Coulomb's Law Electric Field Intensity Field due to a sheet charge Differential VOltage Electric Potential Electric Field of an Electric Dipole

·                    Coulomb's Law

Coulomb found out that each electric point charge exerts a mechanical force on the other, and this force exhibits five characteristics as follows:

1.      The force is proportional to the product of the magnitudes Q1 and Q2 of the two electric charges.

2.      Like charges repel each other while like charges attract each other.

3.      The force is inversely proportional to the square of the distance between the two charges.

4.      The force depends upon the medium in which the two charges are located.

5.      The force always acts along the straight line joining the two point charges Q1 and Q2.

Mathematically:

in MKS :

k        = (4peo)-9

eo      = permittivity of free space

= (1/36p) x 10-9 F/m  = 8.854 pF/m

F                 = Force (N)

Q1 and Q2          = charges (C)

1 coul        = 6.25 x 1018 e

1 e              = 1.6 x 10-19 coul

Note:

I.        The distance R between the charge bodies Q1 and Q2 must be large compared to the linear dimension of the bodies.

II.       Q1 and Q2 must be static.

If we have more than 2 point charges, we can use the principle of superposition to determine the force on a particular charge say Q1.

·                    Electric Field Intensity

Electric Field Intensity is defined as the force per unit charge when placed in the electric field.

where:

QT     - test charge

E        - electric field intensity (V/m)

a1t     - unit vector in the direction of R1T

###### Electric Field due to Several Charges

·                    Field due to a Line Charge

Consider a filament like distribution of volume charge density such as a very fine, sharp beam in a cathode ray tube or a charged conductor of very small radius, we find it convenient to treat the charge as a line charge of density rL (C/m).

subs (2) and (3) in (1)

r        - perpendicular distance of the line charge to the point

ar      - direction of the perpendicular line connecting the line charge to the point

·                    Field due to a Sheet Charge

This charge distribution may often be used to approximate that found on the conductors of a strip transmission line or a parallel plate of capacitor.

At z = z plane:

dEx and dE1x, dEy and dE1y will cancel each other

In general, E due to an infinite sheet of charge:

ar      - direction of the smallest distance connecting the sheet of charge to the point in consideration.

·      Electric Potential Voltage (Voltage Potential of Voltage)

- amount of work, or potential energy, required to move a unit charge between two points.

-the presence of electric field between two points results to the voltage difference between them.

Consider a positive charge "q" in a uniform electric field parallel to the negative y-direction.

Fe = qE (force exerted on the charge in the – y direction)

Fext = -q(-Eay)(dy ay) (force needed to move the charge in the +y direction)

dw = Fext · dl = -qE · dl (joules)

dw = -q (-Eay)(dyay) = qEdy (work or energy needed if a charge is movd a distance dy along ay)

·       Differential electric potential (or differential voltage)

dv = dw/q = -E · d(Joules/C or V)

·                    Potential difference between two points P2 and P1

(a) Electric Dipole

(b) Electric Field Pattern

line integral of  around any closed contour is zero.

Assume P1 is the ground, V1= 0 P1 = ¥

(Electric potential at any point)

·                    Electric Potential

let P1 = R and dl = dR aR

(due to a point charge)

(due to several charges)

(due to volume charge specified over a                       given volume v’)

(due to surface charge specified over a                       given volume s’)

(due to line charge specified over a                       given volume l’)

·                    Electric Field as a Function of Electric Potential

dv = -E dl  also   dv = ÑV dl

E = -ÑV

·                    Electric Field of an Electric Dipole

Electric Dipole consists of two point charges of equal magnitude and opposite polarity, separated by a small distance

since d<<R

assumptions: R1||R2  \R2 – R1 @ dcosq

R1R2 @ R2

qdcosq = qd · aR = p · ar

d – distance vector from –q to +q

aR – unit vector pointing from the center of the dipole toward the observation point P.

P = qd – dipole moment of the electric dipole

from E = -ÑV

(in spherical coordinates)