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Fields & Effects

 

Electrostatic Fields 1

 

law of electrostatics

Coulomb's law

field strength

potential

potential difference

E-V relation

 

 

 

Law of Electrostatics

 

 

the law of electrostatics

 

 

Like charges repel, while unlike charges attract.

 

 

The convention for charge relates to early electrostatic experiments with particular materials.

 

glass rubbed with silk gives a positive charge

 

ebonite rubbed with fur gives a negative charge

 

Using these results other materials and particles are assigned charge:

 

perspex(positive), polythene(negative),

 

electrons(negative), protons(positive) etc.

 

The decision to make the charge from glass and silk positive, and the charge from ebonite and fur negative was completely arbitrary. The two charges could easily have been transposed, resulting in a positive charge on the electron!

 

 

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Coulomb's Law

 

 

Coulomb's law

 

 

The force F between two point charges Q1 & Q2 is directly proportional to the product of the charges and inversely proportional to the square of the distance r between them.

 

Coulomb's Law - equation #1

 

Making the proportionality into an equation by introducing a constant k :

 

Coulomb's Law - equation #2

 

The value of k is given by :

 

Coulomb's law - equation #3

 

Hence the Coulomb's Law equation becomes :

 

 

Coulomb's Law - equation #4

 

 

The quantity ε is called the permittivity. It has a value depending on the medium surrounding the charges.

 

The symbol εo (epsilon nought) is used to describe the permittivity of free space,

i.e. a vacuum.

 

The value of εo can be found from 'Maxwell's equations'. The result includes another constant μo (mu nought).

 

μo is called the magnetic permeability of free space.

 

The two are connected by the following relation, where c is the velocity of light :

 

Coulomb's Law - equation #5

 

 

The unit of charge is the coulomb (C). This is a S.I. derived unit.


By definition, a coulomb is the charge passed when a current of 1 ampere flows for 1 second.

 

1 C = 1 AS-1

 

 

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Electric Field Strength E

 

The electric field strength at a point is equal to the force on a unit charge at the point.

 

 

electric field strength E

 

 

From Coulomb's Law,

 

electric field strength - equation #1

 

where,

 

QT is a unit test charge (1 C )

QP is any charge at a point

 

 

By definition, electric field strength is force/unit charge. So at the point where charge QT is positioned the field strength E is given by :

 

electric field strength - equation #2

 

But QT is a unit charge, therefore E = F .

 

Substituting for F in the initial Coulomb's Law equation,

 

electric field strength - equation #3

 

We can now see how electric field strength E varies with distance r from the point.

 

 

E - r graph

 

 

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Electric Potential V

 

The electrical potential V at a point in an electric field is numerically equal to the

 

work done W in transferring a unit positive charge from infinity to the point.

 

 

definition of potential V

 

 

From Coulomb's law, the force between two point charges Q1 and Q2 is given by :

 

Coulomb's Law - equation #4

 

 

derivation of potential V in terms of Q and r

 

 

Q1 is positive at point S. Q2 is also positive but at point T.

Q1 exerts a repulsive force F on Q2.

Q2 exerts a repulsive force F on Q1 .

 

 

Consider an external force moving Q2 at T an infinitesimal distance δx towards Q1 at S.

 

Because the distance δx is so small, the repulsive force F may be considered to be constant during the movement.

 

Using,

work = force x distance force moves

 

the work done δW is given by :

 

potential V - equation #1

 

The negative sign indicates that work is done against the field. That is, the motion is in the opposite direction to the direction of repulsion.

 

Substituting into this equation for F, from the Coulomb's Law equation (above) :

 

potential V - equation #2

 

Therefore the total work W done in bringing the charge Q2 from infinity to a point a distance r from S (where x = r) is given by:

 

potential V - equation #3

 

integrating between the limits of r and infinity,

 

potential V - equation #4

 

potential V - equation #5

 

potential V - equation #6

 

 

 

V-r graph

 

 

The curve follows a simple inverse relation similar to y = 1/x .

 

 

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Electric Potential Difference

 

The potential difference between two points in an electric field is numerically equal to

 

the work done in moving a unit + charge from the lower potential point to the higher.

 

 

definition of potential difference

 

 

From the definition of potential comes the definition of the volt, with specific units for work and charge.

 

The potential difference between two points is 1 volt if one joule of energy is used in

 

moving 1 Coulomb of charge between the points.

 

 

definition of the volt

 

 

Simply put, the number of volts is equal to the energy involved in moving 1 Coulomb of charge between points. So a 12V battery produces 12J of energy for every Coulomb moved between its terminals.

 

The equation connecting work W, charge Q and potential difference V is as follows :

 

potential difference V - equation #1

 

(joules = Coulombs x volts)

 

 

The diagram below illustrates that:

 

the work done in taking charge around a closed loop is zero.

 

Work W is done by the electric field in moving the charge from V1 to V2 .

 

However, work -W must be done against the field to return the charge back to V1 .

 

So the sum amount of work done = W + (-W) = 0

 

 

pd paths and work done

 

 

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Relation between E and V

 

Consider a charge +Q being moved by a force F from an arbitrary point A to another point B against an electric field of strength E.

 

 

E-V derivation of relation

 

 

The distance moved, δx , is very small, such that the force F may be considered constant.

Hence the work done δW by the force is :

 

E related to V - equation #1

 

The force is equal to the force exerted by the field on the charge, but in the opposite direction(note negative sign).

 

derivation E-V relation - equation #1

 

Substituting in the original equation for F gives :

 

E-V derivation relation - eqaution #2

 

From the definition of potential difference, W = QV .

 

Therefore, if the potential difference between A & B is δV :
( VB > VA )

 

E-v derivation relation - equation #3

 

Substituting for δW ,

 

E-V derivation relation - equation #4

 

cancelling the Q's and rearranging,

 

E-V derivation relation - equation #5

 

In the limit as δV and δx tend to zero,

 

E-V derivation relation - equation #6

 

Multipling both sides by -1 :

 

E-V derivation - equation #7

 

The E-r and V-r graphs below show the relation clearly.


The gradient of the V-r graph is negative. So the negative of its gradient gives a positive value for E in the E-r graph.

 

 

E-V relation graph comparison

 

 

 

 

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