that junction.. The sum of the potential differences across all the elements around any closed circuit loop must be zero.

6 9.3 Magnetic Fields Magnetic fields are a way of describing Coulomb s law forces for moving charges. Moving charges produce magnetic fields. Moving test charges feel magnetic forces. Whether a field is only an electric field or a combination of electric and magnetic fields depends on the motion of the observer. The magnetic force that acts on a charge q moving with velocity v in a magnetic field B has magnitude F qvb sin [9.] where θ is the angle between v and B. To find the direction of this force, use right-hand rule number : point the fingers of your open right hand in the direction of v and then curl them in the direction of B. Your thumb then points in the direction of the magnetic force F. If the charge is negative rather than positive, the force is directed opposite the force given by the righthand rule. The SI unit of the magnetic field is the tesla (T). Section 9.4: Magnetic Force on a Current-Carrying Conductor If a straight conductor of length l carries current I, the magnetic force on that conductor when it is placed in a uniform external magnetic field B is F IBsin (Don t memorize) [9.6] where θ is the angle between the direction of the current and the direction of the magnetic field. Right-hand rule number also gives the direction of the magnetic force on the conductor. In this case, however, you must point your fingers in the direction of the current rather than in the direction of v. Section 9.6: Motion of a Charged Particle in a Magnetic Field If a charged particle moves in a uniform magnetic field so that its initial velocity is perpendicular to the field, it will move in a circular path in a plane perpendicular to the magnetic field. The radius r of the circular path can be found from Newton's second law and centripetal acceleration.

7 Section 9.7: Magnetic Field of a Long, Straight Wire and Ampère's Law The magnetic field at distance r from a long, straight wire carrying current I has the magnitude B I 0 [9.] r where μ 0 is the permeability of free space. The magnetic field lines around a long, straight wire are circles concentric with the wire. The direction of the circles is given by a right-hand rule: Put your thumb in the direction of the current, and the magnetic field lines go around the wire in the direction of your fingers. Section 0.: Induced emf and Magnetic Flux The magnetic flux Φ B through a closed loop is defined as BAcos [0.] B where B is the strength of the uniform magnetic field, A is the cross-sectional area of the loop, and angle between B and a direction perpendicular to the plane of the loop. is the Section 0.: Faraday's Law of Induction Faraday's law of induction states that the instantaneous emf induced in a circuit equals the negative of the rate of change of magnetic flux through the circuit, B N [0.] t where N is the number of loops in the circuit. The magnetic flux Φ B can change with time whenever the magnetic field B, the area A, or the angle θ changes with time. Lenz's law states that the current from the induced emf creates a magnetic field with flux opposing the change in magnetic flux through a circuit. Section 0.5: Generators When a coil of wire with N turns, each of area A, rotates with constant angular speed ω in a uniform magnetic field B, the emf induced in the coil is NAB sint (Use only, don t memorize) [0.7] Such generators naturally produce alternating current (AC), which changes direction with frequency ω/π. The AC current can be transformed to direct current.

8 Section 0.7: RL Circuits When the current in a coil changes with time, an emf is induced in the coil according to Faraday's law. This self-induced emf is defined by the expression I L [0.9] t where L is the inductance of the coil. The SI unit for inductance is the henry (H); H = V s/a. If a resistor and inductor are connected in series to a battery and a switch is closed at t = 0, the current in the circuit doesn't rise instantly to its maximum value. After one time constant τ = L/R, the current in the circuit is 63.% of its final value ε/r. As the current approaches its final, maximum value, the voltage drop across the inductor approaches zero. Section 0.8: Energy Stored in a Magnetic Field The energy stored in the magnetic field of an inductor carrying current I is Section.: Resistors in an AC Circuit PEL LI [0.5] If an AC circuit consists of a generator and a resistor, the current in the circuit is in phase with the voltage, which means the current and voltage reach their maximum values at the same time. In discussions of voltages and currents in AC circuits, rms values of voltages are usually used. The rms values of currents and voltage (I rms and V rms ), are related to the maximum values of these quantities (I max and V max ) as follows: I rms I, V max V max rms [.] [.3] The rms voltage across a resistor is related to the rms current in the resistor by Ohm's law: V rms I rmsr [.4] Section.: Capacitors in an AC Circuit If an AC circuit consists of a generator and a capacitor, the voltage lags behind the current by 90. This means that the voltage reaches its maximum value one-quarter of a period after the current reaches its maximum value. (Positive voltage opposes current flow, as in a resistor.) The impeding effect of a capacitor on current in an AC circuit is given by the capacitive reactance X C, defined as X C f C [.5] where f is the frequency of the AC voltage source.

9 The rms voltage across and the rms current in a capacitor are related by V I X C, rms rms C [.6] Section.3: Inductors in an AC Circuit If an AC circuit consists of a generator and an inductor, the voltage leads the current by 90. This means the voltage reaches its maximum value one-quarter of a period before the current reaches its maximum value. The effective impedance of a coil in an AC circuit is measured by a quantity called the inductive reactance X L, defined as X L f L [.8] The rms voltage across a coil is related to the rms current in the coil by V I X L, rms rms L [.9] Section.4: The RLC Series Circuit If an AC circuit contains a resistor, an inductor, and a capacitor connected in series, the limit they place on the current is given by the impedance Z of the circuit, defined as Z R ( X L X C ) [.3] The relationship between the maximum voltage supplied to an RLC series AC circuit and the maximum current in the circuit, which is the same in every element, is V rms I rmsz [.4] In an RLC series AC circuit, the applied rms voltage and current are out of phase. The phase angle between the current and voltage is given by Be able to draw a phasor diagram for R, XL, XC, and Z. X L X C tan [.5] R Section.6: Resonance in a Series RLC Circuit The current has its maximum value when the impedance has its minimum value, corresponding to X L = X C and Z = R. The frequency f 0 at which this happens is called the resonance frequency of the circuit, given by f [.9] 0 LC

10 Section.7: The Transformer If the primary winding of a transformer has N turns and the secondary winding consists of N turns, then if an input AC voltage V is applied to the primary, the induced voltage in the secondary winding is given by N V [.] V N When N is greater than N, V exceeds V and the transformer is referred to as a step-up transformer. When N is less than N, making V less than V, we have a step-down transformer. In an ideal transformer, the power output equals the power input. Section.8-.3: Electromagnetic Waves and their Properties Electromagnetic waves are created by accelerating electric charges, and have the following properties:. The electric and magnetic fields of electromagnetic waves are perpendicular to the direction of propagation of the waves.. Electromagnetic waves travel at the speed of light. 3. The ratio of the electric field to the magnetic field at a given point in an electromagnetic wave equals the speed of light: E c [.6] B 4. The speed c, frequency f, and wavelength λ of an electromagnetic wave are related by c f [.3] Section.: The Nature of Light Light has a dual nature. In some experiments it acts like a wave, in others like a particle, called a photon by Einstein. The energy of a photon is proportional to its frequency, E hf [.] where h J s is Planck's constant. Light is neither a wave nor a particle. We know how light behaves, but it is difficult for us to know what light is.

11 Section.: Reflection and Refraction In the reflection of light off a flat, smooth surface, the angle of incidence,, with respect to a line perpendicular to the surface is equal to the angle of reflection, : [.] Light that passes into a transparent medium is bent at the boundary and is said to be refracted. The angle of refraction is the angle the ray makes with respect to a line perpendicular to the surface after it has entered the new medium. Section.3: The Law of Refraction The index of refraction of a material, n, is defined as c n [.4] v where c is the speed of light in a vacuum and v is the speed of light in the material. The law of refraction, or Snell's law, states that n [.8] sin n sin where n and n are the indices of refraction in the two media. The incident ray, the reflected ray, the refracted ray, and the normal to the surface all lie in the same plane. Section.4: Dispersion and Prisms Section.7: Total Internal Reflection Total internal reflection can occur when light, traveling in a medium with higher index of refraction, is incident on the boundary of a material with a lower index of refraction. The maximum angle of incidence for which light can move from a medium with index n into a medium with index n, where n is greater c than n, is called the critical angle and is given by n sin c for n n [.9] n Total internal reflection is used in the optical fibers that carry data at high speed around the world.

12 Section 3.-3: Images Formed by Spherical Mirrors The magnification M of a mirror or lens is defined as the ratio of the image height h' to the object height h, which is the negative of the ratio of the image distance q to the object distance p: M h q [3.] h p The object distance and image distance for a mirror or lens are related by: f [3.6] p q where f is the focal length of the lens or mirror. For a mirror, the focal length and radius of curvature are related by f = R/. Sign conventions: f > 0 for converging lenses or mirrors, f < 0 for diverging lenses or mirrors q > 0 for real images, q < 0 for virtual images. p > 0 for all physical objects (the usual case) and for images functioning as objects when they are in front of the lens or mirror. p < 0 for images functioning as objects when they are behind the lens or mirror. Section 3.6: Thin Lenses Equations (3.) and (3.6) also apply to lenses. Section 4.: Conditions for Interference Interference occurs when two or more light waves overlap at a given point. Constructive interference occurs between two waves when they are in phase (crests are on crests and troughs on troughs). Destructive interference occurs when the waves are out phase (crests are on troughs). Section 4.: Young's Double-Slit Experiment In Young's double-slit experiment, two slits separated by distance d are illuminated by a monochromatic (one-wavelength), coherent (all the light has in the beam has the same phase) light source. An interference pattern consisting of bright and dark fringes is observed on a screen a distance L from the slits. The condition for bright (constructive interference) and dark (destructive interference) are: Constructive : Destructive : d sin m d sin m m 0,,, [4.] [4.3] The number m is called the order number of the fringe. The position y m of the bright fringes on the screen can be determined by using the relation sin y m / L, which is true for small angles. This can be substituted into Equations 4. and 4.3 to give the location of the bright fringes on a screen.

13 Section 4.3: Change of Phase Due to Reflection Section 4.4: Interference in Thin Films An electromagnetic wave undergoes a phase change of 80 on reflection from a medium with a higher index of refraction. There is no change when the wave goes into a medium with a lower index of refraction. The wavelength n of light in a medium with index of refraction n is n [4.7] n where λ is the wavelength of the light in free space. Light reflecting from a thin film of thickness t will undergo constructive or destructive interference depending on whether the crests reflected from the upper surface align with crests or troughs from the lower surface. This is summarized in the table below: Equation (m = 0,,...) Section 4.7: Single-Slit Diffraction phase shift 0 or phase shifts t m [4.9] constructive destructive n t m [4.0] destructive constructive n When waves pass through small openings, around obstacles, or by sharp edges, light bends around the edges and interferes with light bending around other parts of the object. The diffraction pattern produced by a single slit on a distant screen consists of a central bright maximum flanked by less bright fringes alternating with dark regions. The angles θ at which the diffraction pattern has zero intensity (regions of destructive interference) are described by Destructive : sin m m,, 3, [4.] a where a is the width of the slit and λis the wavelength of the light incident on the slit. Section 4.8: The Diffraction Grating A diffraction grating consists of many equally spaced, identical slits. The condition for maximum intensity in the interference pattern of a diffraction grating is Constructive : d sin m m 0,,, [4.] where d is the spacing between adjacent slits and m is the order number of the diffraction pattern. A diffraction grating can be made by putting a large number of evenly spaced scratches on a glass slide. The number of such lines per centimeter is the inverse of the spacing d.

14 Section 4.9: Polarization of Light Waves In unpolarized light, the electric field is oriented in random directions perpendicular to the light ray. In polarized light, the electric field is all oriented in the same direction. Unpolarized light can be polarized by selective absorption, reflection, or scattering. When unpolarized light passes through a polarizing sheet, its intensity is reduced by half, and the light becomes polarized. Section 5.: The Eye Hyperopia (farsightedness): An object at 5 cm forms an image at the near point. Myopia: (nearsightedness): An object at infinity forms an image at the far point. The power of a lens in diopters is the inverse of the focal length in meters. Section 6.3: Einstein's Principle of Relativity The two basic postulates of the special theory of relativity are as follows:. The laws of physics are the same in all inertial frames of reference.. The speed of light is the same for all inertial observers, independently of their motion or of the motion of the source of light. Section 6.4: Consequences of Special Relativity Some of the consequences of the special theory of relativity are as follows:. Clocks in motion relative to an observer slow down, a phenomenon known as time dilation. The relationship between time intervals in the moving and at-rest systems is t t [6.7] p where Δt is the time interval measured in the system in relative motion with respect to the clock, [6.8] β = v/c, and Δt p is the proper time interval measured in the system moving with the clock.. The length of an object in motion is contracted in the direction of motion. The equation for length contraction is L / [6.9] L p where L is the length measured by an observer in motion relative to the object and L p is the proper length measured by an observer for whom the object is at rest. 3. Events that are simultaneous for one observer are not simultaneous for another observer in motion relative to the first.

15 Section 6.6: Relativistic Momentum The relativistic expression for the momentum of a particle moving with velocity v is p mv [6.0] Section 6.7: Relativistic Energy and the Equivalence of Mass and Energy The rest energy of an object is E R mc [6.3] The total energy of an object is E E R [6.5] The kinetic energy of an object is the difference between the total energy and the rest energy: K E ER mc ( ) [6.] Momentum is related to the total energy through the equation E p c E [6.6] R Section 7.: Blackbody Radiation and Planck's Hypothesis The characteristics of blackbody radiation can't be explained with classical concepts. The hotter an object, the higher the frequency (the smaller the wavelength) is the peak its blackbody radiation curve. Planck first introduced the quantum concept when he assumed that the subatomic oscillators responsible for blackbody radiation could have only discrete amounts of energy given by E n = nhf [7.] where n is a positive integer called a quantum number and f is the frequency of vibration of the resonator. Section 7.: The Photoelectric Effect and the Particle Theory of Light The photoelectric effect is a process whereby electrons are ejected from a metal surface when light is incident on that surface. Einstein provided a successful explanation of this effect by extending Planck's quantum hypothesis to electromagnetic waves. In this model, light is viewed as a stream of particles called photons, each with energy E = hf, where f is the light frequency and h is Planck's constant. The maximum kinetic energy of the ejected photoelectrons is KE max = hf φ [7.6] where φ is the work function of the metal. Section 7.6: The Dual Nature of Light and Matter

16 Light exhibits both a particle and a wave nature. Louis de Broglie proposed that all matter has both a particle and a wave nature. The de Broglie wavelength of any particle of mass m and speed v is h [7.4] p De Broglie also proposed that the frequencies of the waves associated with particles obey the Einstein relationship E = hf. Section 7.7: The Wave Function In the theory of quantum mechanics, each particle is described by a quantity Ψ called the wave function. The probability density (probability per unit volume) of finding the particle at a particular point at some instant is proportional to Ψ. Quantum mechanics has been highly successful in describing the behavior of atomic and molecular systems. Section 7.8: The Uncertainty Principle According to Heisenberg's uncertainty principle, it is impossible to measure simultaneously the exact position and exact momentum of a particle. If Δx is the uncertainty in the measured position and Δp x the uncertainty in the momentum, the product Δx Δp x satisfied the relationship: Section 8.3: The Bohr Theory of Hydrogen h x p [7.6] x 4 The Bohr model of the atom is successful in describing the spectra of atomic hydrogen and hydrogen-like ions. One of the basic assumptions of the model is that the electron can exist only in certain orbits such that its angular momentum mvr is an integral multiple of ħ, where ħ is Planck's constant divided by π. Assuming circular orbits and a Coulomb force of attraction between electron and proton, the energies of the quantum states for hydrogen are where n is an integer called the principal quantum number. E E n n,, 3, [8.] n If the electron in the hydrogen atom jumps from an orbit having quantum number n i to an orbit having quantum number n f, it emits a photon of frequency f, given by E E [8.4] n f n i The correspondence principle states that quantum mechanics is in agreement with classical physics when the quantum numbers for a system are very large.

17 Section 8.4: Quantum Mechanics and the Hydrogen Atom One of the many successes of quantum mechanics is that the quantum numbers n, l, and m l associated with atomic structure arise directly from the mathematics of the theory. The quantum number n is called the principal quantum number, l is the angular momentum quantum number (or orbital quantum number), and m l is the (orbital) magnetic quantum number. These quantum numbers can take only certain values: n < in integer steps, 0 l n, and l m l l. In addition, a fourth quantum number, called the spin magnetic quantum number m s, is needed to explain a fine doubling of lines in atomic spectra, with m s = ±½. You should know the following equation for the angular momentum in terms of the angular momentum quantum number. Note that this is not in the book! L = ħl(l + ) Section 8.9: The Exclusion Principle and the Periodic Table An understanding of the periodic table of the elements became possible when Pauli formulated the exclusion principle, which states that no two electrons in an atom in the same atom can have the same values for the set of quantum numbers n, l, m l, and m s. A particular set of these quantum numbers is called a quantum state. The exclusion principle explains how different energy levels in atoms are populated. Once one subshell is filled, the next electron goes into the vacant subshell that is lowest in energy. Atoms with similar configurations in their outermost shell have similar chemical properties and are found in the same column of the periodic table. Section 9.: Binding Energy Nuclei are represented symbolically as A Z X, where X represents the chemical symbol for the element. The quantity A is the mass number, which equals the total number of nucleons (neutrons plus protons) in the nucleus. The quantity Z is the atomic number, which equals the number of protons in the nucleus. Nuclei that contain the same number of protons but different numbers of neutrons are called isotopes. In other words, isotopes have the same Z value but different A values. Most nuclei are approximately spherical, with an average radius given by r r A / 3 [9.] 0 where A is the mass number and r 0 is a constant equal to. x 0-5 m. The total mass of a nucleus is always less than the sum of the masses of its individual nucleons. This mass difference Δm, multiplied by c, gives the binding energy of the nucleus. Section 9.3: Radioactivity The spontaneous emission of radiation by certain nuclei is called radioactivity. There are three processes 4 by which a radioactive substance can decay: alpha (α) decay, in which the emitted particles are He nuclei; beta (β) decay, in which the emitted particles are electrons or positrons; and gamma (γ) decay, in which the emitted particles are high-energy photons. Nuclei in a radioactive substance decay in such a way that the number of nuclei present varies with time according to the expression N t N 0 e [9.4a] where N is the number of radioactive nuclei present at time t, λ is the decay constant, N 0 is the number at time t = 0, and e =.78 is the base of the natural logarithms.

18 The half-life T / of a radioactive substance is the time required for half of a given number of radioactive nuclei to decay. The half-life is related to the decay constant by / T (Use only, don t memorize.) [9.5] Section 9.4: The Decay Processes If a nucleus decays by alpha emission, it loses two protons and two neutrons. A typical alpha decay is 34 4 U Th He (Know the pattern, don t memorize this equation.) [9.9] Note that in this decay, as in all radioactive decay processes, the sum of the Z values on the left equals the sum of the Z values on the right; the same is true for the A values. A typical beta decay is 4 6 C 4 7 N e e (Know the pattern, don t memorize this equation.) [9.5] When a nucleus undergoes beta decay, an antineutrino is emitted along with an electron, or a neutrino along with a positron. A neutrino has zero electric charge and a small mass (which may be zero) and interacts weakly with matter. Nuclei are often in an excited state following radioactive decay, and they release their extra energy by emitting a high-energy photon called a gamma ray. A typical gamma ray emission is C* 6 6 C (Know the pattern, don t memorize this equation.) [9.8] where the asterisk indicates that the carbon nucleus was in an excited state before gamma emission. Section 30.: Nuclear Fission In nuclear fission, the total mass of the products is always less than the original mass of the reactants. Nuclear fission occurs when a heavy nucleus splits, or fissions, into two smaller nuclei. The lost mass is transformed into energy, electromagnetic radiation, and the kinetic energy of daughter particles. Section 30.5: The Fundamental Forces of Nature There are four fundamental forces of nature: the strong (hadronic), electromagnetic, weak, and gravitational forces. The strong force is the force between nucleons that keeps the nucleus together. The weak force is responsible for beta decay. The electromagnetic and weak forces are now considered to be manifestations of a single force called the electroweak force. Every fundamental interaction is said to be mediated by the exchange of field particles. The electromagnetic interaction is mediated by the photon, the weak interaction by the W ± and Z 0 bosons, the gravitational interaction by gravitons, and the color interaction by gluons. The strong interaction is the residual effects of the color interaction between two color neutral particles located near each other.

19 Section 30.6: Positrons and Other Antiparticles An antiparticle and a particle have the same mass, but opposite charge, and may also have other properties with opposite values, such as lepton number and baryon number. It is possible to produce particle-antiparticle pairs in nuclear reactions if the available energy is greater than mc, where m is the mass of the particle (or antiparticle). Section 30.8: Classification of Particles Particles other than photons are classified as hadrons or leptons. Hadrons interact primarily through the strong force. There are two types of hadrons: baryons and mesons. Mesons have a baryon number of zero and have either zero or integer spin. Baryons, which generally are the most massive particles, have nonzero baryon numbers and spins of / or 3/. The neutron and proton are examples of baryons. Leptons are not made of more fundamental particles, to the best of our knowledge. Leptons interact only through the weak and electromagnetic forces. There are six leptons: the electron, e ; the muon, μ ; the tau, τ ;and their associated neutrinos, ν e, ν μ, and ν τ. Section 30.9: Conservation Laws In all reactions and decays, quantities such as energy, linear momentum, angular momentum, electric charge, baryon number, lepton number, electron number, and muon number are strictly conserved. Section 30.: Quarks We believe that all hadrons are composed of quarks which have fractional electric charges and baryon numbers of /3 and come in six "flavors": up, down, strange, charmed, top, and bottom. Each baryon contains three quarks, and each meson contains one quark and one antiquark. According to the theory of quantum chromodynamics, quarks have a property called color, and the force between quarks is the color force. Each flavor of quark comes in three colors: red, green, and blue. The color force increases as the distance between particles increases, so quarks are always confined and can never be knocked out of a baryon. In strong and electromagnetic interactions, we can sometimes determine whether a reaction can occur or what the reaction problems should be by applying conservation of quark flavor. For example: