Optics

Q1. (4 marks)

• Snell's law: $n_1\sin\theta_1 = n_2\sin\theta_2$ — ratio of sines equals inverse ratio of refractive indices. (1)
• Critical angle $\theta_c$: the angle of incidence in the denser medium for which the angle of refraction is $90^\circ$. (1)
• Derivation: put $\theta_1 = \theta_c$ (denser $n_1$) and $\theta_2 = 90^\circ$: $n_1\sin\theta_c = n_2\sin90^\circ$. (1)
• So $\sin\theta_c = n_2/n_1$, i.e. $\theta_c = \sin^{-1}(n_2/n_1)$ (for $n_1>n_2$). (1)

Q2. (5 marks)

• Sign convention: $u=-15\,\text{cm}$, $f=-10\,\text{cm}$ (concave). (1)
• Mirror equation $\dfrac1v+\dfrac1u=\dfrac1f \Rightarrow \dfrac1v = \dfrac1f-\dfrac1u = \dfrac{1}{-10}-\dfrac{1}{-15}$. (1)
• $\dfrac1v = -\dfrac1{10}+\dfrac1{15} = \dfrac{-3+2}{30}=-\dfrac1{30}\Rightarrow v=-30\,\text{cm}$. (1)
• Magnification $m=-\dfrac{v}{u}=-\dfrac{-30}{-15}=-2$. (1)
• Image at $30\,\text{cm}$ in front, real, inverted, magnified ($2\times$). (1)

Q3. (4 marks)

• Malus's law: $I = I_0\cos^2\theta$, where $\theta$ is the angle between the polarizer axis and the plane of polarization of incident light. (2)
• $\theta=30^\circ$: $I = I_0\cos^2 30^\circ = I_0\times(0.866)^2 = 0.75\,I_0$. (2)

Q4. (5 marks)

• Lens maker's equation: $\dfrac1f = (n-1)\left(\dfrac1{R_1}-\dfrac1{R_2}\right)$. (1)
• Biconvex: $R_1=+20\,\text{cm}$, $R_2=-20\,\text{cm}$. (1)
• $\dfrac1f = (1.5-1)\left(\dfrac1{20}-\dfrac1{-20}\right)=0.5\times\dfrac{2}{20}=0.5\times0.1=0.05\,\text{cm}^{-1}$. (1)
• $f = 20\,\text{cm}=0.20\,\text{m}$. (1)
• Power $P=\dfrac1{f(\text{m})}=\dfrac1{0.20}=+5\,\text{D}$. (1)

Q5. (5 marks)

• Path difference at point $y$: $\Delta = d\sin\theta \approx \dfrac{d\,y}{D}$. (1)
• Bright fringes: $\dfrac{d\,y_n}{D}=n\lambda \Rightarrow y_n=\dfrac{n\lambda D}{d}$. (1)
• Fringe width $\beta = y_{n+1}-y_n = \dfrac{\lambda D}{d}$. (1)
• Substitute: $\beta = \dfrac{600\times10^{-9}\times1}{0.5\times10^{-3}}$. (1)
• $\beta = 1.2\times10^{-3}\,\text{m}=1.2\,\text{mm}$. (1)

Q6. (4 marks)

• Grating condition: $d\sin\theta = n\lambda$ (principal maxima). (1)
• Grating spacing $d = \dfrac{1\,\text{cm}}{5000}=2\times10^{-4}\,\text{cm}=2\times10^{-6}\,\text{m}$. (1)
• $\sin\theta = \dfrac{n\lambda}{d}=\dfrac{1\times500\times10^{-9}}{2\times10^{-6}}=0.25$. (1)
• $\theta = \sin^{-1}(0.25)=14.48^\circ$. (1)

Q7. (3 marks)

• Powers in contact add: $P = P_1+P_2 = 5+(-2)=+3\,\text{D}$. (2)
• $f = \dfrac1P = \dfrac1{3}\,\text{m}=+33.3\,\text{cm}$. (1)

Q8. (5 marks)

• Brewster's law: when light is incident at the polarizing angle $\theta_B$, the reflected light is completely plane-polarized; reflected and refracted rays are mutually perpendicular. (1)
• Geometry: $\theta_B + 90^\circ + \theta_r = 180^\circ \Rightarrow \theta_r = 90^\circ-\theta_B$. (1)
• Snell: $n = \dfrac{\sin\theta_B}{\sin\theta_r}=\dfrac{\sin\theta_B}{\sin(90^\circ-\theta_B)}$. (1)
• $=\dfrac{\sin\theta_B}{\cos\theta_B}$. (1)
• $\therefore n=\tan\theta_B$. (1)

Q9. (3 marks)

• Chromatic aberration: inability of a lens to focus different colours (wavelengths) at the same point, due to variation of $n$ with $\lambda$; reduced using an achromatic doublet (convex + concave of different glasses). (1.5)
• Spherical aberration: failure of a spherical mirror/lens to focus marginal and paraxial rays at the same point; reduced by using stops/parabolic surfaces or plano-convex orientation. (1.5)

Q10. (2 marks)

• Rayleigh criterion: two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other. (1)
• Limit of resolution: $\Delta\theta = \dfrac{1.22\lambda}{D}$. (1)

[
  {"claim":"Q2 image distance v = -30 cm","code":"u=-15; f=-10; v=1/(1/f-1/u); result = abs(v-(-30))<1e-9"},
  {"claim":"Q2 magnification m = -2","code":"u=-15; v=-30; m=-v/u; result = abs(m-(-2))<1e-9"},
  {"claim":"Q4 focal length 20 cm and power 5 D","code":"n=Rational(3,2); R1=20; R2=-20; f=1/((n-1)*(Rational(1,R1)-Rational(1,R2))); P=1/(f/100); result = (f==20) and (P==5)"},
  {"claim":"Q5 fringe width = 1.2 mm","code":"lam=600e-9; D=1; d=0.5e-3; beta=lam*D/d; result = abs(beta-1.2e-3)<1e-12"},
  {"claim":"Q6 sin theta = 0.25 for first order","code":"d=2e-6; lam=500e-9; s=1*lam/d; result = abs(s-0.25)<1e-12"},
  {"claim":"Q7 combined power 3 D, focal length 1/3 m","code":"P=5+(-2); f=1/P; result = (P==3) and abs(f-Rational(1,3))<1e-12"},
  {"claim":"Q3 Malus 30 deg gives 0.75 I0","code":"val=cos(pi/6)**2; result = simplify(val-Rational(3,4))==0"}
]