doPhysics · Electromagnetism 1
Electric Charge, Coulomb's Law and the Electric Field
Charge and its properties, the force between point charges and its superposition, the electric field and its lines, and the fields of rods, wires, rings, arcs, discs and sheets.
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Why this chapter
Why this chapter exists
A comb lifts paper, a sweater crackles, a photocopier prints, and an atom holds together, all by one force. Electric charge comes in two kinds, in whole numbers of electrons, and is never made or destroyed. Two charges push or pull with a force that falls as the square of their distance, like gravity but enormously stronger. From that force comes the electric field, a property of every point in space that tells what any charge placed there would feel. This chapter builds the field for points, lines, rings and sheets, the base of all of electricity.
You will learn
By the end of this chapter you will
- Describe the two kinds of charge, charging by rubbing, contact and induction, and the difference between conductors and insulators.
- Use the quantisation and the conservation of charge to count electrons and to share charge between touching conductors.
- Find the force between point charges with Coulomb's law in scalar and vector form, in vacuum and in a medium.
- Add electric forces by superposition for charges on a line, at the corners of polygons, and for charged balls on threads, and locate null points.
- Define the electric field as force per unit charge, find the field of point charges, and read and draw field lines.
- Derive and use the fields of a finite rod, a long and a half-infinite wire, a ring on its axis, an arc at its centre, a disc and a large sheet.
Before you begin
Before you begin
You should be able to draw a free-body diagram and add forces as vectors by components, as in Objects in Equilibrium. You should know Newton's law of gravitation and the gravitational field of a rod and a ring, from Gravitational Force and Field. You should be able to use the equations of motion for constant acceleration, and to integrate simple powers and \(\sin\) and \(\cos\).
Lessons
1
Charge and Its Properties
Two kinds of charge, how rubbing, contact and induction move them, and why charge comes in whole numbers of electrons and can never be made or destroyed.
Two kinds of charge and how bodies chargeQuantisation of chargeConservation of charge
2
Coulomb's Law
The force between two point charges grows with each charge and falls as the square of the distance. Its vector form, the pair of equal and opposite forces, the weakening in a medium, and why the electric force dwarfs gravity inside an atom.
Coulombs lawCoulombs law in vector formCoulombs law in a mediumElectric force against gravity
3
Superposition and Equilibrium
The force on a charge from many charges is the vector sum of the forces from each one. Charges on a line, at the corners of triangles, squares and polygons, the point where a third charge feels no force, and charged balls hanging on threads.
The superposition principleCharges at the corners of polygonsWhere a third charge feels no forceCharged balls on threads
4
The Electric Field and Its Lines
The electric field is the force per unit charge at each point of space. The field of a point charge, fields added as vectors, the field-line picture and its rules, and a charge accelerated by a uniform field.
Field as force per unit chargeField of point chargesElectric field linesA charge in a uniform field
5
Fields of Rods and Wires
A charged rod is cut into point-like elements whose fields are added by an integral. The field of a rod beside it and on its own line, the long straight wire whose field falls as 1/r, and the half-infinite wire.
Charge densities and the element dqField of a rod on its perpendicular bisectorThe long wire and the half infinite wireField on the line of a rod
6
Rings, Arcs, Discs and Sheets
Symmetry does most of the work for curved and flat charge. The field on the axis of a ring and where it peaks, the field at the centre of an arc, a disc built from rings, and the uniform field of a large sheet and of two sheets.
Field on the axis of a charged ringField at the centre of an arcField of a disc and a large sheetTwo sheets and a ball beside a sheet
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- charging by induction
- Charging a conductor without touching it. A charged body held near separates the conductor's charges; earthing lets one kind escape, leaving a charge opposite to that of the body.
- conductor
- A material in which charge can move freely, such as a metal, tap water, the human body or the earth.
- conservation of charge
- The law that the total charge of an isolated system, counted with signs, never changes. Charge can move from body to body but is never made or destroyed.
- coulomb
- The SI unit of charge, C. One coulomb is the charge of \(6.25 \times 10^{18}\) electrons.
- Coulomb's law
- The law that two point charges exert forces of size \(k|q_1q_2|/r^2\) on each other, along the line joining them. The forces repel for like charges and attract for unlike ones.
- dielectric breakdown
- The sudden conduction of an insulator when the field in it grows too strong; dry air breaks down at about \(3 \times 10^6\,\text{N/C}\) and sparks.
- dielectric constant
- The number \(K\) by which a medium reduces the force between charges placed in it; its permittivity is \(\varepsilon = K\varepsilon_0\). About 1 for air and about 80 for water.
- earthing
- Connecting a conductor to the ground by a wire. The earth is so large that it can give or take any amount of charge.
- electric charge
- A property of matter that makes bodies exert electric forces on each other. It comes in two kinds, positive and negative; like charges repel and unlike charges attract.
- electric field
- The force per unit charge on a small positive test charge at a point, \(\vec E = \vec F/q_0\), in N/C. A charge \(q\) placed there feels \(q\vec E\).
- elementary charge
- The smallest free charge, the size of the charge of an electron or a proton, \(e = 1.6 \times 10^{-19}\,\text{C}\).
- field line
- A curve whose tangent at every point gives the direction of the electric field. Lines start on positive charges, end on negative ones, and crowd where the field is strong.
- gold-leaf electroscope
- A device with two thin gold leaves hanging from a metal rod in a glass jar. Charge on the rod makes the leaves repel and spread apart.
- insulator
- A material in which charge cannot move freely, such as glass, plastic, rubber or dry air. Charge placed on it stays where it is put.
- line charge
- Charge spread along a thin straight wire, described by its linear charge density \(\lambda\). A long line charge gives a field \(2k\lambda/r\).
- linear charge density
- The charge per unit length of a thin rod, wire or ring, \(\lambda\), in C/m.
- null point
- A point where the forces of two or more charges on a third charge, or their fields, add to zero.
- permittivity of free space
- The constant \(\varepsilon_0 = 8.85 \times 10^{-12}\,\text{C}^2/(\text{N m}^2)\) in Coulomb's law written as \(1/4\pi\varepsilon_0\); it sets how strongly charges act on each other in vacuum.
- point charge
- A charged body whose size is negligible compared with the distances involved, treated as a charge at a single point.
- polarisation
- A slight shift of the charges inside the molecules of an insulator when a charged body is near. It leaves opposite charge on the near face.
- principle of superposition
- The rule that the force due to many charges is the vector sum of the forces due to each charge acting alone. Fields add the same way.
- surface charge density
- The charge per unit area of a sheet or a surface, \(\sigma\), in C/m².
- test charge
- A small positive charge used to measure an electric field, small enough not to disturb the charges that make the field.
- uniform field
- A field with the same size and direction at every point, such as the field between two large parallel plates with opposite charges.
- unit vector
- A vector of length one that only gives a direction; \(\hat r_{12}\) points from one charge towards the other.
- volume charge density
- The charge per unit volume of a solid body, \(\rho\), in C/m³.