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Geometry and Shapes for Physics

Why this chapter

Why this chapter exists

You may go on with the physics now. When a lesson needs this tool, it will tell you and bring you here.

A concave mirror makes an image twice the size, and a lamp under water lights a circle on the surface. A fine spray costs more pumping than a coarse one, and sunlight at Mars is less than half as strong as at the Earth. A magnet pushes an electron beam sideways, not towards itself. Each of these answers is geometry. It takes two triangles of one shape, a radius that is a normal, and a surface that grows as the square. It also takes a cone that cuts ever larger patches, and three axes linked by the right hand. This chapter teaches those tools, each with a physics example from the book and its own numbers.

You will learn

By the end of this chapter you will

Find a shadow, a pinhole image and a mirror image from similar triangles, with sides in the ratio \(k\) and areas in \(k^2\).
Take the radius as the normal to a curved surface, with \(\sin i = h/R\), and use arcs, chords, sectors and the height \(r^2/2R\) of a shallow arc.
Use the areas and volumes of the circle, sphere, cylinder and cone, scale them with size, and use the surface-to-volume ratio \(3/r\).
Measure a solid angle in steradians, use \(4\pi\) and \(2\pi(1 - \cos\theta)\), and explain the inverse square and the sharing of flux by geometry.
Set up right-handed axes in space, find distances with Pythagoras twice, find directions with the two right-hand rules, and argue from symmetry.

Before you begin

You should know Pythagoras' theorem, the sine and cosine of an angle, and angles measured in radians. Each lesson stands alone and starts from a physics situation, so you may open any of them from a physics chapter.

Already know this?

Lessons

Similar Triangles

Two triangles with the same angles have their sides in one ratio. The shadow under a street lamp, the pinhole camera, and the image of a concave mirror found from two pairs of similar triangles.

14 min0%

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Circles, Tangents and Arcs

The tangent to a circle is at right angles to the radius, so the normal to a curved mirror or lens is the radius. Where a ray meets a curved surface, arcs, chords and sectors, and the height of a shallow arc, \(t \approx r^2/2R\).

Circles tangents and arcsArcs chords and sectorsThe height of a shallow arc

15 min0%

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Areas and Volumes of Common Shapes

The areas and volumes of the circle, the sphere, the cylinder and the cone, why areas grow as the square of the size and volumes as the cube, and the surface-to-volume ratio \(3/r\) that makes small drops round and small hot bodies cool fast.

Areas and volumes of common shapesAreas grow as the square volumes as the cubeSurface to volume ratio

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Solid Angle and the Inverse Square

An angle in three dimensions, the steradian, with \(4\pi\) for the whole sphere and \(2\pi(1 - \cos\theta)\) for a cone. Why anything spread evenly from a point falls as \(1/r^2\), and how a point charge shares its flux by geometry alone.

Solid angle and the inverse squareThe solid angle of a coneSharing flux by solid angle

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Three Dimensions and the Right-Hand Rule

Points, axes and planes in space, the distance by Pythagoras used twice, right-handed axes with \(\hat i \times \hat j = \hat k\), the two right-hand rules of physics, and the symmetry arguments of Gauss's law.

Three dimensions and the right hand ruleThe two right hand rulesSymmetry arguments in three dimensions

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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.

Glossary

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

centre of curvature: The centre of the sphere of which a curved mirror or lens face is a part. Every normal to the surface passes through it.
chord: The straight line joining two points of a circle. For an angle \(\theta\) at the centre its length is \(2R\sin(\theta/2)\).
coordinate plane: A plane through two of the three axes, such as the \(xy\) plane, on which the third coordinate is zero.
grip rule: The right-hand rule for turning. Curl the fingers in the sense of a rotation or a circulation, and the thumb gives the direction of its axis vector.
inverse-square law: The rule that whatever a point source sends out evenly, and nothing absorbs, falls in strength as \(1/r^2\) with the distance \(r\).
normal: The line at right angles to a surface at a point. For a spherical surface it is the radius through that point.
pinhole camera: A dark box with a tiny hole in one face and a screen on the opposite face. Straight rays cross at the hole, so an upside-down image forms on the screen.
projected area: The area of the shadow a body casts on a plane at right angles to a parallel beam. For a sphere of radius \(R\) it is the disc \(\pi R^2\).
right-handed axes: Three perpendicular axes for which the right fingers, curled from \(+x\) towards \(+y\), leave the thumb along \(+z\), so that \(\hat i \times \hat j = \hat k\).
sagitta: The height by which an arc stands above its chord. For a chord of half-length \(r\) under an arc of radius \(R\), it is close to \(r^2/2R\) when \(r\) is small.
scale factor: The number \(k\) by which every length of one of two similar figures must be multiplied to give the other. Areas then go as \(k^2\) and volumes as \(k^3\).
sector: The slice of a disc between two radii. For an angle \(\theta\) in radians its area is \(\tfrac12 R^2\theta\).
similar triangles: Two triangles whose three angles are equal. They have the same shape, and each side of one is the same multiple of the matching side of the other.
solid angle: The area that a cone cuts from a sphere centred on its tip, divided by the square of the radius, \(\Omega = A/r^2\). It measures how wide a cone is.
steradian: The unit of solid angle, sr. The whole sphere round a point is \(4\pi\,\text{sr}\), and a hemisphere is \(2\pi\,\text{sr}\).
surface-to-volume ratio: The surface area of a body divided by its volume. For a sphere of radius \(r\) it is \(3/r\), so small bodies have much more surface for their size.
symmetry argument: Reasoning that a field, a flux or a current must look the same after any turn, reflection or shift that leaves its source unchanged.
tangent: A straight line that touches a circle at one point without cutting it. At that point it is perpendicular to the radius.