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\[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\] - \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\] + \[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\] % ----------------------- \subsection*{Magnetic fields} @@ -293,7 +293,10 @@ % \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 $\Phi$ @@ -364,6 +367,7 @@ % ----------------------- $T={1 \over f}\quad$(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} @@ -373,14 +377,14 @@ % ----------------------- \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 @@ -389,10 +393,19 @@ % ----------------------- \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} @@ -424,10 +437,14 @@ 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} @@ -435,31 +452,32 @@ % ----------------------- \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, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$ @@ -473,7 +491,7 @@ \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 $f_0$} @@ -528,7 +546,7 @@ $f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ & 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} -- 2.43.2