Question

A beam of incident parallel light falls on a diverging lens of focal length \(20\) cm in magnitude. If a converging lens of focal length \(15\) cm in magnitude is placed at a distance of \(10\) cm to the right of the diverging lens on the other side, then, the final image formed is:

• A. Virtual and is at a distance of 30 cm to the right of the converging lens.
• B. Virtual and is at a distance of 40 cm to the right of the diverging lens.
• C. Real and is at a distance of 30 cm to the right of the converging lens.
• D. Real and is at a distance of 40 cm to the right of the converging lens.

Hint:

For lens combinations, the image formed by the first lens acts as the object for the second lensUse sign convention carefully for each lens.

Answer 1

The Correct Option is C

Step 1: Identify focal length of diverging lens.
For a diverging lens:
\[ f_1=-20\,\text{cm} \]
Incident rays are parallel, so the diverging lens forms a virtual image at its focus.
\[ v_1=-20\,\text{cm} \]

Step 2: Locate this image with respect to second lens.
The converging lens is placed \(10\) cm to the right of the diverging lens.
The image formed by the first lens is \(20\) cm to the left of the diverging lens.
So distance of this image from second lens:
\[ 20+10=30\,\text{cm} \]
Thus for the second lens:
\[ u=-30\,\text{cm} \]

Step 3: Write focal length of converging lens.
For converging lens:
\[ f_2=+15\,\text{cm} \]

Step 4: Apply lens formula.
\[ \frac{1}{f}=\frac{1}{v}-\frac{1}{u} \]
\[ \frac{1}{15}=\frac{1}{v}-\frac{1}{-30} \]
\[ \frac{1}{15}=\frac{1}{v}+\frac{1}{30} \]

Step 5: Solve for \(v\).
\[ \frac{1}{v}=\frac{1}{15}-\frac{1}{30} \]
\[ \frac{1}{v}=\frac{2-1}{30} \]
\[ \frac{1}{v}=\frac{1}{30} \]
\[ v=30\,\text{cm} \]

Step 6: Interpret the sign of image distance.
Since \(v\) is positive, the final image is formed to the right of the converging lens.
A positive image distance for the second lens indicates a real image.

Step 7: Final conclusion.
\[ \boxed{\text{Real image at }30\,\text{cm to the right of the converging lens}} \]