P (x, y, z)

P3 (-1, -1, 0)

Chapter 2

Coulomb’s law and electric field

intensity

e

e

e

e

P2 (-1, 1, 0)

P1 (1, 1, 0) P4 (1, -1, 0)

P (1, 1, 1)

r-r4

r-r1

r-r2r-r3

(6, 8, z)

(6, 0, 0)

(x, y, z)

(0, 8, 0)

ρL

y

z

x

R = √ x2 +y2P (x, 0, 0)

y

z

x

Y’

2.1 Law of coulomb* Coulomb state the fore between two small objects separated in a vacuum or free space by a distance which is large compared to their size is proportional to the charge on each and inversely proportional to the square of the distance between them.

If the SI units are used, Q is measured in coulombs (C), R is in meters (m), and the force is in Newton (N). Whereas k is the proportionality constant: The new constant is called the permittivity of free space and has a magnitude, measured in farads per meter (F/m).

SI units

Cgs units

EXAMPLE;

A 2nC positive charge is located in vacuum at P1 (3, -2, -4) and a 5µC negative charge at P2 (1, -4, 2). Find the vector force on the negative charge, and the magnitude of the force on the charge at P1.

x

z

yP2 (1, -4, 2)

P1 (3, -2, -4)

2.2 Electric field intensityIs defined as the force per unit charge that would be experienced by a stationary point charge at a given location in the field:

where F is the electric force experienced by the particle.q is its charge, E is the electric field wherein the particle is located.

On Coulomb's Law for, the contribution to the E-field at a point in space due to a single, discrete charge located at another point in space is given by the following: q is the charge of the particle creating the electric force, r is the distance from the particle with charge q to the E-field evaluation point, is the unit vector pointing from the particle with charge q to the E-field evaluation point, Is the electric constant.

Thus;

This expression can be reduced when we use a summation sign ∑ and a summing integer m which takes on all integral values between 1 and n,

E(r)

P

r – r2

Q2

r2

r – r1Q1

r1

y

x

z

E2

a2

r

a1 E1

The vector addition of the total electric field intensity at P due to Q1 and Q2 is made possible by the linearity of Coulomb’s law

EXAMPLE;

Find E at P (1, 1, 1) caused by four identical 3-nC charges located at P1 (1, 1, 0), P2 (-1, 1, 0), P3 (-1, -1, 0), and P4 (1, -1, 0).

P1 (1, 1, 0)

P4 (1, -1, 0)

P3 (-1, -1, 0)P2 (-1, 1, 0)

P (1, 1, 1)

r-r4 r-r1

r-r3 r-r2

y

x

z

FIELD DUE TO CONTINUOUS VOLUME CHARGE DISTRIBUTION

z + dz

z

ρ + dρρ

ρФ

ρ ( Ф + dФ)

dρ = (ρ + dρ) - ρ

dz = (z + dz) -z

ρ [ (Ф + dФ) – Ф ] = ρdФ

Cylindrical

V =

Volume charge density ρv = ∆Q/∆V C/m3 ρv =

V = b x w x h

y

z

x

SPHERICAL COORDIANTES

EXAMPLE;

Find the total charge contained in a 2-cm length of the electron beam.

Z =4cm

Ρv = -5e-105 ρz µC/m3

Z = 2cm

ρ = 1cm

y

z

x

But;FIELD OF A LINE CHARGE

P (0, 0, Y) y

R = r – r’

z

x

r’

(0, 0, z’)ar

dz

dQ = PLdz’

ρL

dEz

dE

EXAMPLE;

1. An infinite long uniform line charge is located at y=3, z=5. If PL = 30 µC/m, find E at Pb (0, 0, 1).

2. A uniform line charge density of 20 nC lies on the z – axis between z = 1 and z = 3 m no other charge present. Find E at the origin.

2.5 FIELD OF A SHEET OF CHARGE• Infinite sheet of charge is used to approximate

that found on the conductors of strip transmission line. ρs is commonly know as surface charge density.

• The line charge density, or charge per unit length, is ρL = ρsdy’, and the distance from this line charge to our general point P on the x axis is R = √ x2 + y2

• Ex at P from this differential-width strip is then

Adding the effects of all the strips;

If the point P is chosen on the negative x axis, then;

For the field is always directed away from the positive charge. In addition of a unit vector aN the sign now will be +;

Y’

dy’ρs

R = √ x2 + y2

x

y

z

P (x, 0, 0)

EXAMPLE;

In the rectangular region -2<x<2, -3<y<3, z=0. The surface charge density is given by ρs = (x2 + y2)3/2 ,If no other change present. Find E at P (0, 0, 1).

STREAM LINE* continuous lines from the test charge which are, at every point, tangential to E and are an indication of the direction of E.* also called flux lines or direction lines.* associated with the arrows which are used to show the directions of E.

Two properties of stream lines.1. these are the continuous lines from the line charge which only shows the direction of E and are everywhere tangent to E.2. The magnitude of the field can be shown to be inversely proportional to the spacing of the streamlines.

Equations of stream lines.

Cartesian.

Cylindrical.

Spherical.

EXAMPLE;

Given the electric field E = (4x – 3y)āx - (2x + 4y)āy . Find the equation of that streamline passing through the point P (2, 3, -4) and a unit vector āE specifying the direction of Ē at Q (3, -2, 5).