Question

Apparent depth for point object $x$ in all three cases are $H_1,H_2\&H_3$ respectively when seen from below given $H=30cm$, $n=1.5$ & $R=3m$, then

• A. $0.8<H_2–H_1<0.9$
• B. $H_2>H_1$
• C. $H_2>H_3$
• D. $H_3>H_1$

Answer 1

Answer: (B), (C)

$H_2>H_3$

The explanation for the correct option(s):

Option B : $H_2>H_1$

Option C: $H_2>H_3$

Case 1: No meniscus is formed

$H=30cm$

$n=1.5$

$\begin{array}{rcl}R& =& 3m\\ & =& 300cm\end{array}$

Apparent depth is given by,

$H_1=\frac{H}{n}$

$\Rightarrow$$H_1=\frac{30}{1.5}$

$\Rightarrow$$H_1=20cm$

Case 2: upper meniscus is formed

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$

Where, $n_2=$Refractive index of air

$n_1=$Refractive index of the material

$v=$Apparent depth

$u=$Real depth

$R=$Radius of aperture

$\Rightarrow$$\frac{1}{-H_2}-\frac{3}{-2\times 30}=\frac{1-{\displaystyle \frac{3}{2}}}{-300}$

$\Rightarrow$ $H_2=20.684cm$

Case 3: lower meniscus is formed

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$

Where, $n_2=$Refractive index of air

$n_1=$Refractive index of the material

$v=$Apparent depth

$u=$Real depth

$R=$Radius of aperture

$\Rightarrow$$\frac{1}{-H_3}-\frac{3}{-2\times 30}=\frac{1-{\displaystyle \frac{3}{2}}}{300}$

$\Rightarrow$ $H_3=19.354cm$

From above values,

$H_2>H_1>H_3$

The explanation for the incorrect option(s):

A: $0.8<H_2–H_1<0.9$

D: $H_3>H_1$

$\because$$H_2–H_1$is $0.648$, option A is incorrect.

$\because$$H_1$is greater than $H_3$ ,option D is incorrect.

Hence, option B & C is correct.