Two transparent media A and B are separated by a plane boundary. The speeds of light in those media are $1.5 \

Question

Two transparent media A and B are separated by a plane boundary. The speeds of light in those media are $1.5 	imes 10^8$ m/s and $2.0 	imes 10^8$ m/s, respectively. The critical angle for a ray of light for these two media is:

Accepted Answer

1. **Refractive Index and Speed:** The refractive index ($n$) of a medium is inversely proportional to the speed of light ($v$) in that medium ($n = c/v$). Thus, the medium with the lower speed is optically denser.
   - Speed in A ($v_A$) = $1.5 	imes 10^8$ m/s (Denser)
   - Speed in B ($v_B$) = $2.0 	imes 10^8$ m/s (Rarer)
2. **Condition for Total Internal Reflection:** Light must travel from the denser medium (A) to the rarer medium (B).
3. **Critical Angle Formula:** The sine of the critical angle ($i_c$) is equal to the ratio of the refractive indices ($n_{rarer}/n_{denser}$) or the ratio of speeds ($v_{denser}/v_{rarer}$).
   $$ \sin i_c = \frac{n_B}{n_A} = \frac{v_A}{v_B} $$
4. **Calculation:**
   $$ \sin i_c = \frac{1.5 	imes 10^8}{2.0 	imes 10^8} = \frac{1.5}{2.0} = 0.75 $$
   $$ i_c = \sin^{-1}(0.750) $$

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