#radii of curvature

Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens has radii of curvature \( 30 \, \text{cm} \) and \( -30 \, \text{cm} \) with refractive index \( 1.

**Spherical refracting surface** power P = (n₂-n₁)/R, lens power sum of two surfaces. Lens maker derivation combines two refractions, sign of R₂ negative for second surface convex opposite direction, yielding 1/f positive for convex lens. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.6 , R₁ = 30 cm , R₂ = -30 cm . (1/f) = (1.6 - 1) ( (1/30) - (1/-30) ) = 0.6 ( (1/30) + (1/30) ) = 0.6 × (2/30) = (1.2/30) = (1/25) . f = 25 cm . Substituting values gives 25 cm, which matches expected image position and

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Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens of refractive index \( 1.5 \) has radii of curvature \( 18 \, \text{cm} \) and \( -18 \, \text{cm}

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.5 , R₁ = 18 cm , R₂ = -18 cm . (1/f) = (1.5 - 1) ( (1/18) - (1/-18) ) = 0.5 ( (1/18) + (1/18) ) = 0.5 × (2/18) = (1/18) . f = 18 cm . Substituting values gives 18 cm, which matches expected image

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Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens of refractive index \( 1.55 \) has radii of curvature \( 22 \, \text{cm} \) and \( -22 \, \text{cm}

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.55 , R₁ = 22 cm , R₂ = -22 cm . (1/f) = (1.55 - 1) ( (1/22) - (1/-22) ) = 0.55 ( (1/22) + (1/22) ) = 0.55 × (2/22) = (1.1/22) = (1/20) . f = 20 cm . Substituting values gives 20 cm, which matches

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Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens of refractive index \( 1.5 \) has radii of curvature \( 40 \, \text{cm} \) and \( -40 \, \text{cm}

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.5 , R₁ = 40 cm , R₂ = -40 cm . (1/f) = (1.5 - 1) ( (1/40) - (1/-40) ) = 0.5 ( (1/40) + (1/40) ) = 0.5 × (2/40) = (1/40) . f = 40 cm . Substituting values gives 40 cm, which matches expected image

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Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens of refractive index \( 1.55 \) has radii of curvature \( 30 \, \text{cm} \) and \( -30 \, \text{cm}

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.55 , R₁ = 30 cm , R₂ = -30 cm . (1/f) = (1.55 - 1) ( (1/30) - (1/-30) ) = 0.55 ( (1/30) + (1/30) ) = 0.55 × (2/30) = (1.1/30) . f = (30/1.1) ≈ 27.27 cm . Substituting values gives 27.3 cm, which matches

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Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens of refractive index \( 1.6 \) has radii of curvature \( 25 \, \text{cm} \) and \( -25 \, \text{cm}

**Lens maker's formula** 1/f = (n-1)(1/R₁ - 1/R₂), n refractive index, R₁,R₂ radii of curvature (m), sign convention R positive if surface convex towards incident light. For double convex R₁=12 cm, R₂=-12 cm, n=1.5, 1/f=(0.5)(1/12 -1/(-12))=(0.5)(2/12)=1/12, f=12 cm, converging. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.6 , R₁ = 25 cm , R₂ = -25 cm . (1/f) = (1.6 - 1) ( (1/25) - (1/-25) ) = 0.6 ( (1/25) + (1/25) ) = 0.6 × (2/25) = (1.2/25) . f = (25/1.2) ≈ 20.83 cm . Substituting values gives 20.8 cm, which matches

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Refraction at Spherical Surfaces and Lens Maker's Formula

A double convex lens of refractive index \( 1.5 \) has radii of curvature \( 12 \, \text{cm} \) and \( -12 \, \text{cm}

**Spherical refracting surface** power P = (n₂-n₁)/R, lens power sum of two surfaces. Lens maker derivation combines two refractions, sign of R₂ negative for second surface convex opposite direction, yielding 1/f positive for convex lens. Lens maker’s formula: (1/f) = (n - 1) ( (1/R₁) - (1/R₂) ) . n = 1.5 , R₁ = 12 cm , R₂ = -12 cm . (1/f) = (1.5 - 1) ( (1/12) - (1/-12) ) = 0.5 ( (1/12) + (1/12) ) = 0.5 × (2/12) = (1/12) . f = 12 cm . Substituting values gives 12 cm, which matches expected image position and magnification from

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