34.14 For a convex spherical mirror that has focal length $\mathbf{f}\mathbf{=}\mathbf{-}\mathbf{12}\mathbf{cm}$, what is the distance of an object from the mirror’s vertex if the height of the image is half the height of the object?

Object image relation for spherical mirror is given by:

$\frac{\mathbf{1}}{\mathbf{u}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{v}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{R}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{f}}$

Where, u= object distance from the vertex of the mirror,

u is (+) when the object is in front of the reflecting surface of mirror,

v is image distance from the vertex of the mirror,

v is (+) when the image is in front of the reflecting surface of mirror,

R is the radius of curvature of the mirror, f= focal length,

R is (+) in concave mirror and (-) in convex mirror,

u and v are negative if the case is opposite to what is mentioned above

Remember that the focal length of a mirror depends only on the curvature of the mirror and not on the material from which the mirror is made because the formation of the image results from rays reflected from the surface of the material.

Apply: The above rule can be applied on either concave and convex mirrors or even plane mirror when we consider the radius to be infinity. The equation gives one the exact location of an image by knowing the radius or the focal length of a mirror and the location of the object. In addition, the sign for the distance of the image can be negative if the image in the opposite direction of the outgoing rays which means the image is not real image but is virtual one and is located inside the mirror.

Lateral magnification for spherical message is given by:

$\mathbf{m}\mathbf{=}\frac{\mathbf{-}\mathbf{v}}{\mathbf{u}}\mathbf{=}\frac{{\mathbf{h}}_{\mathbf{i}}}{{\mathbf{h}}_{\mathbf{o}}}$

where, m is the magnification,

u is object distance from the vertex of the mirror,

u is (+) when the object is in front of the reflecting surface of mirror,

v is image distance from the vertex of the mirror,

v is (+) when the image is in front of the reflecting surface of mirror,

is the height of the image, is the height of the object,

m is (+) when the image is erect and (-) when the image is inverted

For a convex mirror, the image is always virtual and v is negative, so

$m=\frac{-v}{u}$is positive.

So, this$m=\frac{h_i}{h_o}$ also positive, thus both$h_i$ and$h_0$ should be positive or both should be negative.

The height of image is half the height of object, so

role="math" localid="1663906906836" $h_i=\frac{1}{2}{h}_0$

And,

$$\begin{gathered} m=\frac{-v}{u}=\frac{h_i}{h_o}=\frac{{\displaystyle \frac{1}{2}}{h}_o}{h_0}=\frac{1}{2} \\ u=-2v \end{gathered}$$

$$\begin{gathered} \frac{1}{u}+\frac{1}{v}=\frac{2}{R}=\frac{1}{f}, \\ \frac{1}{u}+\frac{1}{v}=\frac{1}{-2v}+\frac{1}{v}=\frac{1}{-2v}+\frac{\left(-2\right)}{\left(-2v\right)}=\frac{-1}{-2v}=\frac{1}{f}, \\ v=\frac{f}{2}=\frac{-12cm}{2}=-6cm \\ u=-2v=-2\times \left(-6cm\right)=12cm \end{gathered}$$

Therefore, the object distance should be 12 cm from the vertex of the mirror.