Question

\(1.6\) is a refractive index of plano-convex lens, then the radius of curvature of the curved surface is \(60 \mathrm{~cm}\). The focal length of the lens is \(\mathrm{cm}\) (B) \(10 \overline{0}\) (A) 50 (C) \(-50\) (D) \(-100\)

Short answer

The focal length of the plano-convex lens is 100 cm. The correct answer is (B) \(10\overline{0}\).

Step-by-step solution

Recall the Lensmaker's Equation

The Lensmaker's Equation is given by: \( \dfrac{1}{f} = (n - 1) \left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right) \) where f is the focal length, n is the refractive index of the lens, \(R_1\) and \(R_2\) are the radii of curvature for the two surfaces of the lens, and \(n - 1\) represents the difference in refractive indices between the lens material and the surrounding medium (air in this case).

Apply the variables provided to the Lensmaker's Equation

We are given that n = 1.6, \(R_1 = 60\) cm (radius of curvature of the convex side), and \(R_2 = \infty\) (radius of curvature for the flat side, since the plano surface is considered to have an infinite radius of curvature). Substitute the values into the equation: \( \dfrac{1}{f} = (1.6 - 1) \left(\dfrac{1}{60} - \dfrac{1}{\infty}\right) \)

Simplify and solve for f

The equation now becomes: \( \dfrac{1}{f} = (0.6) \left(\dfrac{1}{60}\right) \) Multiplying both sides by 60, \( \dfrac{60}{f} = 0.6 \) Now we can solve for f, \( f = \left(\dfrac{60}{0.6}\right) = 100 \) The focal length of the plano-convex lens is 100 cm. From the given options, the correct answer is: (B) \(10\overline{0}\).

Key concepts

Refractive Index

The refractive index is a measure that describes how light travels through different materials. It is calculated as the ratio of the speed of light in a vacuum to the speed of light in the material. This concept is crucial in optics, as it helps in determining how much light will bend, or refract, when entering a material.
A higher refractive index means that light will slow down more significantly and bend more when it enters the material. For a plano-convex lens with a refractive index of 1.6, like in the given problem, this means light slows down considerably as it enters the lens from air, causing it to refract towards the normal.
The refractive index plays a key role in the lensmaker's equation where it impacts the focal length calculation by altering the amount of curvature or bending of light through the lens.

Radius of Curvature

The radius of curvature is the radius of the sphere from which a lens surface is a part. It’s a measure of how curved the lens surface is, expressed in terms of centimeters or meters.
In the problem, we are dealing with a plano-convex lens, where one side is flat (plano) and the other side is curved. The curved side has a radius of curvature of 60 cm. The flat side, being a plane, has an infinite radius of curvature.
Understanding the radius of curvature is essential because it directly affects how lenses refract light. A smaller radius of curvature means a more curved lens surface, leading to stronger light focusing characteristics. Therefore, the radius of curvature is a vital parameter in calculating the focal length using the lensmaker’s equation.

Focal Length

Focal length is the distance between the lens and the point where light rays converge into a single point, known as the focal point. It’s an essential characteristic that determines the lens's optical power.
In the context of the lensmaker’s equation, the focal length is determined by the curvature of the lens surfaces and the refractive index. For the plano-convex lens problem, substituting the known values, 1.6 for the refractive index, and radii of curvature values into the lensmaker’s formula, gives us a focal length of 100 cm. This focal length tells us how strongly the lens can focus incoming parallel light rays.
A shorter focal length indicates a stronger lens, capable of converging light more sharply, which is decisive in various applications such as in glasses, cameras, and microscopes.

Plano-Convex Lens

A plano-convex lens is one that is flat on one side and convex on the other. The shape of the lens is vital because it affects how light is refracted. The plano side is just like a straight wall that lets light pass through unchanged, while the convex side curves outward, bending the light towards a focal point.
Plano-convex lenses are widely used in optical devices where unidirectional light convergence is desired and are especially common in laser applications due to their ability to collimated light beams by focusing or diverging them.
In the exercise, we worked with a plano-convex lens, emphasizing how its specific shape – one flat side and one convex side – along with the refractive index of the material, impacts how the focal length is calculated using the lensmaker's equation. This makes plano-convex lenses particularly useful for condensing light and incr/images/internal/increasing intensity in focused areas.