Final Exam Equations

$Logarithmic\ Entropy = k_B * ln(Total\ Microstates\ (W))$ $Choose\ M\ out\ of\ N\ Objects = \frac{N!}{(N-M)! * M!}$

$[Electric\ Field] = \left[\frac{Force}{Charge}\right] = \left[\frac{Mass * Acceleration}{Charge}\right] = \left[\frac{Mass * Length}{Time^2 *Charge}\right]$ $k_c = \frac{8.99 * 10^9\ Newton\ \cdot meters^2}{1\ Coulomb^2}$ $Point\ Charge\ Electric\ Field = E = k_c *\frac{q}{r^2}$

$1\ Avagadro\ Constant = \frac{Number\ of\ Particles \ (N)\ 6.02214076 * 10^{23} }{1\ mol}$ $1\ Joule = \frac{1\ Kilo\ Gram * 1\ Meter^2}{1\ Second^2}$ $Planck\ Constant\ (h) = \frac{6.62607015*10^{-34}\ Joules}{1\ Second}$ $Reduced\ Planck\ Constant\ (\hbar) = \frac{Plank\ Constant\ (h)}{2*\pi} = \frac{1.054571817*10^{-34}\ Joules}{1\ Second}$ $1\ Hertz = \frac{60\ Revolutions}{1\ Minute} = \frac{2 *\ \pi\ Radians}{1\ Second}$ $Angular\ Veloctiy\ (\omega) = 2 * \pi * Frequency\ (f)\ Hertz$ $Frequency\ (f) = \frac{Angular\ Velocity\ (\omega)}{2 * \pi}$ $Speed\ of\ Light = \frac{299792458\ meters}{1\ Second}$ $1\ Electric\ Constant\ (\epsilon_0)\ (Vacuum\ Permittivity) = \frac{8.854187*10^{-12}\ Farads}{1\ Meter}$ $Force\ Between\ Two\ Separated\ Electric\ Charges\ with\ Spherical\ Symmetry\ in\ a\ Vacuum = F_C = \frac{1}{4*\pi * \epsilon_0}*\frac{q_1*q_2}{r^2}$ $Gravitational\ Constant = \frac{6.67408 * 10^{-11} Meters^3}{1\ Kilogram^3 * 1\ Second^2}$ $Force_{gravity} = \frac{Gravitatonal\ Constant\ (G)* Mass_1 * Mass_2}{Distance\ Between\ Mass_1\ and\ Mass_2\ (r)}$ $\frac{Force_{gravity}}{Force_{electric}} = \frac{Gravitational\ Constant\ (G) * Mass_1 * Mass_2}{Coulomb\ Constant * 1\ Elementary\ Charge\ (e^2)}$ $Acceleration\ Due\ To\ Gravity\ (a) = \frac{Gravitatonal\ Constant\ (G) * Mass}{(Radius\ of\ Earth\ (6.3781 * 10^6\ Meters))^2} = \frac{9.81\ Meters}{1\ Second^2}$ $Force = Mass * Acceleration$ $Energy_{photon} = \frac{Planks\ Constant\ (h) * Speed\ of\ Light\ (c)}{Wavelength\ \lambda_{photon} } = 4.66* 10^{-19}\ Joule$ $Energy_{photon} * Wavelength\ (\lambda)_{photon} = Plank\ Constant\ (h) * Speed\ of\ Light\ (C)$ $Energy_{photon} = Plank\ Constant\ (h) * Frequency_{Proton}\ (f)$ $Energy_{photon} = \frac{6.62607015*10^{-34}\ Joules}{1\ Second} * \frac{4.60 * 10^{14}Frequency_{Proton}\ (f)\ Hertz}{1\ Second}$ $Characterization\ of\ Electromagnetic\ Force , Finate\ Structure\ Constant\ (\alpha) = \frac{2*\pi *\ Coulomb\ Constant (k) * (1\ Elementary\ Charge)^2}{E_{photon}*Wavelength\ (\lambda)_{photon}} = \frac{2* \pi * Coulomb\ Constant (k) * (1\ Elementary\ Charge)^2}{h *c } = \frac{1}{137}$ $Fine\ Structure\ Coupling\ Strength\ (1\ Elementary\ Charge^2) = 4 * \pi * Electric\ Constant\ (\epsilon_0) * Reduced\ Planck\ Constant\ (\hbar) * Elementary\ Charge\ (c) * Gap\ Between\ Spectral\ Lines\ of\ Hydrogen\ (\alpha)$ $1\ Elementary\ Charge = \sqrt{\left(4*\pi *Fine-Structure\ Constant\ (a)\right)} * \sqrt{Reduced\ Planck's\ Constant\ (\hbar ) * Speed\ of\ Light\ (C)}$ $1\ Elementary\ Charge\ (e) = \frac{1\ Farad}{1\ Avagadro\ Constant}$ $1\ Elementary\ Charge\ (e) = 1.602176634*10^{-19} Coulomb$ $1\ Coulomb = 1.036 * 10^{-5}\ mols \cdot Na\ elementary\ charge$ $1\ Ohm\ (\Omega) = \frac{1\ Joule \cdot Second}{1\ Coulomb^2}$ $1\ Ampere\ (A)(Amp) = \frac{1\ Coulomb}{1\ Second}$ $\ Siemen\ (S) = [\Omega^{-1}] = \frac{1\ Ampere\ (A)}{1\ Volt}$ $1\ Siemen\ (S) = \frac{1\ Coulomb^2}{1\ Joule \cdot Second}$ $Conductance\ (G) = \frac{1}{Resistance} = \frac{Current\ (I)}{Voltage\ (V)} = Siemens$ $\left[2\ Plates\ Charge\ Separation\ ,\ (+Q,-Q)\ ,\ Coulomb\ , \frac{1\ Ampere}{1\ Second}\right]$ $\left[Capacitance\ ,\ (C)\ ,\ Farad\ , \frac{1\ Ampere}{1\ Second}\right]$ $\left[Potential\ Difference\ ,\ (\Delta V)\ ,\ Volt\ ,\ \frac{1\ Joule}{1\ Coulomb}\right]$ $Charge\ (Q) = Capacitance\ (C) * Potential\ Difference\ (\Delta V)$ $Capacitance = \frac{Charge\ (Q)}{Potential\ Difference\ (\Delta V)}$ $Potential\ Difference\ (\Delta V) = \frac{Charge\ (Q)}{Capacitance\ (C)}$ $1\ Farad = \frac{1\ Coulomb}{1\ Volt} = \frac{1\ Coulomb}{\frac{1\ Joule}{1\ Coulomb}} = \frac{1\ Coulomb}{\frac{\frac{1\ Kilo\ Gram * 1\ Meter^2}{1\ Second^2}}{1\ Coulomb}} = \frac{1\ Coulomb^2}{\frac{1\ Kilo\ Gram * 1\ Meter^2}{1\ Second^2}} = \frac{1\ Coulomb^2 * 1\ Second^2}{1\ Kilo\ Gram * 1\ Meter^2}$ $Electric\ Field\ (E) = \frac{Potential\ Difference\ (\Delta V)}{Length\ (L)}$ $Charge\ Density\ (\sigma) = \frac{Total\ Charge\ (q)}{Surface\ Area\ (A)}$ $Dielectric\ Constant\ (K) = \frac{Capacitance\ (C)_{dielectric}}{Capacitance\ (C_0)_{vaccum}}$ $(Energy\ Density)_{dielectric}\ (u) = \frac{Dielectric\ Constant\ (K) * Permitivity\ In\ Vacuum\ (\epsilon_0) * Electric\ Field^2}{2}$ $Surface\ Charge\ Density\ (\sigma) = \frac{Total\ Charge\ (q)}{Surface\ Area\ (A)}$

$|Charge\ on\ Each\ Capacitor\ Plate| = Capacitance\ (C) * Voltage\ Difference\ Across\ Capacitor\ Plate\ (\Delta V)$ $V_1 - V_2 = \frac{Q}{C}$ $Total\ Resistance\ in\ Parallel = R_T = \frac{1}{\sum^n_{i=1}\left(\frac{1}{R_i}\right)}$ $Voltage\ Difference\ (\Delta V) = Current\ (I) * Resistance\ (R)$ $I_{in} = I_{out}$ $1\ Ohm\ (\Omega) = \frac{1\ Joule \cdot \ Second}{1\ Coulomb^2}$ $1\ Volt = \frac{1\ Joule}{1\ Coulomb}$ $1\ Watt = \frac{1\ Joule}{1\ Second} = \frac{kg \cdot m^2}{s^3} = Power$ $I = \frac{\Delta V}{R} = \frac{\frac{1\ Joule}{1\ Coulomb}}{\frac{1\ Joule \cdot \ Second}{1\ Coulomb^2}} = \frac{\cancel{1\ Joule}}{\cancel{1\ Coulomb}} * \frac{1\ Coulomb^{\cancel{2}}}{\cancel{1\ Joule}\ \cdot\ Second} = \frac{1\ Coulomb}{1\ Second}$ $Total\ Resistance\ in\ Series = \sum^n_{i=1}\left(R_i\right)$ $Capacitance\ in\ Series = C_{eq} = \frac{1}{\sum^n_{i=1}\left(\frac{1}{C_i}\right)}$ $Capacitance\ in\ Parallel = \sum^n_{i=1}\left(C_i\right)$ $Voltage\ Drop\ on\ Individual\ Resistor = \frac{Resistance_{individual}\ Ω}{ Resistance_{total}\ Ω } \ * (Voltage_{total})$ $\Delta V_{across\ R_1} = Current\ (I) * R_1 = \frac{V * R_1}{Resistance_{total/eq}} = \frac{V* R_1}{R_1 + R_2}$

$Angular\ Frequency\ (\omega) = \sqrt{\frac{Spring\ Constant\ (k)}{Mass\ (m)}}$ $Spring\ Constant\ (k) = Mass\ (m) * Angular\ Frequency\ (\omega)^2$ $Resistance\ (R) = \frac{Potential\ Difference\ (\Delta V)}{Current\ (I)}$ $Capacitance\ (C) = \frac{Charge\ (Q)}{Potential\ Difference\ (\Delta V)}$ $Inductor\ (L) = \frac{Potential\ Difference\ (\Delta V)}{\frac{dI}{dt}}$

Speed\ of\ a\ Wave\ (V) = Frequency\ (f) * Wavelength\ (\lambda)

Phase\ Velocity = \frac{Angular\ Frequency\ (\omega)}{Wave\ Number\ (k)}

Phase\ Velocity = \frac{25\ Hertz\ \left(\frac{cycles}{1\ second}\right)}{\frac{0.125\ Wave\ Lengths}{1\ cm}} = \frac{200\ cm\cdot Cycles}{1\ Wave\ Length\cdot Second}

Wave\ Propagation\ Speed = \sqrt{\frac{Tension}{Linear\ Mass\ Density\ of\ String}}

Wave\ Propagation\ Speed \propto \sqrt{\frac{1}{Linear\ Mass\ Density\ of\ String}}

Wavelength\ (\lambda) = \frac{Velocity}{Frequency}

Wave\ Period\ (T) = \frac{1}{Frequency\ (f)} = \frac{Wavelength\ (\lambda)}{Velocity\ (v)}

Wave\ Propagation\ Velocity = Frequency\ (f) * Wavelength\ (\lambda)

Wave\ Propagation\ Velocity = \sqrt{\frac{Tension\ (T)}{Linear\ Mass\ Density\ of\ String\ (\mu)}}

Angular\ Velocity\ (\omega) = 2 * \pi * Frequency\ (f)

Frequency\ (f) = \frac{Angular\ Velocity\ (\omega)}{2*\pi}

Power\ Transmitted\ By\ String\ Wave = \frac{Linear\ Mass\ Density\ (\mu)*Angular\ Velocity\ (\omega)^2*Wave\ Amplitude\ (A)^2* Wave\ Propagation\ Velocity\ (v)}{2}

Angular\ Wave\ Number\ (k) = \frac{2*\pi}{Wave\ Length\ (\lambda)} = \frac{2*\pi}{0.7795454545454545\ meters} = 8.060062784720168

1.A. What is the energy of the emitted photon (in electron volts)?

Energy = \frac{h*c}{\lambda} = \frac{1239.84193\ eV \cdot nm}{590\ nm} = 2.101427\ eV

1.B. At room temperature (T ~ 300 K), what is the probability that a sodium atom has its electron excited up to the 3p state?

\frac{P(3p)}{P(3s)} = e^{-\left(\frac{\Delta E}{k*T}\right)} = e^{-\left(\frac{2.1\ eV}{\frac{8.61733262145*10^{-5}\ eV}{1\ K} *300\ K}\right)} = 5.266839447935011*10^{-36}

Energy = Planck's\ Constant\ (h) * Frequency\ (f) = \frac{Planck's\ Constant\ (h)*Speed\ of\ Light\ (c)}{Wavelength\ (\lambda)}

Energy = \frac{1239.84193\ eV \cdot nm}{450\ nm} = 2.755204288888889\ eV