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**Electrostatic Fields 1**

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__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!

__Coulomb's Law__

The force * F* between two point charges

*is directly proportional to the product of the charges and inversely proportional to the square of the distance*

**Q**&_{1 }**Q**_{2 }*between them.*

**r**

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

The value of ** k** is given by :

Hence the Coulomb's Law equation becomes :

The quantity ** ε** is called the

**permittivity**. It has a value depending on the medium surrounding the charges.

The symbol ** ε_{o}** (epsilon nought) is used to describe the

**permittivity of free space**,

i.e. a vacuum.

The value of ** ε_{o}** can be found from 'Maxwell's equations'. The result includes another constant

*(mu nought).*

**μ**_{o}

* μ_{o}* is called the

**magnetic permeability**of free space.

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

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}**

__Electric Field Strength E__

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

From Coulomb's Law,

where,

* Q_{T}* is a unit test charge (1 C )

* Q_{P}* is any charge at a point

By definition, electric field strength is force/unit charge. So at the point where charge * Q_{T}* is positioned the field strength

*is given by :*

**E**

But * Q_{T}* is a unit charge, therefore

*=*

**E***.*

**F**

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

We can now see how electric field strength * E* varies with distance

*from the point.*

**r**

__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.**

From Coulomb's law, the force between two point charges * Q_{1}* and

*is given by :*

**Q**_{2}

* Q_{1}* is positive at point

**S**.

*is also positive but at point*

**Q**_{2}**T**.

* Q_{1}* exerts a repulsive force

*on*

**F***.*

**Q**_{2}* Q_{2}* exerts a repulsive force

*on*

**F***.*

**Q**_{1}

Consider an external force moving * Q_{2}* at

**T**an infinitesimal distance

*towards*

**δx***at*

**Q**_{1}**S**.

Because the distance* δx* is so small, the repulsive force

*may be considered to be constant during the movement.*

**F**

Using,

**work = force x distance force moves**

the work done* δW* is given by :

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) :

Therefore the total work * W* done in bringing the charge

*from infinity to a point a distance*

**Q**_{2}*from*

**r****S**(where

*) is given by:*

**x = r**

integrating between the limits of * r* and infinity,

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

__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. **

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. **

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

*and potential difference*

**Q***is as follows :*

**V**

(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 V_{1 } to V_{2 } .

However, work **-W** must be done __against the field__ to return the charge back to V_{1 } .

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

__Relation between E and V__

Consider a charge * +Q* being moved by a force

*from an arbitrary point*

**F****A**to another point

**B**against an electric field of strength

*.*

**E**

The distance moved, * δx* , is very small, such that the force

*may be considered constant.*

**F**Hence the work done * δW * by the force is :

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

Substituting in the original equation for * F* gives :

From the definition of potential difference, * W = QV *.

Therefore, if the potential difference between **A** & **B** is * δV *:

(

**V**_{B}>

**V**_{A})

Substituting for * δW * ,

cancelling the * Q*'s and rearranging,

In the limit as * δV *and

*tend to zero,*

**δx**

Multipling both sides by -1 :

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.

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