Question

A convex lens of refractive index 1.5 has radii of curvature +20 cm and $-$20 cm. The focal length of the lens is:

• A. 10 cm
• B. 20 cm
• C. 40 cm
• D. 80 cm

Hint:

For an equiconvex lens (where $R_1 = +R$ and $R_2 = -R$) made of glass with refractive index $\mu = 1.5$:
Substituting into the formula gives:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{R} + \frac{1}{R} \right) = 0.5 \left( \frac{2}{R} \right) = \frac{1}{R} \implies f = R \]
So, the focal length is exactly equal to the radius of curvature!
Recognizing this pattern helps solve such problems instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from "Ray Optics and Optical Instruments."
We are asked to calculate the focal length ($f$) of a double convex lens of a known refractive index ($\mu$) and radii of curvature of its two surfaces.

Step 2: Key Formula or Approach:
We use the Lens Maker's Formula, which relates the focal length of a lens to its refractive index and the radii of curvature:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where:
$f$ = focal length of the lens
$\mu$ = refractive index of the lens medium with respect to the surrounding medium (air, $\mu = 1$)
$R_1$ = radius of curvature of the first surface
$R_2$ = radius of curvature of the second surface

Step 3: Detailed Explanation:

• We are given the following values:
Refractive index ($\mu$) = $1.5$
Radius of curvature of first surface ($R_1$) = $+20\text{ cm}$
Radius of curvature of second surface ($R_2$) = $-20\text{ cm}$

• Substitute these values into the Lens Maker's Formula:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{+20} - \frac{1}{-20} \right) \]

• Calculate the refractive index term:
\[ 1.5 - 1 = 0.5 = \frac{1}{2} \]

• Calculate the term in the brackets:
\[ \frac{1}{20} - \left(-\frac{1}{20}\right) = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \]

• Multiply the two terms to find the reciprocal of the focal length:
\[ \frac{1}{f} = \frac{1}{2} \times \frac{1}{10} = \frac{1}{20} \]

• Taking the reciprocal gives the focal length:
\[ f = 20\text{ cm} \]

• The positive sign of the focal length indicates that the lens is indeed converging in nature, which matches the properties of a standard convex lens.



Step 4: Final Answer:
The focal length of the convex lens is $20\text{ cm}$, which is option (B).