% -----------------------
\subsection*{Work and energy}
- $W=Fx=\Delta \Sigma E$ (work)
+ $W=Fs=Fs \cos \theta=\Delta \Sigma E$
$E_K = {1 \over 2}mv^2$ (kinetic)
$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
- $\Sigma mv_0=\Sigma mv_1$ (conservation)
+ $\Sigma (mv_0)=(\Sigma m)v_1$ (conservation)
- $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
+ % $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
- $n$-body collisions: $\rho$ of each body is independent
+ % $\Sigma E_K = \Sigma ({1 \over 2} m v^2) = {1 \over 2} (\Sigma m)v_f$
+
+ if elastic:
+ $$\sum _{i{\mathop {=}}1}^{n}E_K (i)=\sum _{i{\mathop {=}}1}^{n}({1 \over 2}m_i v_{i0}^2)={1 \over 2}\sum _{i{\mathop {=}}1}^{n}(m_i) v_f^2$$
+
+ % $n$-body collisions: $\rho$ of each body is independent
% ++++++++++++++++++++++
\section{Relativity}
$\therefore \, t$ must dilate as speed changes
+ {\bf high-altitude particles:} $t$ dilation means more particles reach Earth than expected (half-life greater when obs. from Earth)
+
{\bf Inertial reference frame} $a=0$
{\bf Proper time $t_0$ $\vert$ length $l_0$} measured by observer in same frame as events
% -----------------------
\subsection*{Energy and work}
- $E_0 = mc^2$ (rest)
+ $E_{\text{rest}} = mc^2, \quad E_K = (\gamma-1)mc^2$
- $E_{total} = E_K + E_{rest} = \gamma mc^2$
+ $E_{\text{total}} = E_K + E_{\text{rest}} = \gamma mc^2$
- $E_K = (\gamma 1)mc^2$
-
- $W = \Delta E = \Delta mc^2$
+ $W = \Delta E = \Delta mc^2=(\gamma-1)m_{\text{rest}} c^2$
% -----------------------
\subsection*{Relativistic momentum}
$$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
% -----------------------
- \subsection*{High-altitude muons}
- \begin{itemize}
- {\item $t$ dilation more muons reach Earth than expected}
- {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
- \end{itemize}
% +++++++++++++++++++++++
\section{Fields and power}
\[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
- \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
+ \[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
% -----------------------
\subsection*{Magnetic fields}
% \textbf{Right hand slap:} $B \perp I \perp F$ \\
% ($I$ = thumb)
- \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
+ \includegraphics[width=\columnwidth]{graphics/lenz.png}
+
+ \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}.$
+ If $f$ increases, ampl. \& $f$ of $\mathcal{E}$ increase
\textbf{Transformers:} core strengthens \& focuses $\Phi$
% -----------------------
$T={1 \over f}\quad$(period: time for one cycle)
$v=f \lambda \quad$(speed: displacement / sec)
+ $f={c \over \lambda}\quad\hspace{0.7em}$(for $v=c$)
% -----------------------
\subsection*{Doppler effect}
% -----------------------
\subsection*{Interference}
- \includegraphics[width=4.5cm]{graphics/possons-spot.png}
+ \includegraphics[width=4.5cm]{graphics/poissons-spot.png} \\
Poissons's spot supports wave theory (circular diffraction)
\textbf{Standing waves} - constructive int. at resonant freq
\textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
- \textbf{Incoherent} - e.g. incandescent bulb
+ \textbf{Incoherent} - e.g. incandescent/LED
% -----------------------
\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 \(v\)) \\
- where \(a=2\) for antinodes at both ends, \(a=4\) for antinodes at one end
+ 1st harmonic = fundamental
+
+ \textbf{for nodes at both ends:} \\
+ \(\hspace{2em} \lambda = {{2l} \div n}\)
+ \(\hspace{2em} f = {nv \div 2l} \)
+
+ \textbf{for node at one end ($n$ is odd):} \\
+ \(\hspace{2em} \lambda = {{4l} \div n}\)
+ \(\hspace{2em} f = {nv \div 4l} \) \\
+ alternatively, $\lambda = {4l \over {2n-1}}$ where $n\in \mathbb{Z}$ and $n+1$ is the next possible harmonic
+
+
+ % \(a=2\) for nodes at both ends, \\ \(a=4\) for node at one end
% -----------------------
\subsection*{Polarisation}
angle of incidence $\theta_i =$ angle of reflection $\theta_r$
- Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
+ Critical angle $\theta_c = \sin^{-1}{n_2 \over n_1}$
Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
+ ${v_1 \div v_2} = {\sin\theta_1 \div \sin\theta_2}$
+
+ $n_1 v_1 = n_2 v_2$
+
% +++++++++++++++++++++++
\section{Light and Matter}
% -----------------------
\subsection*{Planck's equation}
- \[ f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c \]
+ \[ \quad E=hf={hc \over \lambda}=\rho c = qV\]
\[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
\[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
- \subsection*{Force of electrons}
- \[ F={2P_{\text{in}}\over c} \]
- % \begin{align*}
- \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
- \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
- % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
- % \end{align*}
-
\subsection*{De Broglie's theory}
\[ \lambda = {h \over \rho} = {h \over mv} \]
\[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
+ \[ v = \sqrt{2E_K \div m} \]
\begin{itemize}
\item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
\item confirmed by e- and x-ray patterns
\end{itemize}
+ \subsection*{Force of electrons}
+ \[ F={2P_{\text{in}}\over c} \]
+ % \begin{align*}
+ \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
+ \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
+ % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
+ % \end{align*}
+
\subsection*{X-ray electron interaction}
\begin{itemize}
- \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
+ \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ and $r$ is radius of orbit
\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 $V_{\operatorname{sup}} > 0$: 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
+ \item max $I$ (not $V$) depends on intensity
\end{itemize}
\subsubsection*{Threshold frequency $f_0$}
measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
- \subsection*{Wave-particle duaity}
+ \subsection*{Wave-particle duality}
\subsubsection*{wave model}
\begin{itemize}