Physics 212 Final Exam Notes

Constants and Formulas

  • Constants:
    • k=8.99×109N⋅m2/C2k = 8.99 \times 10^9 N \cdot m^2/C^2
    • c=3.00×108m/sc = 3.00 \times 10^8 m/s
    • vsound=340m/sv_{sound} = 340 m/s
    • g=9.8m/s2g = 9.8 m/s^2
    • ϵ0=14πk=8.85×10−12C2/N⋅m2\epsilon_0 = \frac{1}{4\pi k} = 8.85 \times 10^{-12} C^2/N \cdot m^2
    • μ0=4π×10−7T⋅m/A\mu_0 = 4\pi \times 10^{-7} T \cdot m/A
    • e=1.60×10−19Ce = 1.60 \times 10^{-19} C
    • 1eV=1.60×10−19J1 eV = 1.60 \times 10^{-19} J
    • kB=1.38×10−23J/K=8.62×10−5eV/Kk_B = 1.38 \times 10^{-23} J/K = 8.62 \times 10^{-5} eV/K
    • Proton:
      • mp=1.67×10−27kgm_p = 1.67 \times 10^{-27} kg
      • mpc2=938MeVm_p c^2 = 938 MeV
      • μp=1.41×10−26J/T\mu_p = 1.41 \times 10^{-26} J/T
    • Electron:
      • me=9.11×10−31kgm_e = 9.11 \times 10^{-31} kg
      • mec2=0.511MeVm_e c^2 = 0.511 MeV
      • μe=9.28×10−24J/T\mu_e = 9.28 \times 10^{-24} J/T
    • h=6.63×10−34J⋅s=4.14×10−15eV⋅sh = 6.63 \times 10^{-34} J \cdot s = 4.14 \times 10^{-15} eV \cdot s
    • ℏ=h2π=1.05×10−34J⋅s=6.58×10−16eV⋅s\hbar = \frac{h}{2\pi} = 1.05 \times 10^{-34} J \cdot s = 6.58 \times 10^{-16} eV \cdot s
    • hc=1240eV⋅nmh c = 1240 eV \cdot nm
    • ℏc=197eV⋅nm\hbar c = 197 eV \cdot nm
    • For hydrogen: E1=−13.6eVE_1 = -13.6 eV
  • Spin Superposition States:
    • ∣+z⟩=12∣+x⟩+12∣−x⟩|+z\rangle = \frac{1}{\sqrt{2}} |+x\rangle + \frac{1}{\sqrt{2}} |-x\rangle
    • ∣−z⟩=12∣+x⟩−12∣−x⟩|-z\rangle = \frac{1}{\sqrt{2}} |+x\rangle - \frac{1}{\sqrt{2}} |-x\rangle
    • ∣+z⟩=12∣+y⟩+i2∣−y⟩|+z\rangle = \frac{1}{\sqrt{2}} |+y\rangle + \frac{i}{\sqrt{2}} |-y\rangle
    • ∣−z⟩=−i12∣+y⟩+i12∣−y⟩|-z\rangle = -i \frac{1}{\sqrt{2}} |+y\rangle + i \frac{1}{\sqrt{2}} |-y\rangle
    • ∣+x⟩=12∣+z⟩+12∣−z⟩|+x\rangle = \frac{1}{\sqrt{2}} |+z\rangle + \frac{1}{\sqrt{2}} |-z\rangle
    • ∣−x⟩=12∣+z⟩−12∣−z⟩|-x\rangle = \frac{1}{\sqrt{2}} |+z\rangle - \frac{1}{\sqrt{2}} |-z\rangle
    • ∣+y⟩=12∣+z⟩+i12∣−z⟩|+y\rangle = \frac{1}{\sqrt{2}} |+z\rangle + i \frac{1}{\sqrt{2}} |-z\rangle
    • ∣−y⟩=12∣+z⟩−i12∣−z⟩|-y\rangle = \frac{1}{\sqrt{2}} |+z\rangle - i \frac{1}{\sqrt{2}} |-z\rangle

Particle Physics

  • Messenger Particles:
    • Photon ($\gamma$): Mass = 0, Spin = 1, Charge = 0, Interaction = Electromagnetic
    • W$^-: Mass = 80,400 MeV/c^2, Spin = 1, Charge = -1, Interaction = Weak
    • Z$^0: Mass = 91,200 MeV/c^2, Spin = 1, Charge = 0, Interaction = Weak
    • Gluon (gl): Mass = 0, Spin = 1, Charge = 0, Interaction = Strong
    • Graviton (g): Mass = 0, Spin = 2, Charge = 0, Interaction = Gravity
    • Note: Antiparticle of W$^- is W$^+$. Antiparticle of a green-antired gluon is a red-antigreen gluon, etc.
  • Leptons:
    • Electron (e$^-$): Mass = 0.511 MeV/c^2, Spin = 1/2, Charge = -1, Le = 1, L$\mu$ = 0, L$\tau$ = 0; Antiparticle: e$^+$
    • Electron-neutrino ($\nue$): Mass ≈ 0, Spin = 1/2, Charge = 0, Le = 1, L$\mu$ = 0, L$\tau$ = 0; Antiparticle: $\bar{\nue}$
    • Muon ($\mu^-$): Mass = 106 MeV/c^2, Spin = 1/2, Charge = -1, Le = 0, L$\mu$ = 1, L$\tau$ = 0; Antiparticle: $\mu^+$
    • Mu-neutrino ($\nu\mu$): Mass ≈ 0, Spin = 1/2, Charge = 0, Le = 0, L$\mu$ = 1, L$\tau$ = 0; Antiparticle: $\bar{\nu\mu}$
    • Tau ($\tau^-$): Mass = 1777 MeV/c^2, Spin = 1/2, Charge = -1, Le = 0, L$\mu$ = 0, L$\tau$ = 1; Antiparticle: $\tau^+$
    • Tau-neutrino ($\nu\tau$): Mass ≈ 0, Spin = 1/2, Charge = 0, Le = 0, L$\mu$ = 0, L$\tau$ = 1; Antiparticle: $\bar{\nu\tau}$
    • All leptons have B = 0 and S = 0.
    • For antiparticles, the signs of the additive properties are reversed.
  • Selected Baryons:
    • Proton (p): Mass = 938.3 MeV/c^2, Spin = 1/2, Charge = +1, B = 1, S = 0
    • Neutron (n): Mass = 939.6 MeV/c^2, Spin = 1/2, Charge = 0, B = 1, S = 0
    • Lambda ($\Lambda$): Mass = 1116 MeV/c^2, Spin = 1/2, Charge = 0, B = 1, S = -1
    • Sigma ($\Sigma^+$): Mass = 1189 MeV/c^2, Spin = 1/2, Charge = +1, B = 1, S = -1
    • Sigma ($\Sigma^0$): Mass = 1192 MeV/c^2, Spin = 1/2, Charge = 0, B = 1, S = -1
    • Sigma ($\Sigma^-$): Mass = 1197 MeV/c^2, Spin = 1/2, Charge = -1, B = 1, S = -1
    • Delta ($\Delta^{++}$): Mass = 1232 MeV/c^2, Spin = 3/2, Charge = +2, B = 1, S = 0
    • Delta ($\Delta^{+}$): Mass = 1232 MeV/c^2, Spin = 3/2, Charge = +1, B = 1, S = 0
    • Delta ($\Delta^{0}$): Mass = 1232 MeV/c^2, Spin = 3/2, Charge = 0, B = 1, S = 0
    • Delta ($\Delta^{-}$): Mass = 1232 MeV/c^2, Spin = 3/2, Charge = -1, B = 1, S = 0
    • Cascade ($\Xi^{0}$): Mass = 1315 MeV/c^2, Spin = 1/2, Charge = 0, B = 1, S = -2
    • Cascade ($\Xi^{-}$): Mass = 1322 MeV/c^2, Spin = 1/2, Charge = -1, B = 1, S = -2
    • Sigma-star ($\Sigma^{*+}$): Mass = 1383 MeV/c^2, Spin = 3/2, Charge = +1, B = 1, S = -1
    • Sigma-star ($\Sigma^{*0}$): Mass = 1384 MeV/c^2, Spin = 3/2, Charge = 0, B = 1, S = -1
    • Sigma-star ($\Sigma^{*-}$): Mass = 1387 MeV/c^2, Spin = 3/2, Charge = -1, B = 1, S = -1
    • Cascade-star ($\Xi^{*0}$): Mass = 1532 MeV/c^2, Spin = 3/2, Charge = 0, B = 1, S = -2
    • Cascade-star ($\Xi^{*-}$): Mass = 1535 MeV/c^2, Spin = 3/2, Charge = -1, B = 1, S = -2
    • Omega-minus ($\Omega^{-}$): Mass = 1672 MeV/c^2, Spin = 3/2, Charge = -1, B = 1, S = -3
    • All baryons have Le = L$\mu$ = L$\tau$ = 0.
    • An antiparticle is indicated with a bar, e.g., $\bar{p}$ is an antiproton.
    • For antiparticles, the signs of the additive properties are reversed.
  • Selected Mesons:
    • Pion ($\pi^0$): Mass = 135 MeV/c^2, Spin = 0, Charge = 0, B = 0, S = 0; Antiparticle: $\pi^0$
    • Pion ($\pi^+$): Mass = 140 MeV/c^2, Spin = 0, Charge = +1, B = 0, S = 0; Antiparticle: $\pi^-$
    • Kaon (K$^+$): Mass = 494 MeV/c^2, Spin = 0, Charge = +1, B = 0, S = +1; Antiparticle: K$^-$
    • Kaon (K$^0$): Mass = 498 MeV/c^2, Spin = 0, Charge = 0, B = 0, S = +1; Antiparticle: $\bar{K^0}$
    • Eta ($\eta$): Mass = 548 MeV/c^2, Spin = 0, Charge = 0, B = 0, S = 0; Antiparticle: $\eta$
    • Eta-prime ($\eta'$): Mass = 958 MeV/c^2, Spin = 0, Charge = 0, B = 0, S = 0; Antiparticle: $\eta'$
    • Rho ($\rho^+$): Mass = 775 MeV/c^2, Spin = 1, Charge = +1, B = 0, S = 0; Antiparticle: $\rho^-$
    • Rho ($\rho^0$): Mass = 775 MeV/c^2, Spin = 1, Charge = 0, B = 0, S = 0; Antiparticle: $\rho^0$
    • Omega ($\omega$): Mass = 783 MeV/c^2, Spin = 1, Charge = 0, B = 0, S = 0; Antiparticle: $\omega$
    • All mesons have Le = L$\mu$ = L$\tau$ = 0.
    • For antiparticles, the signs of the additive properties are reversed.
    • Many mesons are their own antiparticles.
  • Quark Properties:
    • Up (u): Spin = 1/2, Charge = +2/3, B = 1/3, S = 0
    • Down (d): Spin = 1/2, Charge = -1/3, B = 1/3, S = 0
    • Strange (s): Spin = 1/2, Charge = -1/3, B = 1/3, S = -1
    • The antiquarks ($\bar{u}$, $\bar{d}$, $\bar{s}$) have the same spin but the opposite Q, B, and S.

Exam Problems

  • Problem 1:
    • A point charge q=6.8×10−6Cq = 6.8 \times 10^{-6} C is placed at the origin.
    • (a) Determine the strength of the electric field at point P located on the y-axis at y=1.6my = 1.6 m.
    • (b) Indicate the direction of the electric field at point P (into page, out of page).
  • Problem 2:
    • Explain what it means for two particles to be in an entangled state, and why Einstein thought quantum entanglement produced “spooky action at a distance.”
  • Problem 3:
    • A wire consists of two concentric half circles, one with smaller radius a and the other with larger radius b, connected by straight line segments.
    • A current of magnitude I0I_0 circulates through the wire.
    • Determine the magnetic field vector at point P located at the center of the two circular arcs. Express the answer in terms of the given parameters and fundamental constants.
  • Problem 4:
    • A scanning electron microscope uses a beam of electrons with wavelength 1.2nm1.2 nm.
    • Determine the speed of the electrons in this beam.
  • Problem 5:
    • A particle of charge −q0-q_0 and mass m is moving to the right with speed v.
    • The particle is located in a region of uniform magnetic field with strength B0B_0 pointing out of the page.
    • The charge travels in a circular path.
    • (a) Sketch the circular path of the particle.
    • (b) Start from Newton’s Second Law and determine the radius of the particle’s circular path.
  • Problem 6:
    • Bucknell physicists have discovered a new messenger particle with mass 20MeV/c220 MeV/c^2.
    • Determine the range of the force associated with this messenger particle.
  • Problem 7:
    • Light of wavelength 633nm633 nm is incident on a two-slit apparatus with slit spacing of 1.89×10−6m1.89 \times 10^{-6} m, and is projected onto a distant screen.
    • (a) Determine the phase difference Δϕ\Delta \phi for the two beams arriving at point P, at an angle of 3.2∘^{\circ} above the direction towards the central maximum.
    • (b) The light from each of the slits individually reaches the point P with an amplitude A. Draw a phasor diagram and determine the amplitude of the combined waves (answer will contain the constant A).
  • Problem 8:
    • A particle with energy E is confined in a square well potential that is infinite at the left boundary but finite at the right boundary.
    • Sketch a valid wavefunction for this particle in the ψ(x)\psi(x) versus x space.
  • Problem 9:
    • A solid, three-dimensional sphere of radius R has a total positive charge Q spread uniformly throughout its volume.
    • Use Gauss’s law to determine the magnitude of the electric field at point P a distance r from the center of the sphere.
  • Problem 10:
    • An electron is prepared in the following spin superposition state: ∣ψ⟩=110∣+z⟩+310∣−z⟩|\psi\rangle = \frac{1}{\sqrt{10}} |+z\rangle + \frac{3}{\sqrt{10}} |-z\rangle
    • Determine the probability that a measurement of the x-component of this particle’s spin will give the value Sx=−ℏ2S_x = -\frac{\hbar}{2}.
  • Problem 11:
    • Consider the reaction: Σ+→Δ0+e++νe\Sigma^+ \rightarrow \Delta^0 + e^+ + \nu_e.
    • (a) For this reaction, test ALL relevant conservation laws for particle physics.
    • (b) Is this reaction possible? Justify your answer.
  • Problem 12:
    • Consider the reaction n+μ−→Σ−+νμn + \mu^- \rightarrow \Sigma^- + \nu_\mu.
    • (a) Determine the quark content of the baryons and mesons in this reaction.
    • (b) Construct a complete reaction diagram clearly labeling all messenger particles, quarks, and/or leptons, with colors where appropriate.
  • Problem 13:
    • Were positrons abundant in the universe at a time 0.1 s after the Big Bang? Use the results of a quantitative calculation to justify your conclusion.
  • Problem 14:
    • A guitar string vibrates in the third longest wavelength mode, with the string fixed at either end. The length of the string is 0.65 m and the wave speed on the string is 572 m/s.
    • Determine the frequency of vibration of this guitar string.
  • Problem 15:
    • Use Heisenberg’s Uncertainty Principle to determine the minimum kinetic energy for an electron confined to a region of size 5.6×10−10m5.6 \times 10^{-10} m.