Question

A telescope has an objective lens of a focal length of 200 cm and an eyepiece with a focal length of 2 cm. If this telescope is used to see a 50 meter tall building at a distance of 2 km, what is the height of the image of the building formed by the objective lens?
A. 5 cm
B. 10 cm
C. 1 cm
D. 2 cm

Answer 1

Hint: The magnification of the lens is image distance by the object distance. And, the magnification of the lens is image height by the object height. We will equate these equations as the LHS is the same, and then, will substitute the values to find the value of the height of the image of the building.
Formula used:
\[m=\dfrac{I}{O}\]
\[m=\dfrac{v}{u}\]

Complete answer:
From the given information we have the data as follows.
A telescope has an objective lens of a focal length of 200 cm and an eyepiece with a focal length of 2 cm. This telescope is used to see a 50 meter tall building at a distance of 2 km.
\[\begin{align}
  & u=2\,km=2\times {{10}^{3}}m \\
 & v=200\,cm=2\,m \\
 & O=50\,m \\
\end{align}\]
The ray diagram representing the given situation is as follows.

Consider the formula of the magnification of the lens.
\[m=\dfrac{I}{O}\]…… (1)
Where I is the image distance and O is the object distance.
\[m=\dfrac{v}{u}\]…… (2)
Where v is the image height and u is the object height.
Equate the equations (1) and (2), as, the LHS of both the equations are the same, that is, the magnification.
\[\dfrac{I}{O}=\dfrac{v}{u}\]
Substitute the values in the above equation.
\[\dfrac{I}{50}=\dfrac{2}{2\times {{10}^{3}}}\]
Continue further computation.
\[\begin{align}
  & I=50\times \dfrac{2}{2\times {{10}^{3}}} \\
 & \Rightarrow I=0.05\,m \\
 & \therefore I=5\,cm \\
\end{align}\]
\[\therefore \]The height of the image of the building formed by the objective lens is 5 cm.

Thus, option (A) is correct.

Note:
The magnification of the lens: if the magnification of the lens is greater than one, then, the image is magnified, if the magnification of the lens is equal to one, then, the image is of the same size as that of the object and if the magnification of the lens is less than one, then, the image is diminished.