Physics CUET UG 13-05-2025 Shift 2 Paper

• When light travels from an optically denser medium of refractive index $n_1$ into a rarer medium of refractive index $n_2$, it bends away from the normal.
• By Snell's Law: $n_1 \sin i = n_2 \sin r$.
• The critical angle $i_c$ is defined as the angle of incidence in the denser medium for which the angle of refraction in the rarer medium becomes exactly $90^\circ$ ($r = 90^\circ$):
  $$n_1 \sin i_c = n_2 \sin 90^\circ = n_2 \implies \sin i_c = \frac{n_2}{n_1}$$

• Let the transmitted intensity through the first polarizer $P_1$ be $I_1$.
• If the middle polaroid sheet is oriented at an angle $\theta$ with respect to $P_1$, by Malus's Law, the transmitted intensity through it is:
  $$I_2 = I_1 \cos^2\theta$$
• Since $P_1$ and $P_2$ are crossed (at $90^\circ$), the angle between the middle sheet and $P_2$ is $(90^\circ - \theta)$. Applying Malus's Law again:
  $$I_3 = I_2 \cos^2(90^\circ - \theta) = I_1 \cos^2\theta \sin^2\theta = \frac{I_1}{4} (2 \sin\theta \cos\theta)^2 = \frac{I_1}{4} \sin^2(2\theta)$$
• $I_3$ is maximum when $\sin^2(2\theta) = 1 \implies 2\theta = 90^\circ = \pi/2 \implies \theta = 45^\circ = \pi/4$.

Evaluating each statement regarding Total Internal Reflection (TIR):

• Statement (A): TIR occurs strictly when light travels from an optically denser medium to an optically rarer medium at an angle exceeding the critical angle. Hence (A) is incorrect.
• Statement (B): In TIR, $100\%$ of the incident optical energy is reflected back into the denser medium; none propagates into the second medium $\rightarrow$ (B) is correct.
• Statement (C): TIR obeys the fundamental law of reflection ($i = r$) within the denser medium $\rightarrow$ (C) is correct.
• Statement (D): Because no refracted ray enters the second medium, there is no refracted ray and hence no angle of refraction $\rightarrow$ (D) is correct.

By Cartesian sign conventions:

• concave (diverging) lens has a negative focal length: $f = -50\,\text{cm} = -0.5\,\text{m}$.
• The optical power $P$ in dioptres (D) is given by:
  $$P = \frac{1}{f\,(\text{m})} = \frac{1}{-0.5\,\text{m}} = -2\,\text{D}$$