An air bubble in a glass sphere of diameter 4 cm appears to be 1 cm from the surface nearest to the eye when looked at along a diameter. If the refractive index of glass is \( \mu_g=1.5 \), the distance of the bubble from that refracting surface is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1.2 cm
We see the bubble by refraction at the spherical surface nearest to the eye. Light goes from glass into air at this surface. In the Cartesian sign convention, distances are measured from the pole of this surface. They are positive in the direction of the light, that is, towards the eye.
\( n_1=1.5 \qquad\text{(glass)} \)
\( n_2=1 \qquad\text{(air)} \)
\( v=-1\,\mathrm{cm} \qquad\text{(virtual image, on the glass side)} \)
\( R=-2\,\mathrm{cm} \qquad\text{(centre of the sphere on the glass side)} \)
The formula for refraction at a spherical surface gives:
\( \displaystyle \frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R} \)
\( \displaystyle \frac{1}{-1\,\mathrm{cm}}-\frac{1.5}{u}=\frac{1-1.5}{-2\,\mathrm{cm}} \)
\( \displaystyle -\frac{1.5}{u}=1.25\,\mathrm{cm^{-1}} \)
\( \Rightarrow\; u=-1.2\,\mathrm{cm} \)
Therefore, the bubble is 1.2 cm from the refracting surface, inside the glass.