Answer

Verified

81k+ views

**Hint**The minimum diameter of the disc placed on the surface of the liquid is determined by using the two formulas. The first formula used is the critical angle formula and the other formula is radius of the disc formula, then the diameter is determined.

Useful formula:

The critical angle is given by,

$\sin {\theta _c} = \dfrac{1}{\mu }$

Where, ${\theta _c}$ is the critical angle and $\mu $ is the refractive index of the medium.

The radius of the disc is given by,

$r = h \times \tan {\theta _c}$

Where, $r$ is the radius of the disc, $h$ is the height of the source from the surface of the liquid and ${\theta _c}$ is the critical angle.

**Complete step by step answer**

Given that,

The height of the source from the surface of the liquid, $h = 4\,m$,

The refractive index of the liquid is, $\mu = \dfrac{5}{3}$,

Now,

The critical angle is given by,

$\sin {\theta _c} = \dfrac{1}{\mu }\,...................\left( 1 \right)$

By substituting the refractive index of the liquid in the above equation (1), then the above equation (1) is written as,

$\sin {\theta _c} = \dfrac{1}{{\left( {\dfrac{5}{3}} \right)}}$

By rearranging the terms in the above equation, then the above equation is written as,

$\sin {\theta _c} = \dfrac{3}{5}$

By dividing the terms in the above equation, then the above equation is written as,

$\sin {\theta _c} = 0.6$

By rearranging the terms in the above equation, then the above equation is written as,

${\theta _c} = {\sin ^{ - 1}}0.6$

From the trigonometry, the values of the ${\sin ^{ - 1}}0.6 = 36.86$, then the above equation is written as,

${\theta _c} = 36.86$

Now,

The radius of the disc is given by,

$r = h \times \tan {\theta _c}\,...................\left( 2 \right)$

By substituting the height and the critical angle in the above equation (2), then the equation (2) is written as,

$r = 4 \times \tan 36.86$

From the trigonometry, the values of the $\tan 36.86 = 0.75$, then the above equation is written as,

$r = 4 \times 0.75$

By multiplying the terms in the above equation, then the above equation is written as,

$r = 3\,m$

The relation between the radius and the diameter is,

$d = 2r$

Where, $d$ is the diameter and $r$ is the radius.

By substituting the radius in the above equation, then the above equation is written as,

$d = 2 \times 3$

By multiplying the terms in the above equation, then the above equation is written as,

$d = 6\,m$

**Hence, the option (C) is the correct answer.**

**Note**The critical angle is inversely proportional to the refractive index of the medium. As the refractive index of the medium increases, then the critical angle decreases. The radius of the disc is directly proportional to the height and the critical angle.

Recently Updated Pages

Name the scale on which the destructive energy of an class 11 physics JEE_Main

Write an article on the need and importance of sports class 10 english JEE_Main

Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main

Choose the one which best expresses the meaning of class 9 english JEE_Main

What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main

A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main

Other Pages

Electric field due to uniformly charged sphere class 12 physics JEE_Main

A wave is travelling along a string At an instant the class 11 physics JEE_Main

The value of intlimits02pi max left sin xcos x right class 12 maths JEE_Main

Which of the following is not a redox reaction A CaCO3 class 11 chemistry JEE_Main

Man A sitting in a car moving with a speed of 54 kmhr class 11 physics JEE_Main

Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main