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Stage 2 SEM1, Vector calculus (Operations (The curl of a vector field…

- - - - The Laplacian \[\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\]
- - - - i) Find \(s\)
      - ii) Invert \(s=s(t)\Rightarrow t=t(s)\)
      - iii) Substitute \(t(s)\) into \(x(t)\), \(y(t)\) and find new limits
- - - - \(S(U,V,N)\) can be considered a fundamental relation if the three postulates are met
        
        \(S\) is continuous and differentiable wrt \(U\), \(V\) and \(N\)
        
        \(S\) increases monotonically wrt \(U\)
        
        Thermodynamic variables (AKA 'equations of state')
        
        Intensive
        
        \((\frac{\partial U}{\partial N})_{S,V}=\mu\)
        
        \((\frac{\partial U}{\partial V})_{S,N}=-p\)
        
        \((\frac{\partial U}{\partial S})_{V,N}=T\)
        
        Extensive
        
        \((\frac{\partial S}{\partial V})_{U,N}=\frac{p}{T}\)
        
        \((\frac{\partial S}{\partial N})_{U,V}=-\frac{\mu}{T}\)
        
        \((\frac{\partial S}{\partial U})_{V,N}=\frac{1}{T}\)
        
        If \(T(s,v)\) and \(p(s,v)\) are known, the fundamenal relation can be found by integrating
        \(\mathrm{d}u=T\mathrm{d}s - P\mathrm{d}v\)
        
        \(S\) is extensive (i.e. additive over its constituent subsystems)
      - The Euler relation: \(U=TS-pV+\mu N\;\Rightarrow\;\mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V+\mu\mathrm{d}N\)
      - The Gibbs-Duhem relation: \(S\mathrm{d}T-V\mathrm{d}p+\mu\mathrm{d}N\)
      - \(U=cNRT\)
      - \(W=\int_{V_0}^V p\mathrm{d}V\)
- - - - Derivatives
- - - - \(\mathrm{d}Q_c=-\frac{T_c}{T_h}\mathrm{d}Q_h\)
        \(\mathrm{d}W=-[1-\frac{T_c}{T_h}]\mathrm{d}Q_h\) where \(\eta_{max}=-\frac{\mathrm{d}W}{\mathrm{d}Q_h}=[1-\frac{T_c}{T_h}]\)
        
        Another way to find the work done: \(W=\int\eta\mathrm{d}Q_h\)
- - - - \(\Delta>0\Rightarrow \varphi=Ae^{\lambda_1x}+Be^{\lambda_2x}\)
      - \(\Delta<0\Rightarrow\varphi=e^{\alpha x}(A\cos\beta x + B\sin\beta x)\)
      - \(\Delta = 0 \Rightarrow\varphi=Ae^{\lambda x}+Bxe^{\lambda x}\)
  - - - \(T=\frac{k_T}{k_I}|\frac{A_T}{A_I}|^2\), \(R=|\frac{A_R}{A_I}|^2\)
        \(T+R=1\)
- - - - Multiple lens can be used in achromatic doublets, systems which use a biconvex and phono-concave lens to reduce chromatic abberation
      - The thin lens equation:
        \(\frac{n_m}{s_0}+\frac{n_m}{s_i}=(n_l-n_m)(\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{f}\)
        For air (\(n_m=1\)), this shortens to:
        \(\frac{1}{s_0}+\frac{1}{s_i}=(n_l-1)(\frac{1}{R_1}-\frac{1}{R_2})=\frac{1}{f}\)
        
        For biconvex lens, generally \(R_1>0\) and \(R_1=-R_2\)
- - - - Linearly polarised light is a superposition of left-circularly and right-circularly polarized light.
        It is rotated in an optically active medium; this is because left-circularly and right-circularly polarised light have different refractive indices
        \(\theta(z)=\beta z\;\;\;\;\;\;\;\;\;\;\beta = \frac{\pi}{\lambda}(n_R-n_L)\)
        Right handed rotation is dextrorotary; \(n_R>n_L\). Left handed motion is levorotary.
        
        Malus' Law \(I(\theta)=I(0)\cos^2{\theta}\) for light passing through two linear polarizers, where \(\theta=\theta_2-\theta_1\neq 0\)
        
        \(\)
        
        Transmitted light is polarized in the plane of the polarizer
        
        If there is a phase shift in the left or right-handed circular component, in general the direction of the linear polarisation is changed.
        
        Fabry-Perot etalons
        Coefficient of finesse \(F=\frac{4R}{(1-R)^2}\)
        Finesse \(\mathcal{F}=\frac{\Delta\delta}{\bar{\delta}}=\frac{\pi\sqrt{F}}{2}\) is the ratio of peak separation to peak width; a measure of how clearly they are separated
  - - - Standard results
        
        \(\mathrm{rect}(\frac{x}{a}) \Rightarrow a\text{sinc}(au)=\frac{1}{\pi au}\sin{\pi au}\)
      - Convolution
        \(f(x)*g(x)=\int_{-\infty}^{\infty}f(x')g(x-x')\mathrm{d}x'\)
        \(\Rightarrow \mathcal{F}[f(x)*g(x)] = \mathcal{F}[f(x)]\cdot\mathcal{F}[g(x)]\)
        
        Properties of Dirac function
        \(\int_{-\infty}^{\infty}f(x)\cdot\delta(x-a)\mathrm{d}x=f(a)\)
        \(\int_{-\infty}^{\infty}e^{i2\pi (x-a)u}\mathrm{d}u=\delta(x-a)\)
  - - - There are \(N-2\) subsidiary maxima between each main maximum
        As \(N\rightarrow\infty\), the main maxima become narrower, more closely resembling a comb graph, and there are more subsidiary maxima between them
  - - - The Huggens Fresnel principle states that each element of a wavefront acts as a source of secondary wavelets, which diverge spherically. The resultant field is a superposition of these wavelets.
- - - - Forward
        
        The p junction is connected to the positive terminal: \(n_p\) and \(p_n\) are reduced. Carriers are pushed towards and hence shrink the depletion region
        
        \(\phi_b\Rightarrow \phi_b-V_a\)
      - Reverse
        
        The p junction is connected to the negative terminal: \(n_p\) and \(p_n\) are increased. Carriers are pulled away from and hence widen the depletion region
        
        \(\phi_b\Rightarrow \phi_b+V_a\)
    - - Tunnelling between neighbouring valence and conduction bands under high can occur reverse bias
      - In a strong electric field, carriers gain enough energy to make ionising collisions, creating hole-electron pairs - this leads to a spike in \(I_p\)
      - High reverse bias leads to heat dissipation: this further increases the current, leading to thermal runaway