Calculating Lens Power: Double Convex Lens

This problem requires us to calculate the power of a double convex lens given its radii of curvature and the refractive index of the material. The power of a lens, measured in diopters (D), tells us how strongly the lens converges or diverges light. A positive power indicates a converging lens (like a convex lens), and a negative power indicates a diverging lens (like a concave lens).

Understanding the Lensmaker's Formula

The relationship between the focal length (\(f\)) of a thin lens, its refractive index (\(n\)), and the radii of curvature of its two surfaces (\(R_1\) and \(R_2\)) is given by the Lensmaker's formula:

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

The power (\(P\)) of a lens is the reciprocal of its focal length (\(f\)) when the focal length is expressed in meters:

\[ P = \frac{1}{f} \]

where \(P\) is in diopters (D) and \(f\) is in meters.

Sign Convention for Radii of Curvature

A crucial step is applying the correct sign convention for the radii of curvature. A standard convention is the Cartesian sign convention, where light travels from left to right. The radius of curvature is positive if the center of curvature lies on the right side of the lens surface and negative if it lies on the left side.

• For the first surface of the lens (where light enters), if it's convex, the center of curvature is on the right, so \(R_1\) is positive. If it's concave, the center is on the left, so \(R_1\) is negative.
• For the second surface of the lens (where light exits), if it's convex, the center of curvature is on the left, so \(R_2\) is negative. If it's concave, the center is on the right, so \(R_2\) is positive.

In this problem, we have a double convex lens. This means both surfaces are convex. Assuming light enters from the left:

• The first surface is convex, so its radius \(R_1 = +10\) cm.
• The second surface is also convex, so its center of curvature is on the left. Thus, its radius \(R_2 = -20\) cm.

Step-by-Step Calculation of Lens Power

Given values:

• Refractive index of glass, \(n = 1.5\)
• Radius of curvature of the first surface, \(R_1 = +10\) cm
• Radius of curvature of the second surface, \(R_2 = -20\) cm

Using the Lensmaker's formula:

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

Substitute the given values:

\[ \frac{1}{f} = (1.5 - 1)\left(\frac{1}{10 \text{ cm}} - \frac{1}{-20 \text{ cm}}\right) \]

\[ \frac{1}{f} = (0.5)\left(\frac{1}{10} + \frac{1}{20}\right) \text{ cm}^{-1} \]

Find a common denominator for the terms in the parenthesis:

\[ \frac{1}{f} = 0.5\left(\frac{2}{20} + \frac{1}{20}\right) \text{ cm}^{-1} \]

\[ \frac{1}{f} = 0.5\left(\frac{2+1}{20}\right) \text{ cm}^{-1} \]

\[ \frac{1}{f} = 0.5\left(\frac{3}{20}\right) \text{ cm}^{-1} \]

\[ \frac{1}{f} = \frac{1.5}{20} \text{ cm}^{-1} \]

To find the power in diopters, the focal length must be in meters. Since \(1/f\) is in cm\(^{-1}\), we can directly calculate the power in diopters if we work in meters. Let's convert the radii to meters first:

• \(R_1 = 10 \text{ cm} = 0.10 \text{ m}\)
• \(R_2 = -20 \text{ cm} = -0.20 \text{ m}\)

Using the formula for power \(P = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)\) with \(R\) in meters:

\[ P = (1.5 - 1)\left(\frac{1}{0.10 \text{ m}} - \frac{1}{-0.20 \text{ m}}\right) \]

\[ P = (0.5)\left(\frac{1}{0.10} + \frac{1}{0.20}\right) \text{ m}^{-1} \]

\[ P = 0.5\left(10 + 5\right) \text{ m}^{-1} \]

\[ P = 0.5 \times 15 \text{ m}^{-1} \]

\[ P = 7.5 \text{ m}^{-1} \]

Since 1 m\(^{-1}\) = 1 Diopter (D), the power is:

\[ P = +7.5 \text{ D} \]

The positive sign confirms that this is a converging lens, as expected for a double convex lens.

Summary of Calculation Steps

1. Identify the given parameters: refractive index \(n\), radii of curvature \(R_1\) and \(R_2\).
2. Apply the correct sign convention for \(R_1\) and \(R_2\) based on the lens type (double convex) and assumed direction of light.
3. Convert radii to meters if calculating power directly using \(P = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)\).
4. Substitute the values into the Lensmaker's formula (or power formula).
5. Perform the calculation to find the power \(P\) in diopters.

The calculated power of the double convex lens is +7.5 D.

Revision Table: Key Concepts for Lens Power

Additional Information on Optical Lenses and Power

Types of Lenses: Lenses are broadly classified into converging lenses (like convex lenses) and diverging lenses (like concave lenses). A double convex lens is a type of converging lens.

• Convex Lenses: Thicker in the middle than at the edges. They converge parallel rays of light to a real focal point (positive focal length). Power is positive. Examples: double convex, plano-convex, convex-meniscus (if thicker in middle).
• Concave Lenses: Thinner in the middle than at the edges. They diverge parallel rays of light, appearing to come from a virtual focal point (negative focal length). Power is negative. Examples: double concave, plano-concave, concave-meniscus (if thinner in middle).

Diopter: The unit of power, Diopter (D), is defined as the reciprocal of the focal length in meters (\(1 \text{ D} = 1 \text{ m}^{-1}\)). This unit is convenient for opticians and optometrists as the power of multiple thin lenses in contact simply adds up.

Applications of Lens Power: Lens power is fundamental in designing optical instruments like eyeglasses, telescopes, microscopes, and cameras. Eyeglasses are prescribed in diopters to correct vision problems, compensating for the eye's inability to focus light correctly on the retina. A person who is farsighted (hyperopia) needs a converging lens (+ power), while a person who is nearsighted (myopia) needs a diverging lens (- power).