a point of light is placed at hee bottom of a large tank filled with a liquid refractive index n=5/3.what is thwe percentage of light energy that escapes from the liquid surface into the surroundings?

To determine the percentage of light energy that escapes from a liquid with a refractive index of \( n = \frac{5}{3} \), we can use Snell's Law and the concept of critical angle. The critical angle is the angle of incidence above which total internal reflection occurs. When light travels from a medium with a higher refractive index to one with a lower refractive index, some of the light will be refracted out of the liquid, while some will be reflected back into it.

Understanding the Critical Angle

The critical angle (\( \theta_c \)) can be calculated using the formula:

\( \theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right) \)

Here, \( n_1 \) is the refractive index of the liquid (which is \( \frac{5}{3} \)), and \( n_2 \) is the refractive index of air (approximately 1). Plugging in the values:

Calculating the Critical Angle

Substituting the values into the formula:

\( \theta_c = \sin^{-1}\left(\frac{1}{\frac{5}{3}}\right) = \sin^{-1}\left(\frac{3}{5}\right) \)

Using a calculator, we find:

\( \theta_c \approx 36.87^\circ \)

Determining the Percentage of Light Energy Escaping

Now that we have the critical angle, we can analyze how much light escapes. Light that strikes the surface at angles less than the critical angle will partially refract out of the liquid. The percentage of light that escapes can be calculated using the Fresnel equations, which describe how much light is reflected and refracted at an interface.

Using Fresnel's Equations

For normal incidence, the fraction of light transmitted (or refracted) can be approximated by:

\( T = \frac{4n_2}{(n_1 + n_2)^2} \)

Substituting \( n_1 = \frac{5}{3} \) and \( n_2 = 1 \):

\( T = \frac{4 \times 1}{\left(\frac{5}{3} + 1\right)^2} = \frac{4}{\left(\frac{8}{3}\right)^2} = \frac{4}{\frac{64}{9}} = \frac{36}{64} = \frac{9}{16} \)

Calculating the Percentage

To find the percentage of light energy that escapes, we multiply the transmission coefficient by 100:

\( \text{Percentage} = T \times 100 = \frac{9}{16} \times 100 \approx 56.25\% \)

Final Thoughts

Thus, approximately 56.25% of the light energy escapes from the liquid surface into the surroundings. This calculation illustrates how the refractive index of a medium affects light transmission and reflection, which is a fundamental concept in optics.