Question

The human eye can be regarded as a single spherical refractive surface of curvature or cornea \[7.8mm\]. If a parallel beam of light comes to focus at \[3.075cm\] behind the refractive surface, the refractive index of the eye.
(A) \[1.34\]
(B) \[1.72\]
(C) \[1.5\]
(D) \[1.61\]

Answer 1

Hint: We have been told that the parallel beam of light comes to focus at some distance from the lens. If the light rays are parallel, it means the object is at infinity and the distance that has been provided to us is in reality the focal length of the lens. We can use the formula for refraction at a single spherical surface to find out the refractive index.

Formula Used:
\[\left( \dfrac{{{\mu }_{2}}}{v}-\dfrac{{{\mu }_{1}}}{u} \right)=\left( \dfrac{{{\mu }_{2}}-{{\mu }_{1}}}{R} \right)\]

Complete step by step answer:
We are aware of the focal length of the lens, that is \[f=3.075cm\]
Let the refractive index of the human eye be \[\eta \]. The refractive index of air is ideally taken to be \[1\]. Since there are two refractive surfaces of the lens, we will have two curvatures \[{{R}_{1}}\] and \[{{R}_{2}}\] such that \[{{R}_{1}}={{R}_{2}}=R\] where the value of \[R\] is \[7.8mm\].
Hence we can say that radius of curvature \[(R)=0.78cm(\because 1mm={{10}^{-1}}cm)\]
Using the argument in the hint, we can say that the object is at infinity, that is, \[u=\infty \]
The image distance has been provided to us, that is, \[v=3.075cm\]
Now using the formula for refraction at a curved surface, we have
\[\left( \dfrac{\eta }{v}-\dfrac{1}{u} \right)=\left( \dfrac{\eta -1}{R} \right)\] where the symbols have their meaning as discussed above
Substituting the values, we will get
\[\begin{align}
  & \left( \dfrac{\eta }{3.075}-\dfrac{1}{\infty } \right)=\left( \dfrac{\eta -1}{0.78} \right) \\
 & \Rightarrow \dfrac{\eta }{3.075}=\dfrac{\eta -1}{0.78}\left( \because \dfrac{1}{\infty }=0 \right) \\
 & \Rightarrow 1-\dfrac{1}{\eta }=\dfrac{0.78}{3.075} \\
 & \Rightarrow \dfrac{1}{\eta }=1-0.254 \\
 & \Rightarrow \dfrac{1}{\eta }=0.746 \\
 & \Rightarrow \eta =\dfrac{1}{0.746}=1.3404 \\
 & \Rightarrow \eta = 1.34 \\
\end{align}\]
Hence the correct option is (A)

Note: A common misconception in this question among students is that we can apply lens maker formula as all the relevant information needed in the lens maker formula is provided in the question. But we cannot use the lens maker formula because it is applicable when refraction takes place at two surfaces. Although our eye lens has two refracting surfaces, we have been told about the refraction at one surface only. This minute detail makes all the difference.