Physics Q10 – NEET-UG 2023

Light travels a distance

x

in time

t_{1}

in air and

10 x

in time

t_{2}

in another denser medium. What is the critical angle for this medium?

Clue 1 of 3: Speed and Refractive Index

Recall that the speed of light changes in different media, and this change is quantified by the refractive index.

🥷

Ninja Strategy•Physical Constraints on Critical Angle

The argument of $\sin^{- 1}$ for a critical angle must be between 0 and 1. By establishing the relationship $t_{2} > 10 t_{1}$ (from $v_{a i r} > v_{m e d i u m}$ for a denser medium), options (1) and (2) can be immediately eliminated as their arguments become greater than 1.

🧠 Full Solution Path

Step 1: Calculate Speeds of Light✦ Active

The speed of light in air ( $v_{a i r}$ ) and in the denser medium ( $v_{m e d i u m}$ ) can be calculated using the formula speed = distance/time.

v_{a i r} = \frac{x}{t_{1}}

v_{m e d i u m} = \frac{10 x}{t_{2}}

Step 2: Determine Refractive Index of Denser Medium○ Expand

The refractive index of the denser medium ( $n_{m e d i u m}$ ) with respect to air is the ratio of the speed of light in air to the speed of light in the medium. We assume the refractive index of air is approximately 1.

n_{m e d i u m} = \frac{v_{a i r}}{v_{m e d i u m}} = \frac{x / t_{1}}{10 x / t_{2}} = \frac{x}{t_{1}} \times \frac{t_{2}}{10 x} = \frac{t_{2}}{10 t_{1}}

💡 Teacher's Secret Hint

Remember that for a denser medium, $n_{m e d i u m}$ must be greater than 1, which implies $t_{2} > 10 t_{1}$ from our calculation.

Step 3: Calculate Critical Angle○ Expand

The critical angle ( $θ_{c}$ ) for light traveling from the denser medium to air is given by $\sin θ_{c} = \frac{1}{n_{m e d i u m}}$ (since $n_{a i r} \approx 1$ ). Substitute the value of $n_{m e d i u m}$ .

\sin θ_{c} = \frac{1}{\frac{t_{2}}{10 t_{1}}} = \frac{10 t_{1}}{t_{2}}

θ_{c} = \sin^{- 1} (\frac{10 t_{1}}{t_{2}})

💡 Teacher's Secret Hint

The argument of $\sin^{- 1}$ must be between 0 and 1 for a real angle. This means $\frac{10 t_{1}}{t_{2}} \leq 1$ .

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Physics Question 10 – NEET-UG 2023

Clue 1 of 3: Speed and Refractive Index

🧠 Full Solution Path