Spherical Surfaces and Lenses

Why this chapter

Why this chapter exists

A camera, a pair of spectacles, a microscope and the eye itself all form images by refraction at curved surfaces. This chapter starts with one curved surface and a single formula derived from Snell's law. Two surfaces, one after the other, explain a glass ball in sunlight, a bubble in a paperweight and a goldfish in a round bowl. Two surfaces close together make a thin lens, and the same formula becomes the lens maker's formula and then the lens formula. Lenses are then combined, silvered and cut, a focal length is measured, and the two faults of every simple lens are understood and cured.

You will learn

By the end of this chapter you will

State the Cartesian sign convention for surfaces and lenses, and sign \(u\), \(v\) and \(R\) from a figure.
Derive and use \(\mu_2/v - \mu_1/u = (\mu_2 - \mu_1)/R\) and \(m = \mu_1 v/(\mu_2 u)\), and find the two focal lengths of a surface.
Treat two surfaces one after the other: glass spheres, bubbles and fish bowls, and the flat surface as a special case.
Derive the lens maker's formula, and find the focal length and power of a lens in air and in a liquid.
Use \(1/v - 1/u = 1/f\) and \(m = v/u\), draw ray diagrams, and read the graph of \(v\) against \(u\).
Combine thin lenses in contact and with a gap, and treat silvered and cut lenses.
Find \(f\) by the displacement method, and explain spherical and chromatic aberration and the achromatic doublet.

Before you begin

From the chapter on Refraction, you should know Snell's law, apparent depth, and that the refractive index of glass depends on colour. From the chapter on Reflection, you should know the Cartesian sign convention for mirrors and the mirror formula. You should be comfortable with small-angle approximations and with signed numbers.

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Lessons

Refraction at a Spherical Surface

The Cartesian sign convention stated in full for surfaces and lenses, the refraction formula for one curved surface derived from Snell's law, the magnification it gives, and the two focal lengths of a surface.

The sign convention for lenses and surfacesThe refraction formula for a curved surfaceTwo focal lengths of a surface

15 min0%

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Spheres, Bubbles and Fish Bowls

Light that crosses two curved surfaces, one after the other. A glass sphere focusing a beam, a bubble seen inside a glass ball, a fish in a round bowl, and the flat surface as a sphere of infinite radius.

One surface after anotherA bubble in a glass sphereThe fish at the centre of a bowl

14 min0%

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Thin Lenses and the Lens Maker's Formula

Converging and diverging lenses and their shapes, the lens maker's formula derived from two refracting surfaces, a lens in water and in other liquids, and the power of a lens in dioptres.

Converging and diverging lensesThe lens makers formulaA lens in a liquidPower of a lens

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The Lens Formula and Magnification

The thin lens formula \(1/v - 1/u = 1/f\) and the magnification \(m = v/u\), the three rays that locate an image, where the image goes as the object moves, and the graph of \(v\) against \(u\).

The lens formulaMagnification of a lensRay diagrams for lensesGraphs of v against u for a lens

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Combining Lenses: Contact, Gaps, Silvering and Cutting

Thin lenses in contact, whose powers add; two lenses with a gap, \(P = P_1 + P_2 - dP_1P_2\); a lens silvered on one face, which acts as a mirror of power \(2P_L + P_M\); and a lens cut along or across its axis.

Lenses in contactLenses with a gapThe silvered lensCut lenses

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Finding f, and the Faults of Lenses

The displacement method, which finds a focal length from two sharp images; spherical aberration, where the edge of a lens focuses nearer than its middle; and chromatic aberration, cured by an achromatic doublet of two glasses.

The displacement methodSpherical aberrationChromatic aberration and the achromat

14 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

achromatic doublet: A converging lens and a diverging lens of different glasses in contact, chosen with \(\omega_1/f_1 + \omega_2/f_2 = 0\), so that all colours come to one focus.
aperture: The width of the part of a lens or mirror that light can pass through or strike. A larger aperture gathers more light.
Cartesian sign convention: The rule used by the NCERT. Light is drawn from left to right, and every distance is measured from the pole or the optical centre. Distances along the incident light are positive, and heights above the axis are positive.
centre of curvature: The centre of the sphere of which a spherical surface is a part. Its distance from the pole is the radius of curvature \(R\).
chromatic aberration: The failure of a lens to bring all colours to one focus, because the refractive index of glass depends on the colour. Violet focuses nearer than red.
conjugate points: A pair of points of which each is the image of the other. For a lens, an object at one gives an image at the other, and the reverse.
converging lens: A lens that brings a parallel beam to a real focus. A glass lens in air converges when it is thicker in the middle. Its focal length is positive.
dioptre: The unit of power of a lens or surface, D. One dioptre is \(1\,\text{m}^{-1}\), the power of a lens of focal length \(1\,\text{m}\).
dispersive power: The spread of the refractive index over the colours divided by the mean deviation, \(\omega = (\mu_V - \mu_R)/(\mu_Y - 1)\). It measures how strongly a glass separates colours.
displacement method: A way of finding the focal length of a convex lens by moving it between an object and a screen a fixed distance \(D\) apart. The two sharp positions, \(d\) apart, give \(f = (D^2 - d^2)/4D\).
diverging lens: A lens that spreads a parallel beam so that it seems to come from a virtual focus. A glass lens in air diverges when it is thinner in the middle. Its focal length is negative.
equivalent focal length: The focal length \(F\) of the single lens that acts like a combination of lenses.
first focal point: The point on the axis of a refracting surface where an object sends its image to infinity. Its distance from the pole is the first focal length \(f_1\).
focal length: The distance \(f\) from the optical centre of a thin lens to either focus. On the Cartesian convention it is positive for a converging lens and negative for a diverging one.
lens maker's formula: The relation \(1/f = (\mu/\mu_m - 1)(1/R_1 - 1/R_2)\). It gives the focal length of a thin lens from its refractive index, its surroundings and the radii of its faces.
linear magnification: The ratio \(m = h_i/h_o\) of the height of the image to the height of the object. A negative value means an inverted image.
marginal ray: A ray that passes through a lens near its edge, far from the principal axis.
optical centre: The point at the middle of a thin lens, on the principal axis. A ray through it passes without deviation, and all distances are measured from it.
paraxial ray: A ray that stays close to the principal axis and makes a small angle with it. For such a ray, the sine and the tangent of the angle equal the angle itself.
pole: The middle point of a spherical refracting surface or of a mirror, where the principal axis meets it. Distances are measured from it.
power: The ability of a surface or a lens to converge light. It is \(P = 1/f\) for a thin lens in air and \((\mu_2 - \mu_1)/R\) for a surface, in dioptres with lengths in metres.
principal axis: The straight line through the pole and the centre of curvature of a surface, or through the two centres of curvature of a lens.
real image: An image where the rays actually meet after refraction or reflection. It can be caught on a screen.
refractive index: The ratio \(\mu = c/v\) of the speed of light in vacuum to its speed in a medium. It fixes how much light bends when it crosses into that medium.
relative refractive index: The ratio \(\mu/\mu_m\) of the refractive index of a lens to that of the medium around it. The lens bends light only as far as this ratio differs from 1.
second focal point: The point on the axis of a refracting surface where a parallel beam comes to a focus. Its distance from the pole is the second focal length \(f_2\).
silvered lens: A lens with one face coated to reflect. Light crosses the lens twice and reflects once, so the system acts as a mirror of power \(2P_L + P_M\).
spherical aberration: The failure of a lens or mirror with spherical faces to bring all the rays of a parallel beam to one point. Marginal rays focus nearer than paraxial rays.
stop: An opaque plate with an opening, placed in front of a lens to let only part of the beam through.
thin lens: A lens whose thickness is small compared with the radii of its faces and with the distances of object and image. Its two poles merge into one point.
virtual image: An image from which the rays only appear to come. The rays never pass through it, so it cannot be caught on a screen.
virtual object: A point towards which light is converging when it meets a surface or a lens, before it can form an image there. Its distance \(u\) is positive.