Question

Two plano-convex lenses of radius of curvature \(R\) and refractive index \(\mathrm{n}=1.5\) will have equivalent focal length equal to \(\mathrm{R}\), when they are placed (A) at distance \(\mathrm{R}\) (B) at distance \((\mathrm{R} / 2)\) (C) at distance \((\mathrm{R} / 4)\) (D) in contact with each other

Short answer

The two plano-convex lenses with radius of curvature R and refractive index n=1.5 will have an equivalent focal length equal to R when they are placed at a distance of (R/4). Hence, the correct answer is (C) at distance (R/4).

Step-by-step solution

Lens maker's formula for a single plano-convex lens

The lens maker's formula relates the focal length (F) of a lens with the radii of curvature (R1 and R2) and the refractive index (n) of the lens: \(\frac{1}{F} = (n-1) \left(\frac{1}{R1} - \frac{1}{R2}\right)\) For a plano-convex lens, one surface is flat (plane) and the other is curved (convex). That means, R1=R (curved surface) and R2=\(\infty\) (flat surface). So, the lens maker's formula becomes: \(\frac{1}{F} = (n-1) \left(\frac{1}{R} - \frac{1}{\infty}\right)\) #Step 2: Calculate the focal length of a single plano-convex lens#

Focal length of a single plano-convex lens

By substituting n=1.5 and R1=R into the lens maker's formula, we can find the focal length (F) of the given plano-convex lenses: \(\frac{1}{F} = (1.5-1) \left(\frac{1}{R} - \frac{1}{\infty}\right)\) \(\frac{1}{F} = \frac{1}{2R}\) \(F = 2R\) #Step 3: Thin lens formula for the combination of the two lenses#

Thin lens formula for the combination

The thin lens formula relates the equivalent focal length of a combination of two lenses (F_eq) with their individual focal lengths (F₁ and F₂) and the distance between them (d): \(\frac{1}{F_{eq}} = \frac{1}{F_1} + \frac{1}{F_2} - \frac{d}{F_1F_2}\) Since both lenses have the same focal length (F = 2R), we can simplify the formula: \(\frac{1}{F_{eq}} = 2 \left(\frac{1}{2R}\right) - \frac{d}{(2R)^2}\) #Step 4: Determine the possible value of d#

Compare the equivalent focal length with R and find d

Given that the equivalent focal length should equal R, we now need to compare the value of F_eq with R and solve for d: \(R = F_{eq}\) \(R = \frac{1}{2 \left(\frac{1}{2R}\right) - \frac{d}{(2R)^2}}\) After solving for d, we get: \(d = \frac{R}{4}\) Hence, the correct answer is (C) at distance (R/4).

Key concepts

Lens Maker's Formula

The lens maker's formula is a critical tool for understanding how lenses behave and refract light. It connects the focal length of a lens to its curvature and the glass's refractive index. For a single plano-convex lens, which has one flat surface and one curved, this formula is simplified:

• The flat surface is assigned infinite curvature, meaning its radius of curvature is \(R_2 = \infty\).
• The curved surface has radius \(R_1 = R\).

This results in the modified lens maker's formula:\[\frac{1}{F} = (n-1) \left(\frac{1}{R} - \frac{1}{\infty}\right)\]This shows the relationship between the focal length \(F\), refractive index \(n\), and the radius of curvature \(R\), allowing us to calculate the focal length when the lens material and shape are known.

Thin Lens Formula

The thin lens formula is essential for determining how combined lenses act together. This is especially useful in systems where two lenses are used in tandem, as in the problem with the plano-convex lenses.The formula is:\[\frac{1}{F_{eq}} = \frac{1}{F_1} + \frac{1}{F_2} - \frac{d}{F_1F_2}\]Where:

• \(F_{eq}\) is the equivalent focal length of the lens combination,
• \(F_1\) and \(F_2\) are the focal lengths of each lens,
• \(d\) is the distance between the lenses.

This formula helps in finding the composite focal length when lenses are either in contact or at a certain distance apart. For two identical plano-convex lenses in the exercise, we solve this equation with specific \(F_1 = F_2 = 2R\) and derive the ideal inter-lens distance \(d\) for a desired focal length.

Radius of Curvature

Understanding the radius of curvature is key to predicting lens behavior. The radius of curvature relates to how curved a lens surface is. For a plano-convex lens:

• The plane side has an infinite radius of curvature, as it's flat \(R_2 = \infty\).
• The convex side's curvature is finite, providing \(R_1 = R\), a direct input into the lens maker's formula.

The radius of curvature is crucial because it defines how strongly the lens converges or diverges light. A smaller radius indicates a more pronounced curvature, which affects how the lens focuses light and ultimately its focal length.

Refractive Index

The refractive index \(n\) is a measure of how much a material slows down light passing through it. This property influences how light bends when entering or exiting a lens. In plano-convex lenses used in the exercise, the refractive index is \(1.5\).

• A higher refractive index means light is slowed more, bending towards the normal, resulting in stronger focusing power.
• The refractive index impacts the lens maker's formula, as it determines the extent of light bending for a given curvature.

This knowledge allows us to predict how a lens will perform in practical applications, whether it's a camera lens or a scientific instrument, and is fundamental in calculating focal lengths in optical systems.