Definition of Electric Field

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Definition of Electric Field

• If we know at , find for any charge
  at
• We assume that the test charge used to find
  is small so it does not change

Better definition of electric field
2
Electric Field for Point Charges

• is a vector! Must add x, y, z, components
• Apply later to continuously charged objects by
  breaking object up into several small pieces each
  with a charge ?qi

Electric Force , acting on a charge q is
3
Electric Line of Force

• Draw lines pointing in the direction of at
  various points in space
• Lines show the direction a positive point charge
  would move if placed at that point
• Used to represent but they are NOT
• Lines start on charges, end on - charges
• No two field lines cross
• of lines drawn leaving a charge is
  proportional to magnitude of the charge
• of lines drawn per unit area through a surface
  perpendicular to the lines is proportional to
  strength of E field at that region
• Stronger E field means field lines are closer
  together

4
Examples of Electric Field Lines

• The tangent to the field line gives the
  direction of E field
• The SHELL THEOREM allows us to draw the electric
  field for a charged sphere (effective charge is
  at the center)
• The electric field lines are perpendicular to
  the surface of conductors

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Principle of Superposition
Total electric field is
Example Line of charges
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Electric Field for Charge Distributions

• Break object up into small pieces each with a
  charge ?qi
• Set up an integral to sum up fields ?Ei for each
  ?qi apply symmetry to simplify

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Acceleration of Charged Particles

• Consider a particle of mass m and charge q
• Use exactly the same equations as in Mechanics
• Newtons Law

• If there is an electric field
• These forces may depend on velocity and usually
  one can apply the Principle of Superposition
• Later we will add Magnetic forces to this list
• Magnetic forces depend on velocity

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Inkjet Printer
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Gausss Law

• Coulombs law
• Applies to all electrostatics
• Valid but not very practical !
• Why?
• Evaluation of tedious integrals
• No obvious simplifications

Some use of symmetries (Cancellation of
transverse components for the ring of charge)
Gauss law lt---gt Coulombs law
Very useful for systems with high symmetry
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Gaussian Surfaces
A Gaussian surface is any CLOSED surface

• A Gaussian surface defines a VOLUME with an
  inside and an outside
• AREA is a VECTOR quantity whose direction is in
  the OUTWARD sense and NORMAL to the surface
• Useful Gaussian surfaces (define a volume)
• sphere (point charges)
• cylinder (line of charge)
• Gaussian surface will conform to any symmetry
  exhibited by a system

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Concept of Electric Flux and Electric Flux
Density

• Consider Gaussian surface in broken into
  small section of area , and electric
  flux density
• Define electric flux

• Electric flux contributions from individual
  areas adds up to total electric flux

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Gausss Law

• Gausss law says
• The NET electric flux through any closed surface
  (Gaussian surface) is ALWAYS the same as the
  charge inside.

• qvol is the net charge contained in the volume
  defined by the Gaussian Surface
• D is the field due to ALL charges in space
• Location of the charges in the Gaussian surface
  is irrelevant

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Gauss Law and Coulombs Law

• Can we derive one from the other?

Spherical Gaussian surface centered at q From
the symmetry of the situation the area vector and
electric field vector are perpendicular to the
surface
Gauss law is
E is a constant for a given radius r. The
integral is the area of a sphere, then
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Gauss Law and Coulombs Law

• Gauss law from Coulombs law

Integrae Coulombslaw law as following,
For qi inside S,
For qi outside S,
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Charged Isolated Conductor

• Like charges repel.
• Charges in a conductor can move
• (as far from each other as possible)
• An excess charge placed on an isolated conductor
  will move entirely to the surface of the
  conductor
• None of the excess charge will be found within
  the body of the conductor

• The equilibrium field inside a conductor MUST be
  ZERO!
• Why?
• If there is a field the conduction electrons
  would move and currents would exist
• If conduction electrons do not move - The
  electric field MUST be ZERO.

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How to apply Gauss Law

• It is very useful in finding E when the physical
  situation exhibits massive SYMMETRY.
• To solve the equation, you have to be able to
  CHOOSE a closed surface such that the integral is
  TRIVIAL
• Direction surface must be chosen such that E
  is known to be either parallel or perpendicular
  to each piece of the surface
• Magnitude surface must be chosen such that E
  has the same value at all points on the surface
  when E is perpendicular to the surface
• Therefore that allows you to bring E outside
  of the integral

• Remember!
• Principle of Superposition
• Shell theorems (Spherical symmetry)

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Applications of Gauss Law

• Use Gauss law where there is easily identified
  Symmetry of the charge distribution
• Cylindrical symmetry (axis aligned with charge)
• lines of charges
• Cylindrical symmetry (axis perpendicular to
  surface)
• planar systems
• Spherical symmetry for
• point charges
• spherical surfaces

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Divergence Theorem and Differential Form of
Gauss Law
Assume the charge distribution ? inside S,
Apply the divergence theorem,
Then,
Differential form of Gauss law becomes