Question

find the focal length of a concave mirror which gives a real image of same size when the object is placed at 50 cm from the mirror ​

Answer 1

Given :

In concave mirror,

Image height = Object height.

Object distance = -50 cm.

To find :

The focal length of the concave mirror,

Solution :

In concave mirror focal length and image distance is negative.

» The Magnification produced by a mirror is equal to the ratio of the image distance to the object distance with a minus sign and is also equal to the ratio of height of the image to the height of the object .i.e.,

$\boxed{ \bf m = - \dfrac{v}{u} = \dfrac{h'}{h}}$

where,

• v denotes image distance
• u denotes object distance
• h' denotes image height
• h denotes object height

Now, substituting all the given values in the formula,

$\leadsto{\sf -\dfrac{v}{u} = \dfrac{h'}{h}}$

$\leadsto{\sf \dfrac{ - ( - v)}{( - 50)} = 1}$

$\leadsto{\sf v = - 50cm}$

now, using mirror formula that is,

» A formula which gives the relationship between image distance, object distance and focal length of a sperical mirror is known as the mirror formula .i.e.,

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

where,

• v denotes Image distance
• u denotes object distance
• f denotes focal length

now, substituting all the given values,

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

$\leadsto{\sf \dfrac{1}{f} = \dfrac{1}{( - 50)} + \dfrac{1}{( - 50)}}$

$\leadsto{\sf \dfrac{1}{f} = \dfrac{ - 1 - 1}{50} }$

$\leadsto{\sf \dfrac{1}{f} = \dfrac{ - 2}{50} }$

$\leadsto{\sf \dfrac{1}{f} = \dfrac{ - 1}{25} }$

$\leadsto \underline{ \boxed{\sf f = - 25 \: cm}}$

thus, the focal length of the concave mirror is -25cm.