Question

A convex lens of focal length f produces a real image x times the size of an object; the distance of the object from the lens is:
A) $\left( {x - 1} \right)f$.
B)$\left( {x + 1} \right)f$.
C) $\dfrac{{\left( {x + 1} \right)f}}{x}$.
D) $\dfrac{{\left( {x - 1} \right)f}}{x}$.

Answer 1

The lens is defined as the piece of which is transparent and has curved sides such that it helps in concentrating and dispersing the light rays. The two types of glasses are convex and concave lenses, the convex lens concentrates the light whereas a concave lens is used to disperse the coming light.

Formula used: The formula of the lens is given by,
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}$
Where focal lens of the lens is equal to f, the object distance is equal to u and the image distance is equal to v.
The formula of the magnification of the lens is by,
$\Rightarrow m = \dfrac{{ - v}}{u}$
Where magnification is m the image distance is v and the object distance is u.

Complete step by step solution:
It is given in the problem that the focal length of a convex lens is f and it produces a real image x times the size of the object and we need to find the object distance from the lens.
The formula of the magnification of the lens is by,
$\Rightarrow m = \dfrac{{ - v}}{u}$
Where magnification is m the image distance is v and the object distance is u.
As the magnification is given as x.
$\Rightarrow m = \dfrac{{ - v}}{u}$
$\Rightarrow x = \dfrac{{ - v}}{u}$
$\Rightarrow v = - u \times x$………eq. (1)
The formula of the lens is given by,
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}$
Where focal lens of the lens is equal to f, the object distance is equal to u and the image distance is equal to v.
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}$………eq. (2)
Replacing the value of the v from equation (1) to equation (2) we get,
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}$
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{{\left( { - u \times x} \right)}} - \dfrac{1}{u}$
$\Rightarrow \dfrac{1}{f} = - \dfrac{1}{{u \times x}} - \dfrac{1}{u}$
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{u}\left( { - \dfrac{1}{x} - 1} \right)$
$\Rightarrow u = f\left( { - \dfrac{1}{x} - 1} \right)$
$\Rightarrow u = - f\left( {\dfrac{1}{x} + 1} \right)$
$\Rightarrow u = - f\left( {\dfrac{{1 + x}}{x}} \right)$.
The object distance is equal to$u = f\left( {\dfrac{{1 + x}}{x}} \right)$.

The correct answer for this problem is option C.

Note: It is advisable for students to remember the lens formula as it is very helpful in solving the problem. Also the magnification formula should be remembered as it is very helpful in solving such problems. If the object is formed on the left side of the lens then it is taken as negative and if the image is formed on the right side then it is taken as positive.