Peter's Physics Pages

Physics for Industrial Design with Peter Eyland

Lecture 1 Coulomb’s Law

In daily life there seems to be a diversity of forces:

• the resistance of a rusty nut on a bolt,

• the stickiness of honey,

• the pressure of waves on surfers,

• the osmotic pressure causing sap to rise in plants, and

• the impulses that drive a loudspeaker.

All of these arise from three basic interactions:

• Gravitational,

• Electric, and

• Magnetic.

This course will look at Electric and Magnetic interactions only.

Electric forces originate with electric charges, and magnetic forces emerge from the movement of electric charges.

In this lecture the following are introduced

• electric charge,

• conductors and insulators, and

• Coulomb's law.

Conductors and Insulators

Charles Du Fay (~1736) found that to explain attractive and repulsive electric forces, two types of electricity are needed.

These were (later) called "positive charge" and "negative charge".

"Like" charges repel and "unlike" charges attract.

Normally, objects have a balance of positive and negative charge, but some types of material can support a charge separation.

Charge flows easily through conductors so it is difficult to keep the charges separate.

Conductors are materials that have lots of charges that are relatively free to move around within the material.

These "free" charges are called conduction electrons and holes (holes are like bubbles - the absence of electrons).

For example, the metal Copper, has a carrier density ~ 10+23 cm-3.

Insulators (or Dielectrics) do not have many charges that move easily and so you can have an imbalance of charge at different places in the material.

For example, glasses and plastic have a carrier density ~ 1 cm-3.

Semiconductors such as Germanium and Silicon have intermediate carrier densities i.e. ~ 10+10 ... 10+12cm-3.

Semiconductors are useful because their carrier density can be changed drastically by adding impurities, or changing the temperature, or the incident light, etc.

Electric charge separation

Charges are transferred on or off insulators by physical contact.

When some insulators are rubbed together friction can transfer charge from one insulator to the other.

The charges will stick at the places where they were separated to or from.

For example, rubbing glass with silk will transfer electrons from the glass to the silk.

The silk acquires a negative net charge and the glass is left with an equal net positive charge.

The force between the static charges on the materials is called electrostatic force.

Industrial applications of electrostatic force cover such things as, powder coating, ash precipitation, ink-jet printing, and photocopying.

Coulomb’s law

Auguste Coulomb's experiments worked out the quantitative relationship between force, charge and separation.

Coulomb’s law states that:

The size of the electric force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of their separation and acts along the line joining them.

In symbols, where F is the electric force, q is the electric charge and r is the charge separation,

To get the units right in the S.I. system, for stationary charges in air or vacuum:

F is in Newton,

q is in Coulomb and

r is in metre,

The Coulomb is a large unit.

If two 1 Coulomb charges were placed a metre apart, then the force between them would be 9 GN (equivalent to the weight of 900 million kg).

The size of the force is the same all over the surface area of a sphere (4πr2).

To emphasise this spherical symmetry, Coulomb's law is often written as:

The constant ε0 (= 8.85x10-12) is called the permittivity of free space i.e. it says how much electric force is permitted through air.

If there is an insulator or dielectric between the charges, then the electric force is reduced by a factor that is called the dielectric constant k , ie

Sample Dielectric constants

 Material Dielectric constant k Temperature 0C Vacuum 1 Dry Air (1 Atm) 1.00054 20 Carbon Tetrachloride 2.2 20 Hard Rubber 2.8 20 Plastics 3 - 20 20 Paper 3.5 20 Glass 5 - 10 25 Water 61 80 Water 78 25 Titanium Dioxide 100 20

There is a list of dielectric constants.

Example

Find the force of attraction between a -5 mC charge and a +4 mC charge when they are 30 mm apart in air.

Example

Find the force of repulsion between a +9 mC charge and a +5 mC charge when they are 60 mm apart in Titanium Dioxide.

Electric force as a vector

Electric force is a vector quantity so the direction as well as the size needs to be taken into account.

reminder:

The position vector r is

(r,θ) = (r.cos θ, r.sin θ) = (r.cos θ)i + (r.sin θ)j

Here's a link to review or learn about vectors (Peter's physics page on Vectors)

In vector form, Coulomb's law is:

F12 is the force of charge 1 on charge 2.

has unit length and is directed from charge 1 towards charge 2.

Example

A point charge of +1mC is fixed at the origin, O.

A +2mC point charge is fixed 0.12m from O on the +x axis.

A -3mC point charge is fixed 0.42m from O on the +x axis.

Find the electric force on the +2mC charge.

Where has unit length in the positive r direction.

The resultant force is 1.85 N in the positive x direction.

Example

For the fixed point charges shown in the diagram, find the electric force on the -2mC charge from the other two charges.

The forces between the negative charge and each of the positive charges are attractive.

From Pythagoras' theorem

Summarising:

• Two types of electric charge are needed are needed to explain electric attraction and repulsion.
• Conductors have lots of electric charges that are relatively free to move around within the material.
• Insulators, or Dielectrics, do not have many charges that move easily and so you can have an imbalance of charge at different places in the material.
• The size of the electric force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of their separation and acts along the line joining them.

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