Physics
Refraction at Spherical Surfaces
Sign Convention for Refraction at Spherical Surface
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⚡ Quick Summary
The sign convention for refraction at spherical surfaces is similar to that for spherical mirrors. The line joining the object and the center of curvature is the X-axis, with the positive direction along the incident rays. The intersection of the surface and the axis is the origin. u, v, and R are the x-coordinates of the object, image, and center of curvature, respectively. Quantities are positive if they lie on the positive side of the origin, and negative otherwise.
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The line joining the object and the centre is taken as the X-axis. The positive direction of the axis is generally chosen along the direction of the incident rays. The point of intersection of the spherical surface with the axis is taken as the origin. The quantities u, v and R denote the x-coordinates of the object, the image and the centre of curvature respectively. Any of these quantities is positive if the corresponding point lies on the positive side of the origin and is negative if it is on the negative side. Similarly for the lengths perpendicular to the X-axis.
Relation between u, v, and R
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⚡ Quick Summary
The formula relating the object distance (u), image distance (v), and radius of curvature (R) for refraction at a spherical surface is: (μ₂/v) - (μ₁/u) = (μ₂ - μ₁)/R, where μ₁ and μ₂ are the refractive indices of the two media.
(m₂/v) - (m₁/u) = (m₂ - m₁)/R
Derivation based on Snell's law and paraxial rays, leading to the formula (m2/v) - (m1/u) = (m2 - m1)/R. Proper sign convention must be used to ensure the formula's validity in all situations. u, v, and R are the x-coordinates of the object, image, and center of curvature, respectively.
Extended Objects: Lateral Magnification
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⚡ Quick Summary
For an extended object placed perpendicular to the principal axis, the image of a point on the object lying on the normal to the spherical surface will also lie on the same normal. This concept is used to analyze lateral magnification, but the specific formula for lateral magnification is not explicitly provided in this section.
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Considers an extended object OQ placed perpendicular to the line OPC. A ray originating from Q and going towards QC is incident normally on the spherical surface AB and goes undeviated in medium 2. The image of Q must be on the line QC.