Question

A convex mirror has a focal length f. A real object is placed at a distance of f/2 from the pole. Find out the position, magnification and nature of the image.

Answer 1

Hint: To calculate the image distance and magnification, we are using the mirror image and magnification formula. The mirror image formula gives a relation between the magnification, focal length and the distance of the object.
Formula used:

Mirror image: $\dfrac { 1 }{ f } =\dfrac { 1 }{ v } +\dfrac { 1 }{ u } $
Lateral magnification, $ m=\dfrac { -v }{ u }$

Complete step by step answer:
Focal length is the distance between the center of a convex lens and the focal point of the lens.

The given focal length of the mirror is ‘f’.
Given object distance from the pole, $u=\dfrac { -f }{ 2 }$

The distance between the object and the pole of mirror is called object distance (u)
For a convex lens, the virtual distance is negative and the real distance is positive.

By using mirror formula,

$\dfrac { 1 }{ f } =\dfrac { 1 }{ v } +\dfrac { 1 }{ u }$

We shall substitute the value of ‘u’ in the above formula. On solving, we get

$\dfrac { 1 }{ v } +\dfrac { 1 }{ -\left( \dfrac { f }{ 2 } \right) } =\dfrac { 1 }{ f }$
$\dfrac { 1 }{ v } -\dfrac { 2 }{ f } =\dfrac { 1 }{ f }$
$\dfrac { 1 }{ v } =\dfrac { 1 }{ f } +\dfrac { 2 }{ f }$
$\dfrac { 1 }{ v } =\dfrac { 3 }{ f }$
On reciprocating, it becomes $v=\dfrac { f }{ 3 }$

The distance between the image and the pole of the mirror is called image distance (v).
Using magnification formula and applying the values of ‘u’ and ‘v’, we get

$m=\dfrac { -v }{ u }$
$m=\dfrac { -\left( \dfrac { f }{ 3 } \right) }{ -\left( \dfrac { f }{ 2 } \right) }$
$m=\dfrac { f }{ 3 } \times \dfrac { 2 }{ f }$
$m=\dfrac { 2 }{ 3 }$

Therefore, the image formed is virtual, upright and diminished.

Note: It is important to remember the formulas in optics because only with that, the nature and position of the objects can be determined. It is also to be noted that for a convex mirror, ‘u’ is always negative and ‘f’ is always positive.