\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{enumitem}
+\usepackage{supertabular}
+\usepackage{tabularx}
\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
$\theta = \tan^{-1} {{v^2} \over rg}$
- $\Sigma F$ always acts towards centre, but not necessarily horizontally
+ $\Sigma F$ always acts towards centre (horizontally)
$\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
% -----------------------
\subsection*{Projectile motion}
\begin{itemize}
- \item{horizontal component of velocity is constant if no air resistance}
- \item{vertical component affected by gravity: $a_y = -g$}
+ \item $v_x$ is constant: $v_x = {s \over t}$
+ \item use suvat to find $t$ from $y$-component
+ \item vertical component gravity: $a_y = -g$
\end{itemize}
\begin{align*}
$\text{elastic potential energy} = {1 \over 2}kx^2$
+ $x={2mg \over k}$
+
% -----------------------
\subsection*{Motion equations}
\[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
\[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
\[F=qvB\tag{$F$ on moving $q$}\]
\[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[B={mv \over qr}\tag{field strength on e-}\]
\[r={mv \over qB} \tag{radius of $q$ in $B$}\]
if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
\[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
\[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
- \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$
+ \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
+ (emf creates $I$ with associated field that opposes $\Delta \phi$)
\textbf{Eddy currents:} counter movement within a field
% \begin{wrapfigure}{r}{-0.1\textwidth}
\includegraphics[height=4cm]{graphics/dc-motor-2.png}
- \includegraphics[height=3cm]{graphics/ac-motor.png} \\
+ \includegraphics[height=3cm]{graphics/ac-motor.png} \\
+
+ Force on current-carying wire, not copper \\
+ $F=0$ for front & back of coil (parallel) \\
+ Any angle $> 0$ will produce force \\
% \end{wrapfigure}
\textbf{DC:} split ring (two halves)
% -----------------------
\subsection*{Doppler effect}
+
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" $\lambda$).
% -----------------------
\subsection*{Interference}
-
+ \includegraphics[width=4.5cm]{graphics/possons-spot.png}
+ Poissons's spot supports wave theory (circular diffraction)
\textbf{Standing waves} - constructive int. at resonant freq
- \subsection*{Harmonics}
+ \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
+
+ \textbf{Incoherent} - e.g. incandescent bulb
+
+
+
+ % -----------------------
+ \subsection*{Harmonics}
+
\(\lambda = {{al} \div n}\quad\) (\(\lambda\) for \(n^{th}\) harmonic)\\
\(f = {nv \div al}\quad\) (\(f\) for \(n_{th}\) harmonic at length
\(l\) and speed \(v\)) \\
\includegraphics[width=6cm]{graphics/diffraction.jpg}
\includegraphics[width=6cm]{graphics/diffraction-2.png}
\begin{itemize}
- \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
+ % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
\item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
\item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
- \item Fringe separation: \(\Delta x = {{\lambda l }\over d}\) where \\
- \(\Delta x\) = fringe spacing \\
- \(l\) = distance from slits to screen\\
- \(d\) = slit separation (\(=S_1-S_2\))
+ \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
+ % \(\Delta x\) = fringe spacing \\
+ \(l\) = distance from source to observer\\
+ \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
+ \item diffraction $\propto {\lambda \over d}$
\item significant diffraction when ${\lambda \over \Delta x} \ge 1$
+ \item diffraction creates distortion (electron $>$ optical microscopes)
\end{itemize}
-
% -----------------------
\subsection*{Refraction}
\includegraphics[height=3.5cm]{graphics/refraction.png}
% ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
% \end{align*}
+ \subsection*{De Broglie's theory}
+
+ \[ \lambda = {h \over \rho} = {h \over mv} \]
+ \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
+ \begin{itemize}
+ \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
+ \item confirmed by e- and x-ray patterns
+ \end{itemize}
+
+ \subsection*{X-ray electron interaction}
+
+ \begin{itemize}
+ \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
+ \item $\therefore 2\pi r = n{h \over mv} = n \lambda$ (circumference)
+ \item if $2\pi r \ne n{h \over mv}$, no standing wave
+ \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
+ % \item calculating $h$: $\lambda = {h \over \rho}$
+ \end{itemize}
+
\subsection*{Photoelectric effect}
\begin{itemize}
\item $V_{\operatorname{supply}}$ does not affect photocurrent
- \item $V_{\operatorname{sup}} > 0$: e- attracted to collector anode
- \item $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$
- \item $v$ of depends on ionisation energy (shell)
+ \item $V_{\operatorname{sup}} > 0$: attracted to +ve
+ \item $V_{\operatorname{sup}} < 0$: attracted to -ve, $I\rightarrow 0$
+ \item $v$ of e- depends on shell
\item max current depends on intensity
\end{itemize}
- \textbf{Threshold frequency $f_0$}
+ \subsubsection*{Threshold frequency $f_0$}
- Minimum $f$ for photoelectrons to be ejected. $x$-intercept of frequency vs $E_K$ graph. if $f < f_0$, no photoelectrons are detected.
+ min $f$ for photoelectron release. if $f < f_0$, no photoelectrons.
- \textbf{Work function $\phi$}
+ \subsubsection*{Work function $\phi=hf_0$}
- Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of frequency vs $E_K$ graph. $\phi$ is determined by strength of bonding.
+ min $E$ for photoelectron release. determined by strength of bonding. Units: eV or J.
- $\phi=hf_0$
+ \subsubsection*{Kinetic energy E_K=hf - \phi = qV_0}
- \textbf{Kinetic energy}
- E_{\operatorname{k-max}}=hf - \phi
+ $V_0 = E_K$ in eV \\
+ % $E_K = x$-int of $V\cdot I$ graph (in eV) \\
+ dashed line below $E_K=0$
- voltage in circuit or stopping voltage = max $E_K$ in eV
- equal to $x$-intercept of volts vs current graph (in eV)
- \textbf{Stopping potential $V$ for min $I$}
+ \subsubsection*{Stopping potential $V_0$ for min $I$}
- $V=h_{\text{eV}}(f-f_0)$
+ $$V_0=h_{\text{eV}}(f-f_0)$$
- % \columnbreak
+ \subsubsection*{Graph features}
- \subsection*{De Broglie's theory}
+ \newcolumntype{b}{>{\hsize=.75\hsize}X}
+\newcolumntype{s}{>{\hsize=.3\hsize}X}
- \[ \lambda = {h \over \rho} = {h \over mv} \]
- \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
- \begin{itemize}
- \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
- \item confirmed by similar e- and x-ray diff patterns
- \end{itemize}
+ \begin{tabularx}{\columnwidth}{bbbb}
+\hline
+&$m$&$x$-int&$y$-int \\
+\hline
+\hline
+$f \cdot E_K$ & $h$ & $f_0$ & $-\phi$ \\
+$V \cdot I$ & & $V_0$ & intensity\\
+$f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
+\hline
+\end{tabularx}
- \subsection*{X-ray electron interaction}
- \begin{itemize}
- \item e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
- \item rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference)
- \item if $2\pi r \ne n{h \over mv}$, no standing wave
- \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
- \item calculating $h$: $\lambda = {h \over \rho}$
- \end{itemize}
\subsection*{Spectral analysis}
\item $f$ is irrelevant to photocurrent
\item predicts delay between incidence and ejection
\item speed depends on medium
+ \item supported by bright spot in centre
\end{itemize}
\subsubsection*{particle model}
Uncertainty of a measurement is $1 \over 2$ the smallest division
\textbf{Precision} - concordance of values \\
- \textbf{Accuracy} - closeness to actual value
-
- \columnbreak
-
- \quad
-
-
-
-
-
+ \textbf{Accuracy} - closeness to actual value\\
+ \textbf{Random errors} - unpredictable, reduced by more tests \\
+ \textbf{Systematic errors} - not reduced by more tests \\
+ \textbf{Uncertainty} - margin of potential error \\
+ \textbf{Error} - actual difference \\
+ \textbf{Hypothesis} - can be tested experimentally \\
+ \textbf{Model} - evidence-based but indirect representation
\end{multicols}
-% \includegraphics[height=5cm]{graphics/em-spectrum.png}
-
\end{document}