Question

A thin convex lens of focal length ‘f’ made of crown glass is immersed in a liquid of refractive index $\mu_1(\mu_1 > \mu_c) $ where $\mu_c$ is the refractive index of the crown glass. The convex lens now is:
A. A convex lens of longer focal length
B. A convex lens of shorter focal lengths
C. A diverging lens
D. A convex lens of focal length ($\mu_c > \mu_1$)f

Answer 1

Hint: Apply the lens maker's formula and substitute the values of the refractive indices and then finally find the value of the focal length of the lens in the liquid. Check if the focal length value of the lens in the liquid is greater or less than zero then determine the nature of the lens.

Step by step solution:

We know that the lens maker formula is given by
Focal length of the lens which are inserted in liquid is given as $\dfrac{1}{F_L} = (\dfrac{\mu_c}{\mu_l} - 1 )(\dfrac{2}{R} )$
Where the value of the focal length of the lens in the liquid is $F_L$
The value of the refractive index of the liquid is $\mu_l$ and
The value of the refractive index of the material of the lens is given as: $\mu_c$
Now we know that the given that the refractive index of the lens material as $\mu_c$
The refractive index of the liquid is given as: $\mu_1$
Now substituting we get the focal length of the lens in the liquid becomes :
$\dfrac{1}{F_L} = (\dfrac{\mu_c}{\mu_1} - 1 )(\dfrac{2}{R} )$
Now we have got the expression for the focal length of the lens in liquid.
Now let us implement the condition given for the refractive index in the above equation we get as below:
We see that the given condition is $\mu_1 > \mu_c$ so we get that
The RHS of the equation becomes that fraction is less than one and hence we get that the RHS is less than zero and hence we see that the RHS of the equation is negative
Hence the value of the LHS of the equation is also negative and hence the value of the $F_L$ is also negative.
So we got the negative value of the focal length.
So we can infer from the above result that the focal length we got is of the diverging lens and hence we got the negative focal length.
So we conclude that the convex lens becomes a divergent lens from the above discussion.

Note: The point where one tends to make mistake is that deducing the correct conclusion from the obtained result that the focal length of the lens is negative in the immersed liquid we get that the value of the focal length as negative for the lens whose nature is divergent because the divergent rays always converge behind the pole of the lens.