Understanding the Lens Maker's Formula

The Lens Maker's Formula is a crucial equation in optics that relates the focal length of a lens to its refractive index and the radii of curvature of its two surfaces. It is given by:

\[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

Where:

• \(f\) is the focal length of the lens.
• \(\mu\) (mu) is the refractive index of the material from which the lens is made.
• \(R_1\) is the radius of curvature of the first surface (the one on which light first falls).
• \(R_2\) is the radius of curvature of the second surface.

It is important to apply the proper sign convention for the radii of curvature. For a double convex lens, the first surface encountered by light is convex, and its center of curvature is on the side of light propagation, so \(R_1\) is positive. The second surface is also convex, but its center of curvature is on the side opposite to light propagation (or same side as incident light if viewed from the other side), so \(R_2\) is considered negative.

Double Convex Lens Parameters

Let's identify the given parameters for the double convex lens from the question:

• The radius of curvature of the first face, \(R_1 = 20\) cm. Since it's a convex surface facing the incident light, \(R_1\) is positive.
• The radius of curvature of the second face, \(R_2 = 30\) cm. For the second convex surface, following the Cartesian sign convention, its center of curvature is to the left if light is incident from the left, making \(R_2\) negative. So, \(R_2 = -30\) cm.
• The focal length of the lens, \(f = 36\) cm. For a convex lens, the focal length is positive.

Refractive Index Calculation Steps

Now, we can substitute these values into the Lens Maker's Formula to find the refractive index (\(\mu\)) of the lens material. Our goal is to isolate \(\mu\).

Starting with the formula:

\[ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

Substitute the given values:

\[ \frac{1}{36} = (\mu - 1)\left(\frac{1}{20} - \frac{1}{-30}\right) \]

Simplify the term inside the parenthesis:

\[ \frac{1}{36} = (\mu - 1)\left(\frac{1}{20} + \frac{1}{30}\right) \]

Find a common denominator for the fractions (20 and 30), which is 60:

\[ \frac{1}{36} = (\mu - 1)\left(\frac{3}{60} + \frac{2}{60}\right) \]

Add the fractions:

\[ \frac{1}{36} = (\mu - 1)\left(\frac{5}{60}\right) \]

Simplify the fraction \(\frac{5}{60}\) to \(\frac{1}{12}\):

\[ \frac{1}{36} = (\mu - 1)\left(\frac{1}{12}\right) \]

Now, to solve for \((\mu - 1)\), multiply both sides of the equation by 12:

\[ \frac{12}{36} = \mu - 1 \]

Simplify the fraction \(\frac{12}{36}\) to \(\frac{1}{3}\):

\[ \frac{1}{3} = \mu - 1 \]

To find \(\mu\), add 1 to both sides of the equation:

\[ \mu = 1 + \frac{1}{3} \]

Convert 1 to a fraction with denominator 3, i.e., \(\frac{3}{3}\):

\[ \mu = \frac{3}{3} + \frac{1}{3} \]

\[ \mu = \frac{4}{3} \]

Convert the fraction to a decimal:

\[ \mu = 1.333... \]

Determining the Lens Material's Refractive Index

The calculated refractive index of the material of the lens is approximately 1.33. This value corresponds to one of the given options.