doPhysics · Electromagnetism 1

Electric Flux and Gauss's Law

Why this chapter

Why this chapter exists

Coulomb's law gives the field of any charge, but for a wire or a ball it means adding the fields of countless small pieces. For bodies with symmetry there is a much shorter road. This chapter counts how much field passes through a surface, its flux. It then shows that the flux through any closed surface is fixed by the charge inside. The shape of the surface and the charges outside do not matter. With that law, the fields of a long wire, a large sheet, a shell and a solid sphere each take a few lines.

You will learn

By the end of this chapter you will

Treat an area as a vector along its normal, outward on a closed surface, and find its shadow \(A\cos\theta\).
Find the electric flux through flat, curved and closed surfaces, with its sign.
Use solid angle to share a point charge's flux among cones, discs and the faces of a cube.
State Gauss's law, show why it follows from Coulomb's law, and say what an outside charge does and does not change.
Choose a Gaussian surface by symmetry, and say when the law cannot give the field.
Find the fields of a long wire, a large sheet, a thin shell, a solid sphere and nested shells, and read the E-r graph.

Before you begin

From the chapter Electric Charge, Coulomb's Law and the Electric Field, you should know the field of a point charge, \(E = kq/r^2\), field lines, and superposition. From Work, Kinetic Energy and Power you should know the dot product, \(Fd\cos\theta\). The shell theorems of the chapter on Gravitational Force and Field help, but are not needed.

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Lessons

Area as a Vector and Electric Flux

A flat surface has a size and a direction, so its area is a vector along its normal, outward on a closed surface. Electric flux counts the field through a surface, \(\Phi = EA\cos\theta\), and a closed surface in a uniform field carries none.

Area as a vectorFlux through a flat surfaceFlux through a curved or closed surface

13 min0%

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Solid Angle and the Flux of a Point Charge

An angle in three dimensions, measured in steradians. A point charge sends \(q/\varepsilon_0\) through every sphere around it, shared out by solid angle, so a cone, a disc or a face of a cube takes a fixed fraction.

Solid angleFlux of a point charge through a sphereFlux through a cone and a discFlux through the faces of a cube

14 min0%

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Gauss's Law

The flux through any closed surface is the charge inside it divided by \(\varepsilon_0\). Charges outside change the field on the surface but not the flux. The law follows from Coulomb's inverse square, and symmetry decides when it gives the field.

Gausss law statedWhy gausss law follows from coulombs lawChoosing a gaussian surfaceMoving the charge inside the surface

14 min0%

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Fields of Wires, Sheets and Shells

Three fields in a few lines each. A long charged wire gives \(\lambda/2\pi\varepsilon_0 r\), a large charged sheet gives \(\sigma/2\varepsilon_0\) at every distance, and a thin charged shell gives no field inside and a point charge's field outside.

Field of a long straight wireField of a large sheetField of a thin spherical shell

14 min0%

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The Solid Sphere and Concentric Shells

Inside a uniformly charged solid sphere the field grows in a straight line, \(E = kQr/R^3 = \rho r/3\varepsilon_0\); outside it falls as \(1/r^2\). Reading the E-r graph, nested shells, and the cases where Gauss's law is true but cannot give the field.

Field inside a uniformly charged sphereThe e r graph of a charged sphereTwo concentric shellsWhen gauss cannot give e

15 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.

area vector: A vector along the normal to a small flat surface, with a size equal to its area. On a closed surface it points outward.
closed surface: A surface with no edge that encloses a region of space, such as a sphere, a box or a closed cylinder.
electric flux: The amount of electric field passing through a surface, \(\Phi = \int \vec E\cdot d\vec A\); for a uniform field through a flat surface, \(EA\cos\theta\). Its unit is N m²/C.
enclosed charge: The algebraic sum of all the charges inside a closed surface, \(q_\text{in}\). Charges outside are not counted.
Gauss's law: The total electric flux out through any closed surface equals the net charge enclosed divided by \(\varepsilon_0\).
Gaussian surface: An imagined closed surface on which Gauss's law is applied, chosen by the symmetry of the charges.
linear charge density: The charge on each metre of a line or wire, \(\lambda\), in C/m.
normal: The direction at right angles to a surface. A surface has two normals, one on each side; a closed surface takes the outward one.
pillbox: A short closed cylinder that crosses a charged surface, with one flat face on each side. It is the Gaussian surface for a large sheet.
solid angle: The area a cone cuts from a sphere centred on its tip, divided by the square of the radius, \(\Omega = A/r^2\). A whole sphere is \(4\pi\,\text{sr}\).
spherical shell: A thin hollow sphere carrying its charge evenly over its surface. It has no field inside and a point charge's field outside.
steradian: The unit of solid angle: the solid angle that cuts an area \(r^2\) from a sphere of radius \(r\). Symbol sr.
surface charge density: The charge on each square metre of a surface, \(\sigma\), in C/m².
volume charge density: The charge in each cubic metre of a body, \(\rho\), in C/m³.