Question

In an experiment a convex lens of focal length $15\, cm$ is placed coaxially on an optical bench in front of a convex mirror at a distance of $5 \,cm$ from it. It is found that an object and its image coincide, if the object is placed at a distance of $20 \,cm $ from the lens. The focal length of the convex mirror is :

• A. $27.5 \,cm$
• B. $20.0 \,cm$
• C. $25.0\,cm$
• D. $30.5\, cm$

Answer 1

The Correct Option is A

The correct answer is (A) : 27.5 cm
Given: u= -20cm
            f= 15 cm
            distance (d) of convex lens= 5cm

By substituting the values in equation (1) we get,
\(\frac{1}{v}+\frac{1}{20}=\frac{1}{15}\)
\(⇒\frac{1}{v}=\frac{1}{15}-\frac{1}{20}\)
\(⇒\frac{1}{v}=\frac{5}{300}\)
\(⇒v=\frac{300}{5}\)
\(⇒v=60\ cm\)
Thus, the image will be formed at 60cm to the right of the lens and it will be inverted.
Therefore, distance between lens and mirror will be
d = image distance – radius of curvature of convex mirror
\(d = v -2f\)
By substituting the values in above equation we get,
\(5 = 60 - R\)
\(⇒ R = 55\)
But, the radius of curvature (R) is given by,
\(R=2f\)
\(⇒f=\frac{55}{2}\)
\(⇒f=27.5\)
Hence , the focal length of the convex mirror is 27.5cm.