Biconvex & Plano-Concave Lens PYQ | JEE Main Optics 2025

A Bi-Convex Lens and a Plano-Concave Lens Have the Same Magnitude of Power | JEE Main Physics

In this problem, we use the Lens Maker's Formula to compare the powers of two different lenses. Since both lenses have the same magnitude of power, we can equate their powers and determine the required ratio of the radii of curvature.

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❓ Question

A bi-convex lens of refractive index 1.5 and a plano-concave lens of refractive index 1.7 have the same magnitude of power.

The second radius of curvature of the bi-convex lens is equal to the radius of curvature of the plano-concave lens.

Find the ratio of the first radius of curvature to the second radius of curvature of the bi-convex lens.

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📘 Concept Used

• Lens Maker's Formula
• Power of a Lens
• Sign Convention for Lenses
• Convex and Concave Lenses

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✍️ Solution

Step 1: Lens Maker's Formula

For a thin lens in air,

P = (μ − 1) (1/R₁ − 1/R₂)

where,

• P = Power of the lens
• μ = Refractive index of the lens material
• R₁ and R₂ are the radii of curvature

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Step 2: Power of the Bi-Convex Lens

For the bi-convex lens,

μ = 1.5

Using the sign convention,

R₁ = +R₁

R₂ = −R₂

Therefore,

P₁ = (1.5 − 1) [1/R₁ − 1/(−R₂)]

= 0.5 (1/R₁ + 1/R₂)

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Step 3: Power of the Plano-Concave Lens

For the plano-concave lens,

μ = 1.7

One surface is plane, therefore

R = ∞

The curved surface has radius

R = −R₂

Hence,

P₂ = (1.7 − 1) [−1/R₂ + 1/∞]

= −0.7/R₂

Since the question states that the magnitudes of the powers are equal,

|P₁| = |P₂|

Therefore,

0.5 (1/R₁ + 1/R₂) = 0.7/R₂

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Step 4: Simplify the Equation

Multiply both sides by 10:

5 (1/R₁ + 1/R₂) = 7/R₂

5/R₁ + 5/R₂ = 7/R₂

5/R₁ = 2/R₂

Cross multiplying,

5R₂ = 2R₁

Therefore,

R₁/R₂ = 5/2

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💡 Key Learning

• For a convex lens, the first radius is positive and the second radius is negative.
• For a plano lens, the radius of the plane surface is considered infinite.
• Equal magnitude of power means the numerical values of power are equal, irrespective of sign.
• Always apply the correct sign convention before substituting values into the Lens Maker's Formula.

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📝 Final Answer

Required Ratio:

R₁ : R₂ = 5 : 2

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📚 Conclusion

Using the Lens Maker's Formula and the given condition that both lenses have the same magnitude of power, we obtain the ratio of the first and second radii of curvature of the bi-convex lens as 5 : 2. This problem highlights the importance of using the correct sign convention and understanding how the power of different types of lenses is calculated.