Question

Two converging lenses of the same focal length $f$ are separated by a distance $2f$. The axis of the second lens is inclined at angle $\theta ={60}^{\circ }$ with respect to the axis of the first lens. A parallel paraxial beam of light is incident from left side of the lens. Find the $x-$coordinates of the final image with respect to the centre of the first lens taken as origin.

• A. $(0,f)$
• B. $(0,0)$
• C. $(f,0)$
• D. $(0,\frac{f}{2})$

Answer 1

The correct option is B $(0,0)$

Image due to lens forms at $I_1$, and $I_1$ acts as object for second lens.


As second lens is tilted, we can consider object distance for lens as $f\cos {60}^{\circ }=\frac{f}{2}$
Using lens formula for $L_2$
$\frac{1}{v}-\frac{1}{-\frac{f}{2}}=\frac{1}{f}$
$v=-f$
Image will be along $P{P}^{\prime }$ from point $O^{\prime }$
(towards P)
$\therefore O^{\prime }{I}_2^{\prime }=f$
On $x-$ axis this distance will be $f\sec {60}^{\circ }=2f$.
Since, $O{O}^{\prime }=2f$, therefore image will be formed at origin.

Note: A ray passing through $O$ and then $O^{\prime }$ goes undeviated. Therefore, $I_1$ and $I_2$ both should be on this line, which is also the $x-$ axis. That's why for final image we have taken projection of $O^{\prime }{I}_2^{\prime }$ on $x-$ axis