Question

An air bubble in a sphere having 4 cm diameter appears 1 cm from the surface nearest to the eye when looked along diameter. If ${}^a{\mu _g} = 1.5$ , the distance of the bubble from refracting the surface is?
A. 1.2 cm
B. 3.2 cm
C. 2.8 cm
D. 1.6 cm

Answer 1

Hint: For solving this question use the formula, $\dfrac{{{\mu _2}}}{v} - \dfrac{{{\mu _1}}}{u} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$ , here u is the object distance from a pole of spherical surface and v is the image distance from a pole of spherical surface and ${\mu _2},{\mu _1}$ are the refractive index of air and glass. Put the value of v, R and other parameters to find the value of the distance of the bubble from refracting the surface, i.e., u.
Formula used - $\dfrac{{{\mu _2}}}{v} - \dfrac{{{\mu _1}}}{u} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$ , ${}^a{\mu _g} = \dfrac{{{\mu _g}}}{{{\mu _a}}}$

Complete Step-by-Step solution:
We have been given that the diameter of the sphere is 4 cm.
So, the radius will $R = \dfrac{4}{2} = 2$ cm
Also, the air bubble appears 1 cm from the surface nearest to the eye when looked along the diameter.
We have ${}^a{\mu _g} = 1.5$
So, as we know the refractive index of air is 1.
So, we have refractive index of glass with respect to air, i.e., ${}^a{\mu _g} = 1.5$
We know by formula, ${}^a{\mu _g} = \dfrac{{{\mu _g}}}{{{\mu _a}}}$
Here, ${\mu _g},{\mu _a}$ are the refractive index of air and glass.
We know, ${\mu _a} = 1$ and ${}^a{\mu _g} = 1.5$
So, refractive index of glass will be-
$
  {}^a{\mu _g} = \dfrac{{{\mu _g}}}{{{\mu _a}}} \\
   \Rightarrow {}^a{\mu _g} \times {\mu _a} = {\mu _g} \\
$
Putting the values, we get-
$
  1.5 \times 1 = {\mu _g} \\
   \Rightarrow {\mu _g} = 1.5 \\
$
Now, to distance of the bubble from refracting surface, we use the formula-
$\dfrac{{{\mu _2}}}{v} - \dfrac{{{\mu _1}}}{u} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$
Here, ${\mu _2},{\mu _1}$ are the refractive index of air and glass.
And v is the distance at which bubble appears when looked along the diameter which is given to be 1 cm.
Also, radius R = 2 cm and u is the distance of the bubble from the refracting surface.
Putting these values in the above formula, we get-
$
  \dfrac{{{\mu _2}}}{v} - \dfrac{{{\mu _1}}}{u} = \dfrac{{{\mu _2} - {\mu _1}}}{R} \\
   \Rightarrow \dfrac{1}{{ - 1}} - \dfrac{{1.5}}{u} = \dfrac{{1 - 1.5}}{{ - 2}} \\
   \Rightarrow - 1 - \dfrac{{1.5}}{u} = \dfrac{{0.5}}{2} \\
   \Rightarrow - \dfrac{{1.5}}{u} = \dfrac{{0.5}}{2} + 1 \\
   \Rightarrow - \dfrac{{1.5}}{u} = \dfrac{{2.5}}{2} \\
   \Rightarrow u = - \dfrac{{1.5 \times 2}}{{2.5}} = - 1.2cm \\
$
Therefore, the distance of the bubble from the refracting surface is 1.2 cm.
Hence, the correct answer is option A.

Note – Whenever such types of questions appear, then first write down the things given in the question and then as mentioned in the solution use the formula to find the distance of the bubble from the refracting surface. Also, don’t make mistakes while keeping the values along with their signs in the formula, the sign of v and R will be negative according to sign convention, and we got the value of u also negative.