Electrostatic Fields:

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CHAPTER 2
ELECTROMAGNETIC FIELDS THEORY

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

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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

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Electric Potential

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

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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
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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
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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
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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
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Electric Potential (due to point charge)

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

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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

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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.

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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

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Electric Potential

• For continuous charge distributions

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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

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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.

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Solution

• Let

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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

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2nd Maxwell Equation

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

2nd Maxwell Equation
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Relationship between E and V

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

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Relationship between E and V

• Thus

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

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Solution

• (a)
• But
• at point (2, p/2, 0)

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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

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Energy Density in Electrostatic Field

• Lets say the order is reversed, thus

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Energy Density in Electrostatic Field
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Energy Density in Electrostatic Field

• So we can say

The unit for W is joules
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Energy Density in Electrostatic Field
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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

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Energy Density in Electrostatic Field

• The first integral tends to zero an the surface S
  becomes large. Thus, the equation is reduced to

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Energy Density in Electrostatic Field

• So the electrostatic energy density can be
  defined as

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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

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Solution
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