physics / final.texon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (62daddf)
   1\documentclass[a4paper]{article}
   2\usepackage{multicol}
   3\usepackage[cm]{fullpage}
   4\usepackage{amsmath}
   5\usepackage{amssymb}
   6\setlength{\parindent}{0cm}
   7\usepackage[nodisplayskipstretch]{setspace}
   8\setstretch{1.3}
   9\usepackage{graphicx}
  10\usepackage{wrapfig}
  11\usepackage{enumitem}
  12\usepackage{supertabular}
  13\usepackage{tabularx}
  14\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
  15
  16
  17\begin{document}
  18
  19\pagenumbering{gobble}
  20\begin{multicols}{3}
  21
  22% +++++++++++++++++++++++
  23
  24{\huge Physics}\hfill Andrew Lorimer\hspace{2em}
  25
  26% +++++++++++++++++++++++
  27\section{Motion}
  28
  29  $\operatorname{m/s} \, \times \, 3.6 = \operatorname{km/h}$
  30
  31  \subsection*{Inclined planes}
  32    $F = m g \sin\theta - F_{\text{frict}} = m a$
  33
  34% -----------------------
  35  \subsection*{Banked tracks}
  36
  37    \includegraphics[height=4cm]{graphics/banked-track.png}
  38
  39    $\theta = \tan^{-1} {{v^2} \over rg}$
  40
  41    $\Sigma F$ always acts towards centre (horizontally)
  42
  43    $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
  44
  45    Design speed $v = \sqrt{gr\tan\theta}$
  46
  47    $n\sin \theta = {mv^2 \div r}, \quad n\cos \theta = mg$
  48
  49% -----------------------
  50  \subsection*{Work and energy}
  51
  52    $W=Fs=Fs \cos \theta=\Delta \Sigma E$
  53
  54    $E_K = {1 \over 2}mv^2$ (kinetic)
  55
  56    $E_G = mgh$ (potential)
  57
  58    $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
  59
  60% -----------------------
  61  \subsection*{Horizontal circular motion}
  62
  63    $v = {{2 \pi r} \over T}$
  64
  65    $f = {1 \over T}, \quad T = {1 \over f}$
  66
  67    $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
  68
  69    $\Sigma F, a$ towards centre, $v$ tangential
  70
  71    $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
  72
  73    \includegraphics[height=4cm]{graphics/circ-forces.png}
  74
  75% -----------------------
  76  \subsection*{Vertical circular motion}
  77
  78    $T =$ tension, e.g. circular pendulum
  79
  80    $T+mg = {{mv^2}\over r}$ at highest point
  81
  82    $T-mg = {{mv^2} \over r}$ at lowest point
  83
  84    $E_K_{\text{bottom}}=E_K_{\text{top}}+mgh$
  85
  86% -----------------------
  87  \subsection*{Projectile motion}
  88    \begin{itemize}
  89      \item $v_x$ is constant: $v_x = {s \over t}$
  90      \item use suvat to find $t$ from $y$-component
  91      \item vertical component gravity: $a_y = -g$
  92    \end{itemize}
  93
  94    % \begin{align*}
  95      $v=\sqrt{v^2_x + v^2_y}$ \hfill vectors \\
  96      $h={{u^2\sin \theta ^2}\over 2g}$ \hfill max height \\
  97      $x=ut\cos\theta$ \hfill $\Delta x$ at $t$ \\
  98      $y=ut \sin \theta-{1 \over 2}gt^2$ \hfill height at $t$ \\
  99      $t={{2u\sin\theta}\over g}$ \hfill time of flight \\
 100      $d={v^2 \over g}\sin \theta$ \hfill horiz. range \\
 101    % \end{align*}
 102
 103    \includegraphics[height=3.2cm]{graphics/projectile-motion.png}
 104
 105% -----------------------
 106  \subsection*{Pulley-mass system}
 107
 108    $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
 109
 110    $\Sigma F = m_2g-m_1g=\Sigma ma$ (solve)
 111
 112% -----------------------
 113  \subsection*{Graphs}
 114    \begin{itemize}
 115      \item{Force-time: $A=\Delta \rho$}
 116      \item{Force-disp: $A=W$}
 117      \item{Force-ext: $m=k,\quad A=E_{spr}$}
 118      \item{Force-dist: $A=\Delta \operatorname{gpe}$}
 119      \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
 120    \end{itemize}
 121
 122% -----------------------
 123  \subsection*{Hooke's law}
 124
 125  $F=-kx$ (intercepts origin)
 126
 127  $\text{elastic potential energy} = {1 \over 2}kx^2$
 128
 129  $x={2mg \over k}$
 130
 131  Vertical: $\Delta E = {1 \over 2}kx^2 + mgh
 132
 133% -----------------------
 134  \subsection*{Motion equations}
 135
 136    \begin{tabular}{ l r }
 137      & no \\
 138      $v=u+at$ & $x$ \\
 139      $x = {1 \over 2}(v+u)t$ & $a$ \\
 140      $x=ut+{1 \over 2}at^2$ & $v$ \\
 141      $x=vt-{1 \over 2}at^2$ & $u$ \\
 142      $v^2=u^2+2ax$ & $t$ \\
 143    \end{tabular}
 144
 145% -----------------------
 146  \subsection*{Momentum}
 147
 148    $\rho = mv$
 149
 150    $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
 151
 152    $\Sigma (mv_0)=(\Sigma m)v_1$ (conservation)
 153
 154    % $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
 155
 156    % $\Sigma E_K = \Sigma ({1 \over 2} m v^2) = {1 \over 2} (\Sigma m)v_f$
 157
 158    if elastic:
 159    $$\sum _{i{\mathop {=}}1}^{n}E_K (i)=\sum _{i{\mathop {=}}1}^{n}({1 \over 2}m_i v_{i0}^2)={1 \over 2}\sum _{i{\mathop {=}}1}^{n}(m_i) v_f^2$$
 160
 161    % $n$-body collisions: $\rho$ of each body is independent
 162
 163% ++++++++++++++++++++++
 164\section{Relativity}
 165
 166  \subsection*{Postulates}
 167    1. Laws of physics are constant in all intertial reference frames
 168
 169    2. Speed of light $c$ is the same to all observers (Michelson-Morley)
 170
 171    $\therefore \, t$ must dilate as speed changes
 172
 173    {\bf high-altitude particles:} $t$ dilation means more particles reach Earth than expected (half-life greater when obs. from Earth)
 174
 175    {\bf Inertial reference frame} $a=0$
 176
 177    {\bf Proper time $t_0$ $\vert$ length $l_0$} measured by observer in same frame as events
 178
 179% -----------------------
 180  \subsection*{Lorentz factor}
 181
 182    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}, \quad v = c\sqrt{1-{1 \over \gamma^2}}$$
 183
 184    $t=t_0 \gamma$ ($t$ longer in moving frame)
 185
 186    $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)
 187
 188    $m=m_0 \gamma$ (mass dilation)
 189
 190% -----------------------
 191  \subsection*{Energy and work}
 192
 193    $E_{\text{rest}} = mc^2, \quad E_K = (\gamma-1)mc^2$
 194
 195    $E_{\text{total}} = E_K + E_{\text{rest}} = \gamma mc^2$
 196
 197    $W = \Delta E = \Delta mc^2=(\gamma-1)m_{\text{rest}} c^2$
 198
 199% -----------------------
 200  \subsection*{Relativistic momentum}
 201
 202    $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$
 203
 204    $\rho \rightarrow \infty$ as $v \rightarrow c$
 205
 206    $v=c$ is impossible (requires $E=\infty$)
 207
 208    $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
 209
 210% -----------------------
 211
 212% +++++++++++++++++++++++
 213\section{Fields and power}
 214
 215  \subsection*{Non-contact forces}
 216    \begin{itemize}
 217      {\item electric (dipoles \& monopoles)}
 218      {\item magnetic (dipoles only)}
 219      {\item gravitational (monopoles only, $F_g=0$ at mid, attractive only)}
 220    \end{itemize}
 221
 222    \vspace{1em}
 223
 224    \begin{itemize}
 225      \item monopoles: lines towards centre
 226      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (two magnets) or $\rightarrow$ N (single)
 227      \item closer field lines means larger force
 228      \item dot: out of page, cross: into page
 229      \item +ve corresponds to N pole
 230      \item Inv. sq. ${E_1 \over E_2} = ({r_2 \over r_1})^2$
 231    \end{itemize}
 232
 233    \includegraphics[height=2cm]{graphics/field-lines.png}
 234    % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.png}
 235
 236% -----------------------
 237  \subsection*{Gravity}
 238
 239    \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
 240    \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
 241    \[E_g = mg \Delta h\tag{gpe}\]
 242    \[W = \Delta E_g = Fx\tag{work}\]
 243    \[w=m(g-a) \tag{app. weight}\]
 244
 245    % \columnbreak
 246
 247% -----------------------
 248  \subsection*{Satellites}
 249
 250    \[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
 251
 252    \[T={\sqrt{4 \pi^2 r^3 \over {GM}}}\tag{period}\]
 253
 254    \[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
 255
 256% -----------------------
 257  \subsection*{Magnetic fields}
 258    \begin{itemize}
 259      \item field strength $B$ measured in tesla
 260      \item magnetic flux $\Phi$ measured in weber
 261      \item charge $q$ measured in coulombs
 262      \item emf $\mathcal{E}$ measured in volts
 263    \end{itemize}
 264
 265    % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
 266
 267    \[F=qvB\tag{$F$ on moving $q$}\]
 268    \[F=IlB\tag{$F$ of $B$ on $I$}\]
 269    \[B={mv \over qr}\tag{field strength on e-}\]
 270    \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
 271
 272    if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
 273
 274% -----------------------
 275  \subsection*{Electric fields}
 276
 277    \[F=qE(=ma) \tag{strength} \]
 278    \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
 279    \[E=k{q \over r^2} \tag{field on point charge} \]
 280    \[E={V \over d} \tag{field between plates}\]
 281    \[F=BInl \tag{force on a coil} \]
 282    \[\Phi = B_{\perp}A\tag{magnetic flux} \]
 283    \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} = Blv\tag{induced emf} \]
 284    \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
 285
 286    \textbf{Lenz's law:}  $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
 287    (emf creates $I$ with associated field that opposes $\Delta \Phi$)
 288
 289    \textbf{Eddy currents:} counter movement within a field
 290
 291    \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
 292
 293    \includegraphics[height=2cm]{graphics/slap-2.jpeg}
 294    \includegraphics[height=3cm]{graphics/grip.png}
 295
 296    % \textbf{Right hand slap:} $B \perp I \perp F$ \\
 297    % ($I$ = thumb)
 298
 299    \includegraphics[width=\columnwidth]{graphics/lenz.png}
 300
 301    \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}.$
 302    If $f$ increases, ampl. \& $f$ of $\mathcal{E}$ increase
 303
 304    \textbf{Xfmr} core strengthens \& focuses $\Phi$
 305
 306% -----------------------
 307  \subsection*{Particle acceleration}
 308
 309    $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
 310
 311    e- accelerated with $x$ V is given $x$ eV
 312
 313    \[W={1\over2}mv^2=qV \tag{field or points}\]
 314    \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
 315
 316    Circular path: $F\perp B \perp v$
 317
 318% -----------------------
 319  \subsection*{Power transmission}
 320
 321    % \begin{align*}
 322      \[V_{\operatorname{rms}}={V_{\operatorname{p}}\over \sqrt{2}}={V_{\operatorname{p\rightarrow p}}\over {2 \sqrt{2}}} \]
 323      \[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
 324      \[V_{\operatorname{loss}}=IR \]
 325    % \end{align*}
 326
 327    Use high-$V$ side for correct $|V_{drop}|$
 328
 329    \begin{itemize}
 330      {\item Parallel $V$ is constant}
 331      {\item Series $V$ shared within branch}
 332    \end{itemize}
 333
 334    \includegraphics[height=4cm]{graphics/ac-generator.png}
 335
 336% -----------------------
 337  \subsection*{Motors}
 338% \begin{wrapfigure}{r}{-0.1\textwidth}
 339
 340    \includegraphics[height=4cm]{graphics/dc-motor-2.png}
 341    \includegraphics[height=3cm]{graphics/ac-motor.png} \\
 342
 343    Force on current-carying wire, not copper \\
 344    $F=0$ for front & back of coil (parallel) \\
 345    Any angle $> 0$ will produce force \\
 346% \end{wrapfigure}
 347    \textbf{DC:} split ring (two halves)
 348
 349% \begin{wrapfigure}{r}{0.3\textwidth}
 350
 351% \end{wrapfigure}
 352    \textbf{AC:} slip ring (separate rings with constant contact)
 353
 354% \pagebreak
 355
 356% +++++++++++++++++++++++
 357\section{Waves}
 358
 359  \textbf{nodes:} fixed on graph \\
 360  \textbf{amplitude:} max disp. from $y=0$ \\
 361  \textbf{rarefactions} and \textbf{compressions} \\
 362  \textbf{mechanical:} transfer of energy without net transfer of matter \\
 363
 364
 365  \textbf{Longitudinal (motion $||$ wave)}
 366  \includegraphics[width=6cm]{graphics/longitudinal-waves.png}
 367
 368  \textbf{Transverse (motion $\perp$ wave)}
 369  \includegraphics[width=6cm]{graphics/transverse-waves.png}
 370
 371  % -----------------------
 372  $T={1 \over f}\quad$(period: time for one cycle)
 373  $v=f \lambda \quad$(speed: displacement / sec)
 374  $f={c \over \lambda}\quad\hspace{0.7em}$(for $v=c$)
 375
 376  % -----------------------
 377  \subsection*{Doppler effect}
 378
 379  When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}$. $w_n$ reaches observer sooner than $w_{n-1}$ ("apparent" $\lambda$).
 380
 381  % -----------------------
 382  \subsection*{Interference}
 383
 384  \includegraphics[width=4.5cm]{graphics/poissons-spot.png} \\
 385  Poissons's spot supports wave theory (circular diffraction)
 386
 387  \textbf{Standing waves} - constructive int. at resonant freq. Rebound from ends.
 388
 389  \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
 390
 391  \textbf{Incoherent} - e.g. incandescent/LED
 392
 393
 394
 395
 396
 397  % -----------------------
 398  \subsection*{Harmonics}
 399
 400  1st harmonic = fundamental
 401
 402  \textbf{for nodes at both ends:} \\
 403  \(\hspace{2em} \lambda = {{2l} \div n}\)
 404  \(\hspace{2em} f = {nv \div 2l} \)
 405
 406  \textbf{for node at one end ($n$ is odd):} \\
 407  \(\hspace{2em} \lambda = {{4l} \div n}\)
 408  \(\hspace{2em} f = {nv \div 4l} \) \\
 409  alternatively, $\lambda = {4l \over {2n-1}}$ where $n\in \mathbb{Z}$ and $n+1$ is the next possible harmonic
 410
 411
 412  % \(a=2\) for nodes at both ends, \\ \(a=4\) for node at one end
 413
 414  % -----------------------
 415  \subsection*{Polarisation}
 416  \includegraphics[height=3.5cm]{graphics/polarisation.png} \\
 417  Transverse only. Reduces total $A$.
 418
 419  % -----------------------
 420  \subsection*{Diffraction}
 421  \includegraphics[width=6cm]{graphics/diffraction.jpg}
 422  \includegraphics[width=6cm]{graphics/diffraction-2.png}
 423  \begin{itemize}
 424    % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
 425    \item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
 426    \item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
 427    \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
 428    % \(\Delta x\) = fringe spacing \\
 429    \(l\) = distance from source to observer\\
 430    \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
 431    \item diffraction $\propto {\lambda \over d}$
 432    \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
 433    \item diffraction creates distortion (electron $>$ optical microscopes)
 434  \end{itemize}
 435
 436
 437  % -----------------------
 438  \subsection*{Refraction}
 439  \includegraphics[height=3.5cm]{graphics/refraction.png}
 440
 441  When a medium changes character, light is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}. $\lambda$ changes, not $f$.
 442
 443  angle of incidence $\theta_i =$ angle of reflection $\theta_r$
 444
 445  Critical angle $\theta_c = \sin^{-1}{n_2 \over n_1}$
 446
 447  Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
 448
 449  ${v_1 \div v_2} = {\sin\theta_1 \div \sin\theta_2}$
 450
 451  $n_1 v_1 = n_2 v_2$
 452
 453  $n={c \over v}$
 454
 455
 456% +++++++++++++++++++++++
 457\section{Light and Matter}
 458
 459  % -----------------------
 460  \subsection*{Planck's equation}
 461
 462  \[ \quad E=hf={hc \over \lambda}=\rho c = qV\]
 463  \[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
 464  \[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
 465
 466  \subsection*{De Broglie's theory}
 467
 468  \[ \lambda = {h \over \rho} = {h \over mv} = {h \over {m \sqrt{2W \over m}}}\]
 469  \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
 470  \[ v = \sqrt{2E_K \div m} \]
 471
 472  \begin{itemize}
 473    \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
 474    \item confirmed by e- and x-ray patterns
 475  \end{itemize}
 476
 477  \subsection*{Force of electrons}
 478  \[ F={2P_{\text{in}}\over c} \]
 479  % \begin{align*}
 480  \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
 481  \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
 482  % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
 483  % \end{align*}
 484
 485  \subsection*{X-ray electron interaction}
 486
 487  \begin{itemize}
 488    \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ and $r$ is radius of orbit
 489    \item $\therefore 2\pi r = n{h \over mv} = n \lambda$ (circumference)
 490    \item if $2\pi r \ne n{h \over mv}$, no standing wave
 491    \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
 492    % \item calculating $h$: $\lambda = {h \over \rho}$
 493  \end{itemize}
 494
 495  \subsection*{Photoelectric effect}
 496
 497  \begin{itemize}
 498    \item $V_{\operatorname{supply}}$ does not affect photocurrent
 499    \item $V_{\operatorname{sup}} > 0$: attracted to +ve
 500    \item $V_{\operatorname{sup}} < 0$: attracted to -ve, $I\rightarrow 0$
 501    \item $v$ of e- depends on shell
 502    \item max $I$ (not $V$) depends on intensity
 503  \end{itemize}
 504
 505  \subsubsection*{Threshold frequency $f_0$}
 506
 507  min $f$ for photoelectron release. if $f < f_0$, no photoelectrons.
 508
 509  \subsubsection*{Work function $\phi=hf_0$}
 510
 511  min $E$ for photoelectron release. determined by strength of bonding. Units: eV or J.
 512
 513  \subsubsection*{Kinetic energy E_K=hf - \phi = qV_0}
 514
 515
 516  $V_0 = E_K$ in eV \\
 517  % $E_K = x$-int of $V\cdot I$ graph (in eV) \\
 518  dashed line below $E_K=0$
 519
 520
 521  \subsubsection*{Stopping potential $V_0$ for min $I$}
 522
 523  $$V_0=h_{\text{eV}}(f-f_0)$$
 524  Opposes induced photocurrent
 525
 526  \subsubsection*{Graph features}
 527
 528  \newcolumntype{b}{>{\hsize=.75\hsize}X}
 529\newcolumntype{s}{>{\hsize=.3\hsize}X}
 530
 531  \begin{tabularx}{\columnwidth}{bbbb}
 532\hline
 533&$m$&$x$-int&$y$-int \\
 534\hline
 535\hline
 536$f \cdot E_K$ & $h$ & $f_0$ & $-\phi$ \\
 537$V \cdot I$ &  & $V_0$ & intensity\\
 538$f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
 539\hline
 540\end{tabularx}
 541
 542
 543
 544  \subsection*{Spectral analysis}
 545
 546  \begin{itemize}
 547    \item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
 548    \item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
 549    \item Ionisation energy - min $E$ required to remove e-
 550    \item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 551    \item No. of lines - include all possible states
 552  \end{itemize}
 553
 554  \subsection*{Uncertainty principle}
 555
 556  measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
 557
 558  \subsection*{Wave-particle duality}
 559
 560  \subsubsection*{wave model}
 561  \begin{itemize}
 562    \item cannot explain photoelectric effect
 563    \item $f$ is irrelevant to photocurrent
 564    \item predicts delay between incidence and ejection
 565    \item speed depends on medium
 566    \item supported by bright spot in centre
 567    \item $\lambda = {hc \over E}$
 568  \end{itemize}
 569
 570  \subsubsection*{particle model}
 571
 572  \begin{itemize}
 573    \item explains photoelectric effect
 574    \item rate of photoelectron release $\propto$ intensity
 575    \item no time delay - one photon releases one electron
 576    \item double slit: photons interact. interference pattern still appears when a dim light source is used so that only one photon can pass at a time
 577    \item light exerts force
 578    \item light bent by gravity
 579    \item quantised energy
 580    \item $\lambda = {h \over \rho}$
 581  \end{itemize}
 582
 583  % +++++++++++++++++++++++
 584  \section{Experimental \\ design}
 585
 586  \textbf{Absolute uncertainty} $\Delta$ \\
 587  (same units as quantity)
 588  \[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
 589  \[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
 590  \[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
 591  \[ c(A \pm \Delta A) = cA \pm c \Delta A \]
 592
 593  \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
 594  \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
 595  \[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
 596  \[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
 597  \[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
 598  \[ c(A \pm \mathcal{E} A)=cA \pm \mathcal{E} A \]
 599
 600  Uncertainty of a measurement is $1 \over 2$ the smallest division
 601
 602  \textbf{Precision} - concordance of values \\
 603  \textbf{Accuracy} - closeness to actual value\\
 604  \textbf{Random errors} - unpredictable, reduced by more tests \\
 605  \textbf{Systematic errors} - not reduced by more tests \\
 606  \textbf{Uncertainty} - margin of potential error \\
 607  \textbf{Error} - actual difference \\
 608  \textbf{Hypothesis} - can be tested experimentally \\
 609  \textbf{Model} - evidence-based but indirect representation
 610
 611\end{multicols}
 612
 613\begin{center}
 614  \includegraphics[height=2.95cm]{graphics/spectrum.png}
 615\end{center}
 616
 617\end{document}