An object is placed in front of a convex lens of focal length 12 cm. If the size of the real image formed is half the size of the object, then the distance of the object from the lens is:
Show Hint
Use the magnification formula along with the lens equation to find the object distance when the image size is related to the object size.
The magnification \( M \) for a convex lens is calculated using:
\[\nM = \frac{\text{image height}}{\text{object height}} = \frac{v}{u}\n\]
Where:
- \( M = \frac{1}{2} \) (image size is half the object size)
- \( u \) is object distance, and \( v \) is image distance.
The lens equation is:
\[\n\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\n\]
With \( f = 12 \, \text{cm} \) (focal length).
From the magnification equation:
\[\n\frac{1}{2} = \frac{v}{u}\n\]
Therefore:
\[\nv = \frac{u}{2}\n\]
Substituting into the lens equation:
\[\n\frac{1}{12} = \frac{2}{u} - \frac{1}{u}\n\]
Solving for \( u \):
\[\n\frac{1}{12} = \frac{1}{u}\n\]
\[\nu = 36 \, \text{cm}\n\]
The object distance is 36 cm.