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It is useful to design lenses of derived focal length using surface of suitable radii of curvature. Lens maker formula helps in finding various unknowns and is widely applied.
The radii of curvature of faces of a double concave lens are 20 cm and 30 cm. If refractive index of glass is \( \displaystyle\frac{3}{2} \), then the focal length of the lens will be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( -24\,\mathrm{cm} \)
Use the lens maker's formula with the Cartesian sign convention (light travelling from left to right). In a double concave lens the first surface has its centre of curvature on the left (negative) and the second on the right (positive).
\( R_1 = -20\,\mathrm{cm} \)
\( R_2 = +30\,\mathrm{cm} \)
\( \mu = 1.5 \)
\( \displaystyle \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = (1.5 - 1)\left(-\frac{1}{20} - \frac{1}{30}\right) \)
\( \displaystyle \frac{1}{f} = 0.5 \times \left(-\frac{3 + 2}{60}\right) = -\frac{1}{24}\,\mathrm{cm^{-1}} \)
\( \displaystyle f = -24\,\mathrm{cm} \)
The negative sign shows that the lens is diverging, as a concave lens must be.