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CHAPTER 2

ELECTROMAGNETIC FIELDS THEORY

- Electrostatic Fields
- Electric flux and electric flux density
- Gauss's law

In this chapter you will learn

- Electrostatic Fields
- -Charge, charge density
- -Coulomb's law
- -Electric field intensity
- - Electric flux and electric flux density
- -Gauss's law
- -Divergence and Divergence Theorem
- -Energy exchange, Potential difference, gradient
- -Ohm's law
- -Conductor, resistance, dielectric and

capacitance - - Uniqueness theorem, solution of Laplace and

Poisson equation

Electric Potential

- Another way of obtaining E, besides Coulombs law

and Gausss law is using electric scalar

potential, V

Electric Potential

In moving the object from point A to B, the work

can be expressed by

dL is differential length vector along some

portion of the path between A and B

Electric Potential

The work done by the field in moving a charge

from A to B is

If an external force moves the charge against the

field, the work done is negative

Electric Potential

- The work done by an external agent to move a

point charge Q from A to B in an electrical field

E is - Total work done (or potential energy)

Negative sign indicates that the work is done by

an external agent

Electric Potential

- Divide W by Q, we get the electric potential

energy per unit charge or potential difference

between points A and B - A is initial point while B is the final point
- In other word, to find the potential difference

between two points, we must integrate Edl along

the path between point A to point B.

The work done per unit charge by an external

agent to transfer a test charge from infinity to

that particular point

Electric Potential (due to point charge)

- E field due to a point Q at origin is
- So VAB is

Electric Potential

- Or we can write VAB as
- Where VA and VB are the potentials (or absolute

potentials) at B and A, respectively. VA is

potential at B with reference to A. - For point charge, choose reference point at

infinity. - rA ? 8 VA 0, then the potential at any point

(rB ?r) due to point charge Q at origin is

Electric Potential

- E points in radial direction ? displacement in

the ? or F direction is wiped out by the dot

product. - Thus VAB is independent of the path taken.

Electric Potential

- If point charge Q is not located at origin, but

at another point with position vector r, thus

V(x, y, z) or simply V(r) at r becomes - For more than one point charges, the

superposition principle applies

Electric Potential

- For continuous charge distributions

Electric Potential

- If another point is chosen as a reference instead

of infinity, V becomes - where C is a constant that is determined at the

chosen point of reference. - The potential difference VAB can be written as

Example

- Two point charges, -4 µC and 5 µC are located at

(2,-1,3) and (0, 4, -2) respectively. Find

the potential at (1, 0, 1) assuming zero

potential at infinity.

Solution

- Let

2nd Maxwell Equation

- Potential difference is independent of path

taken. - So VBA - VAB
- That is
- Or
- The line integral of E along a closed path must

be zero. - No net work is done when moving a charge along a

closed path in an electrostatic field

2nd Maxwell Equation

- Applying Stokes theorem, becomes
- The field E is conservative, or irrotational

2nd Maxwell Equation

Relationship between E and V

- We know that
- But
- Compare (1) and (2), we get

Relationship between E and V

- Thus

Electric field intensity, E is the gradient of V

with opposite direction of increasing V

Example

- Given the potential
- Find the electric flux density D at (2, p/2, 0)
- Calculate the work done in moving a 10µC charge

from point A (1,30,120) to B(4,90,60)

Solution

- (a)
- But
- at point (2, p/2, 0)

Solution

- (b) The work done can be found using either E or

V

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Energy Density in Electrostatic Field

- The total work done in positioning the three

charges

Energy Density in Electrostatic Field

- Lets say the order is reversed, thus

Energy Density in Electrostatic Field

Energy Density in Electrostatic Field

- So we can say

The unit for W is joules

Energy Density in Electrostatic Field

Energy Density in Electrostatic Field

- Thus we can apply the identities and get
- Apply divergence theorem on the first term on the

right-hand side of this equation, we get

Energy Density in Electrostatic Field

- The first integral tends to zero an the surface S

becomes large. Thus, the equation is reduced to

Energy Density in Electrostatic Field

- So the electrostatic energy density can be

defined as

Example

- Three point charges -1 nC, 4 nC, and 3 nC are

located at (0,0,0), (0,0,1) and (1,0,0)

respectively. Find the energy in the system

Solution

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