Question

The radius of curvature of the concave mirror is $24cm$ and the real image is magnified by $1.5$ times. The object distance is
A. 20 cm
B. 8 cm
C. 16 cm
D. 24 cm

Answer 1

Hint: First, find the focal length of the concave mirror. Then use the formula for the magnification due to a spherical mirror and the relation between the positions of the image and the object to find an equation for the object distance in terms of magnification and focal length.

Formula used:
$f=\dfrac{R}{2}$
$\Rightarrow m=-\dfrac{v}{u}$
$\Rightarrow\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}$
$v$ and $u$ are the positions of the image and object with respect to the pole of the mirror. $R$ is the radius of curvature of the mirror.

Complete step by step answer:
Let us first find the focal length of the concave mirror. The focal length of a spherical concave mirror is given as $f=\dfrac{R}{2}$, where R is the radius of curvature of the mirror.
It is given that $R=24cm$. This means that $f=\dfrac{24}{2}=12cm$.
But according to sign convection, the position of the focus for a concave mirror is negative.
Hence, $f=-12cm$.
The magnification due to a spherical mirror is given as $m=-\dfrac{v}{u}$ …. (i)
v and u are the positions of the image and object with respect to the pole of the mirror and according to the sign convection.
 Also, the relation between u, v and f is given as $\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}$…. (ii)
From (i) we get that $v=-mu$.
Substitute the value of v in (ii).
$\dfrac{1}{f}=\dfrac{1}{-mu}+\dfrac{1}{u}$
$\Rightarrow u=f\left( 1-\dfrac{1}{m} \right)$ …. (iii).
It is given that the image is magnified by 1.5 times. But in case of real image m is negative. This means that $m=-1.5$.
Substitute the values of f and m in (iii).
$u=-12\left( 1-\dfrac{1}{-1.5} \right)$
$\Rightarrow u=-12\left( \dfrac{1.5+1}{1.5} \right)\\
\Rightarrow u =-12\left( \dfrac{2.5}{1.5} \right)\\
\therefore u =-20cm$
Therefore, the position of the object is $-20cm$. Then this means that the object distance is equal to 20cm.

Hence, the correct option is A.

Note: For virtual images, the value of m is positive. Note that the formula for the focal length of a spherical mirror that we used is valid only when the radius of curvature of the mirror is much larger than the aperture of the mirror. A mirror is defined as reflecting surface and can be explained by the law of reflection, which states that when a ray of light is made to fall on the reflecting surface, the reflected ray has its angle of reflection, incident ray, and the reflected ray are normal to the surface at a point of incidence.