Question

An object is placed at a distance of \[36{\text{ }}cm\] from a diverging mirror of radius \[54{\text{ }}cm\] . Find the position of the image and focal length?

Answer 1

Hint: In order to answer this question, we will first find out the focal length and then after finding the focal length we will conclude what type of mirror it is. Then we will proceed further to find out the position of the image by implementing the mirror formula.

Formula used:
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$
where, $f$ = focal length, $v$ = image distance and $u$ = Object distance.

Complete step by step answer:
The mirror is a gleaming surface that reflects incident light to create an image. The wavelength and many other physical properties of reflected light will be nearly identical to those of incident light. An image is created when an object is placed in front of a mirror. The image generated when light rays from an object strike the mirror, are reflected, and converge to form an image called a genuine image. A virtual image is formed when the reflected light beams do not converge and must be extrapolated backwards to generate an image.

Given that: Object distance from the diverging mirror, $u$ = \[36{\text{ }}cm\]
Radius of curvature, $R$ = \[54{\text{ }}cm\]
Focal length $f$ = $\dfrac{R}{2} = \dfrac{{54}}{2} = 27\,cm$
As we know, a diverging mirror is a convex mirror. Now, we will apply the mirror formula to calculate the image distance from the given diverging mirror. By, using mirror formula;
$\because \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$

Now, we will put the given facts:
$\dfrac{1}{{27}} = \dfrac{1}{v} - \dfrac{1}{u}$
so, we have to solve for $v$ , we will keep $v$ term at the one hand side of the equation to find its value:
$\dfrac{1}{v} = \dfrac{1}{{27}} + \dfrac{1}{{36}}$
Now, we will do L.C.M of the denominator to solve the fractional addition at R.H.S-
$\dfrac{1}{v} = \dfrac{{4 + 3}}{{108}} \\
\Rightarrow \dfrac{1}{v} = \dfrac{7}{{108}} $
Now, we will reciprocal the whole above equation to get the value of $v$ :
$\therefore v = \dfrac{{108}}{7} = 15.428\,cm$

Hence, the position of the image is at $15.428\,cm$.

Note:The object is always to the left of the mirror. All distances are calculated from the mirror's pole. Distances measured in the incident ray's direction are positive, whereas distances measured in the direction opposite the incident ray's direction are negative.