Physics - Geometrical Optics Question with Solution | TestHub

The mirror is convex, so the radius of curvature $R$ is positive, $R=20$ cm. The focal length $f$ is half the radius of curvature: $f=\frac{R}{2}=10$ cm.

The magnification $m$ is given as $m=\frac{1}{2}$. The magnification formula is:

$$m=-\frac{v}{u}$$

where $v$ is the image distance and $u$ is the object distance. Since the image is upright for a convex mirror, the magnification is positive. Thus,

$$\frac{1}{2}=-\frac{v}{u}⟹v=-\frac{u}{2}$$

The mirror formula is:

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

Substituting $f=10$ cm and $v=-\frac{u}{2}$ into the mirror formula:

$$\frac{1}{10}=\frac{1}{u}+\frac{1}{-\frac{u}{2}}=\frac{1}{u}-\frac{2}{u}=-\frac{1}{u}$$

$$u=-10cm$$

Since object distance is always negative, the magnitude of the object distance is 10 cm.

Therefore, the object distance must be 10 cm.