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Electrostatic Fields:

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dL is differential length vector along some portion of the path between A and B ... (b) The work done can be found using either E or V. Chapter 1 ... – PowerPoint PPT presentation

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Title: Electrostatic Fields:


1
CHAPTER 2
ELECTROMAGNETIC FIELDS THEORY
  • Electrostatic Fields
  • Electric flux and electric flux density
  • Gauss's law

2
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

3
Electric Potential
  • Another way of obtaining E, besides Coulombs law
    and Gausss law is using electric scalar
    potential, V

4
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
5
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
6
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
7
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
8
Electric Potential (due to point charge)
  • E field due to a point Q at origin is
  • So VAB is

9
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

10
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.

11
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

12
Electric Potential
  • For continuous charge distributions

13
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

14
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.

15
Solution
  • Let

16
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

17
2nd Maxwell Equation
  • Applying Stokes theorem, becomes
  • The field E is conservative, or irrotational

2nd Maxwell Equation
18
Relationship between E and V
  • We know that
  • But
  • Compare (1) and (2), we get

19
Relationship between E and V
  • Thus

Electric field intensity, E is the gradient of V
with opposite direction of increasing V
20
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)

21
Solution
  • (a)
  • But
  • at point (2, p/2, 0)

22
Solution
  • (b) The work done can be found using either E or
    V

23
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25
Energy Density in Electrostatic Field
  • The total work done in positioning the three
    charges

26
Energy Density in Electrostatic Field
  • Lets say the order is reversed, thus

27
Energy Density in Electrostatic Field
28
Energy Density in Electrostatic Field
  • So we can say

The unit for W is joules
29
Energy Density in Electrostatic Field
30
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

31
Energy Density in Electrostatic Field
  • The first integral tends to zero an the surface S
    becomes large. Thus, the equation is reduced to

32
Energy Density in Electrostatic Field
  • So the electrostatic energy density can be
    defined as

33
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

34
Solution
35
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