Question

The least distance of vision of a long sighted person is \[60\]\[cm\]. By using a spectacle lens, this distance is reduced to \[12\]\[cm\]. The power of the lens is:
A) \[ + 5.0D\]
B) \[ + \left( {\dfrac{{20}}{3}} \right)D\]
C) \[ - \left( {\dfrac{{10}}{3}} \right)D\]
D) \[ + 2.0D\]

Answer 1

Hint: In ray optics, the power of an optical system is defined as its ability to bend light. Power is also known as dioptric power, refractive power, focusing power, or convergence power. A lens is said to be powerful, if it has a shorter focal length it. Lenses of shorter focal length will more converge or divert light rays. Converging lenses have positive focal length, so they have positive values of power. Diverging (concave) lenses have negative values of power because they have negative focal length. Power of a lens is defined as the reciprocal of focal length (in meters) of the lens. In the given question first we find the value of focal length f, by lens formula.

Formula used:
First of all we calculate focal length of of lens by lens formula \[\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}\]
After then, we use power formula, \[P = \dfrac{1}{f}\]

Complete step by step solution:
Lens formula is given by \[\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}\]
Given: \[v = - 60\]\[cm\], \[u = - 12\]\[cm\]
Hence, we get, \[\dfrac{1}{f} = \dfrac{1}{{( - 60)}} - \dfrac{1}{{( - 12)}}\]
$\Rightarrow$ \[\dfrac{1}{f} = - \dfrac{1}{{60}} + \dfrac{1}{{12}}\]
$\Rightarrow$ \[\dfrac{1}{f} = \dfrac{1}{{12}} - \dfrac{1}{{60}}\]
$\Rightarrow$ \[\dfrac{1}{f} = \dfrac{{5 - 1}}{{60}}\]
$\Rightarrow$ \[\dfrac{1}{f} = \dfrac{4}{{60}}\]
$\Rightarrow$ \[\dfrac{1}{f} = \dfrac{1}{{15}}\]
\[\therefore f = 15\]\[cm\]
$\Rightarrow$ \[f = \dfrac{{15}}{{100}}\]\[meter\]
$\Rightarrow$ \[f = \dfrac{3}{{20}}\]\[meter\]
Power of a lens is given by \[P = \dfrac{1}{f}\].
$\Rightarrow$ \[P = \dfrac{{20}}{3}D\]

Hence, power of a lens is \[P = + \dfrac{{20}}{3}D\].

Note: When calculating power in the diopter, the unit of focal length (f) should be in meters. If a unit of f is given in cm or other, first convert it to meter. Students also make mistakes by taking signs of u( distance of object placed) and v(distance of image formed). An important thing understood for students is the power of a lens may be positive or negative. The sign of power depends upon the nature of the lens. Its value is positive for Convex lens and negative for concave lens .Power of a plane glass plate is 0 , because its focal length is infinite.