Question 21: Ray Optics

Question

One side of an equilateral prism is painted by a transparent material of refractive index $n_2$. The refractive index of prism is $1.6$. The minimum value of $n_2$ required for total internal reflection from painted face is ________.

Accepted Answer

We use the Lens Maker's Formula for each planoconvex lens :
$\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}ight)$
For left Plano-convex lens $\left(\mathrm{L}_1ight)$ :
The refractive index of the lens is $\mu_1=1.5$
Radii of curvatures are $\mathrm{R}_1=15 \mathrm{~cm}$ (curved surface), $\mathrm{R}_2=\infty$ (plane surface)
$$ \frac{1}{f_1}=(1.5-1)\left(\frac{1}{15}-\frac{1}{\infty}ight)=0.5 	imes \frac{1}{15}=\frac{1}{30} $$
$$\Rightarrow $$ $\mathrm{f}_1=+30 \mathrm{~cm}$
For right Plano-convex lens $\left(\mathrm{L}_2ight)$ :
The refractive index of lens $\mu_2=1.2$
Radii of curvatures are $\mathrm{R}_1=\infty$, (Plane surface) and $\mathrm{R}_2=-12 \mathrm{~cm}$ ( (curved surface)
$$ \frac{1}{f_2}=(1.2-1)\left(\frac{1}{\infty}-\frac{1}{-12}ight)=0.2 	imes \frac{1}{12}=\frac{1}{60} $$
$$\Rightarrow $$ $\mathrm{f}_1=+60 \mathrm{~cm}$
Since the lenses are thin and in contact, the total power is the sum of individual powers :
$$ \frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2} $$
$$\Rightarrow $$ $\frac{1}{F}=\frac{1}{30}+\frac{1}{60}=\frac{2+1}{60}=\frac{3}{60}=\frac{1}{20}$
$$\Rightarrow $$ $\mathrm{F}=+20 \mathrm{~cm}$
Given object distance $\mathrm{u}=-30 \mathrm{~cm}$
Using the Lens Formula: $\frac{1}{\mathrm{v}}-\frac{1}{\mathrm{u}}=\frac{1}{\mathrm{~F}}$
$$ \frac{1}{v}-\frac{1}{-30}=\frac{1}{20} $$
$\Rightarrow \frac{1}{v}+\frac{1}{30}=\frac{1}{20}$
$$\Rightarrow $$ $\frac{1}{v}=\frac{1}{20}-\frac{1}{30}=\frac{3-2}{60}=\frac{1}{60}$
$$ \mathrm{v}=+60 \mathrm{~cm} $$
Magnification for a lens is given by : $\mathrm{m}=\mathrm{v} / \mathrm{u}$
$$ m=\frac{60}{-30}=-2 $$
The negative sign indicates that the image is real and inverted, and its size is twice the size of the object. Hence, the correct option is (A).

Answer

$\dfrac{3\sqrt{3}}{1.6}$

Answer

$\sqrt{3}$

Answer

$\dfrac{3.2}{\sqrt{3}}$

Answer

$\dfrac{4\sqrt{3}}{5}$

Ray Optics

One side of an equilateral prism is painted by a transparent material of refractive index n_2. The refractive index of prism is 1.6. The minimum value of n_2 required for total internal reflection from painted face is ________.

Choose the correct answer: