In the case of a thin lens, how are the centres of curvature typically positioned relative to the optical cent

Question

In the case of a thin lens, how are the centres of curvature typically positioned relative to the optical centre?

Examoogle · Accepted Answer

The correct answer is Option A: They are equidistant from the optical centre.. In a thin lens, the two centres of curvature, $C_1$ and $C_2$, are the centres of the imaginary spheres that form the two refractive surfaces of the lens. For a symmetric lens, these points are positioned at equal distances from the optical centre along the principal axis.

Step-by-Step Solution & Concepts:
In a thin lens, the two centres of curvature, $C_1$ and $C_2$, are the centres of the imaginary spheres that form the two refractive surfaces of the lens. For a symmetric lens, these points are positioned at equal distances from the optical centre along the principal axis.

Working Principle: The lens is defined by two spherical surfaces with radii of curvature $R_1$ and $R_2$. The optical centre is the geometric center of a thin lens through which light passes without deviation. In a biconvex or biconcave lens, the lens is constructed such that both surfaces exhibit curvature, resulting in centres of curvature equidistant from the optical center in symmetric designs.

Key Formulas: $\frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2})$ — The Lens Maker's Formula relating focal length $f$, refractive index $n$, and radii $R_1, R_2$

Key Points: The principal axis passes through both centres of curvature and the optical centre.; For a thin lens, the optical centre is the origin for coordinate measurements.; The distance to $C_1$ and $C_2$ represents the radius of the spherical surface.

Advantages: Simplifies ray tracing for image formation.

Disadvantages: Does not account for spherical aberration in thick lenses.

Applications: Spectacle lens design; Camera lens calibration

Additional Info: In a thin lens, the thickness is negligible compared to the radii of curvature.. Option B is incorrect because the centre of curvature is the center of the sphere, not the point of intersection on the surface.

Answer: A is correct — The centres of curvature are equidistant from the optical centre in symmetric thin lenses.

Exam Tip: Remember that for all thin lenses, the optical centre acts as the reference point for the sign convention in the mirror/lens formula.

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Option B: They are located on the lens surface itself.

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Option C: They are closer to the edge than to the optical centre.

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Option D: They coincide with the optical centre.

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