This problem requires us to determine the radius of curvature of a double-convex lens, given its refractive index and desired focal length. We will utilize the fundamental Lens Maker's Formula to accurately solve this optics problem.

Double-Convex Lens Characteristics

A double-convex lens is a type of converging lens characterized by two spherical surfaces that bulge outwards. For such a lens, light rays converge after passing through it. When applying the Lens Maker's Formula, it's crucial to correctly assign the signs to the radii of curvature of its surfaces.

• The first surface, which the incident light strikes, is convex. According to the Cartesian sign convention, its radius of curvature (\(R_1\)) is considered positive.
• The second surface is also convex. However, its center of curvature lies on the side from which light typically exits the lens. For a double-convex lens where both faces have the same magnitude of curvature, if the first radius is \(+R\), the second radius (\(R_2\)) is taken as \(-R\). This ensures consistency with the formula's derivation for converging lenses.

Lens Maker's Formula Derivation

The Lens Maker's Formula is a powerful tool in optics that connects the focal length of a thin lens to the refractive index of its material and the curvatures of its surfaces. The formula is expressed as:

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

Where:

• \(f\) represents the focal length of the lens.
• \(n\) is the refractive index of the lens material (glass in this case) with respect to the surrounding medium (assumed to be air).
• \(R_1\) is the radius of curvature of the first lens surface.
• \(R_2\) is the radius of curvature of the second lens surface.

For a double-convex lens with both faces having the same radius of curvature, let this common magnitude be \(R\). Based on our sign conventions discussed earlier:

• \(R_1 = +R\)
• \(R_2 = -R\)

Substituting these into the Lens Maker's Formula, we get:

\[ \frac{1}{f} = (n-1) \left(\frac{1}{R} - \frac{1}{(-R)}\right) \]

\[ \frac{1}{f} = (n-1) \left(\frac{1}{R} + \frac{1}{R}\right) \]

\[ \frac{1}{f} = (n-1) \left(\frac{2}{R}\right) \]

This simplified form is very useful for double-convex lenses with equal curvature radii.

Radius of Curvature Calculation Steps

Now, let's plug in the given values into the simplified formula to find the required radius of curvature. The problem provides the following information:

• Refractive index of the glass, \(n = 1.55\)
• Desired focal length, \(f = 30 \text{ cm}\)

Using the derived formula \(\frac{1}{f} = (n-1) \left(\frac{2}{R}\right)\), we can rearrange it to solve for \(R\):

\[ R = 2(n-1)f \]

Substitute the numerical values:

\[ R = 2(1.55 - 1) \times 30 \]

\[ R = 2(0.55) \times 30 \]

\[ R = 1.1 \times 30 \]

\[ R = 33 \text{ cm} \]

Therefore, to achieve a focal length of 30 cm, each face of the double-convex lens must have a radius of curvature of 33 cm.