ELEC 3105 Lecture 1

Coulomb

4. ElectrostaticsApplied EM by Ulaby, Michielssen and Ravaioli

Chapter 4 Overview

Maxwell’s Equations

God said:

And there was light!

Current Density

For a surface with any orientation:

J is called the current density

ELEC 3105 Lecture 1

Coulomb’s Law

Electric field at point P due to single charge

Electric force on a test charge placed at P

Electric flux density D

Coulomb’s force law (point charges)q1

q2 F

origin

1r

2r

1212 rrr

122

12

2112 r

r

qkqF

[F]-force; Newtons {N}

[q]-charge; Coulomb {C}

[r]-distance; meters {m}

[]-permittivity; Farad/meter {F/m}

Property of the medium

Coulomb’s force law (permittivity)

o

mediumr

Relative permittivity

omedium 0006.1For a medium like air

Coulomb’s force law (permittivity)

omedium

122

12

2112 r

r

qkqF

medium

k41

FORCE IN MEDIUM SMALLER THAN FORCE IN VACUUM

Lecture 1 (ELEC 3105)Basic E&M and Power

Engineering

Coulomb's LawThe force exerted by one point charge on another acts along the line joining the charges. It varies inversely as the square of the distance

separating the charges and is proportional to the product of the charges. The force is repulsive if the charges have the same sign and attractive if the

charges have opposite signs.

Action at a distance

Electric Field Due to 2 Charges

Example of (4.18) next

Electric Field due to Multiple Charges

Electric field (charge distribution)

x

y

z

q1

q2

P

N

i i

ii

rr

rrqkE

13

Large number N of point charges

q3

q4

q5

qN

qi

ir irr

r

Given a group of charges we find the net electric field at any point in space by using the principle of superposition. This is a general principle that says a net effect is the sum of the individual effects. Here, the principle means that we first compute the electric field at the point in space due to each of the charges, in turn. We then find the net electric field by adding these electric fields vectorially, as usual.

PRINCIPLE OF SUPERPOSITION

Charge Distributions

Volume charge density:

Total Charge in a Volume

Surface and Line Charge Densities

Electric Field Due to Charge Distributions

Field due to:

Electric field (charge distribution)

qCharge always occurs in integer multiples of the electric charge e = 1.6X10-19C.

It is often useful to imagine that there is a continuous distribution of charge

Charged volume

Charged surface

Charged line

Electric field (charge distribution)

q

The electric field at the point P is obtained by summing the electric field contribution from from each volume element dV.

Charged volume

P

Charge volume element dV

V Volume charge density

V Units; {C/m3 }

dVV Charge in dV

When the volume element dV--> 0

Sum --> Integral

Electric field (charge distribution)

Charged volume

P

VdV r

Ed

V

Field for one element

2r

kdqrEd

dVdq VWith

2r

dVkrEd V

Integration overvolume V

V

V

V r

dVkrEdE 2

Electric field (charge distribution)

V

V

V r

dVkrEdE 2

,.....dxdydzdV

,....222 zyxr

V may be a function of the coordinates usually a constant

41

k usually a constant when medium is uniform

unit vectorfunction of (x,y,z),….

Electric field (charge distribution)

The electric field produced at the point P is:

Charged surface

Charge surface element dS

s Surface charge density

s Units; {C/m2}

dSs Charge on dS

P

S

s

S r

dSkrEdE 2

q

dS

Electric field (charge distribution)

,.....dxdydS

,....222 zyxr

s may be a function of the coordinates usually a constant

41

k usually a constant when medium is uniform

unit vectorfunction of (x,y,z),….

S

s

S r

dSkrEdE 2

Electric field (charge distribution)

The electric field produced at the point P is:

Charged line element d

Linear charge density

Units; {C/m}

d Charge on

P

LL r

dkrEdE 2

Charged line

q

d

d

Electric field (charge distribution)

,.....dxd

,....222 zyxr

may be a function of the coordinates usually a constant

41

k usually a constant when medium is uniform

unit vectorfunction of (x,y,z),….

LL r

dkrEdE 2

Cont.

Cont.

Example 4-5 cont.