Question

The refractive index of a lens material is 1.5 and focal length f due to some chemical changes in the material its refractive index has increased by 2% the percentage change in its focal length is
A. +4.5%
B. -4.5%
C. +5.67%
D. -5.67%

Answer 1

Hint: For this question we will use lens’s maker formula. We will form two equations, one for refractive index and other for refractive index. Solving both the equation and substituting the values given in the equation we can find the change in its focal length.

Formula used:
Lens makers formula \[\dfrac{1}{f}=(\mu -1)\left( \dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}} \right)\]

Complete answer:
Let us consider a lens having \[\mu \] refractive index, radius of curvature of both surfaces R1 and R2 respectively. Let ‘f’ be the focal length of the lens. The formula has its name because it is used by lens manufacturers for making lenses of a particular power and focal length.

According to lens maker formula
 \[\dfrac{1}{f}=(\mu -1)\left( \dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}} \right)\]…………(1)

Let us assume the new focal length be f’
\[\dfrac{1}{f'}=(\mu '-1)\left( \dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}} \right)\]…………(2)
Where \[\mu '\] is the new refractive index such that
\[\begin{align}
  & \mu '=\mu +\dfrac{2}{100}\mu \\
 & =1.5+\dfrac{2}{100}\times 1.5=1.53 \\
\end{align}\]

Dividing equation (1) by equation (2)
\[\begin{align}
  & \dfrac{f}{f'}=\dfrac{1.53-1}{1.5-1}=\dfrac{0.53}{0.50} \\
 & \dfrac{f}{f'}=\dfrac{50}{53} \\
 & f'=\dfrac{50}{53}f
\end{align}\]

According to question the refractive index increases by 2%
Therefore percentage change
\[\dfrac{f-\dfrac{50}{53}f}{f}\times 100\cong -5.67%\]

Negative sign represent that focal length decreases by -5.76%

So, the correct answer is “Option D”.

Additional Information:
Lens is an optical device which is used to converge or diverge the light rays that fall into the surface. Generally there are two types of lens converging lens also known as convex lens and diverging lens also known as concave lens.

Note:
Lens maker formula describes a relation between the refraction index of its material to the focal length of lens and radii of curvature of the two surfaces. To make a lens of particular power from a given material, a lens maker's formula is used. This formula is also used to manufacture lenses of particular focal length.