I-4 Electric Fields

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I-4 Electric Fields
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Main Topics

• Relation of the Potential and Intensity
• The Gradient
• Electric Field Lines and Equipotential Surfaces.
• Motion of Charged Particles in Electrostatic
  Fields.

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A Spherically Symmetric Field I

• A spherically symmetric field e.g. a field of a
  point charge is another important field where the
  relation between ? and E can easily be
  calculated.
• Lets have a single point charge Q in the origin.
  We already know that the field is radial and has
  a spherical symmetry
• E(r)r0kQ/r2

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A Spherically Symmetric Field II

• The magnitude E depends only on r
• E(r)kQ/r2
• If we move a test charge q equal to unity from
  some point A to another point B. The change of
  potential actually depends only on how the radius
  has changed. This is because during the shifts at
  a constant radius work is not done.

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A Spherically Symmetric Field III

• The conclusion potential ? of a spherically
  symmetric field depends only on r and it
  decreases as 1/r
• ?(r)kQ/r
• If we move a non-unity charge q we have again to
  deal with its potential energy
• U(r)kQq/r

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The General Formula E(?)

• The general formula is very simple
• E - grad(?)
• Gradient of a scalar function f in some point is
  a vector
• It points to the direction of the fastest growth
  of the function f.
• Its magnitude is equal to the change of the
  function f, if we move a unit length into this
  particular direction.

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The Relation E(?) in Uniform Fields

• In a uniform field the potential can change only
  in the direction along the field lines. If we
  identify this direction with the x-axis of our
  coordinate system the general formula simplifies
  to
• E - d?/dx
• F - dU/dx

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The Relation E(?) in Centrosymmetric Fields

• When the field has a spherical symmetry the
  general formula simplifies to
• E - d?/dr
• F - dU/dr
• This can for instance be used to illustrate the
  general shape of potential energy and its impact
  to forces between particles in matter.

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The Equipotential Surfaces

• Equipotential surfaces are surfaces on which the
  potential is constant.
• If a charged particle moves on a equipotential
  surface the work done by the field as well as by
  the external agent is zero. This is possible only
  in the direction perpendicular to the field
  lines.

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Equipotentials and the Field Lines

• We can visualize every electric field by a set of
  equipotential surfaces and fieldlines.
• In uniform fields equipotentials are planes
  perpendicular to the fieldlines.
• In spherically symmetric fields equipotentials
  are spherical surfaces centered on the center of
  symmetry.
• Real and imaginary parts of an ordinary complex
  function has the same relations.

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Motion of Charged Particles in Electrostatic
Fields I

• Free charged particles tend to move along the
  field lines in the direction in which their
  potential energy decreases.
• From the second Newtons law
• d(p)/dt q E
• In non-relativistic case
• ma qE ? a E q/m

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Motion of Charged Particles in Electrostatic
Fields II

• The ratio q/m, called the specific charge is an
  important property of the particle.
• electron, positron q/m 1.76 1011 C/kg
• proton, antiproton q/m 9.58 107 C/kg
  (1836 x)
• ?-particle (He core) q/m 4.79 107 C/kg
  (2 x)
• other ions
• The acceleration of elementary particles can be
  enormous!
• Relativistic speeds can be easily reached!

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Motion of Charged Particles in Electrostatic
Fields III

• Either the force or the energetic approach is
  employed.
• Usually, the energetic approach is more
  convenient. It uses the law of conservation of
  the energy and takes the advantage of the
  existence of the potential energy.

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Motion IV Energetic Approach

• If in the electrostatic point a free charged
  particle is at a certain time in the point A and
  after some time we find it in a point B the total
  energy in both points must be the same,
  regardless of the time, path and complexity of
  the field
• EKA UA EKB UB

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Motion V Energetic Approach

• We can also say that changes in potential energy
  must be compensated by changes in kinetic energy
  
• (EKB - EKA) (UB - UA) 0
• (EKB - EKA) q(?B - ?A) 0
• (EKB - EKA) qVBA 0
• In high energy physics 1eV is used as a unit of
  energy 1eV 1.6 10-19 J.

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Homework

• The homework from yesterday is due Monday!

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Things to read

• Chapter 21-10, 23-5, 23-8

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Potential of the Spherically Symmetric Field A-gtB

• We just substitute for E(r) and integrate

• We see that ? decreases with 1/r !

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The Gradient I

• It is a vector constructed from differentials of
  the function f into the directions of each
  coordinate axis.
• It is used to estimate change of the function f
  if we make an elementary shift dl.

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The Gradient II

• The change is the last term. It is a dot product.
  It is the biggest if the elementary shift dl is
  parallel to the grad.
• In other words the grad has the direction of the
  biggest change of the function f !

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The Acceleration of an Electron

• What is the acceleration of an electron in the
  electric field E 2 104 V/m ?
• a E q/m 2 104 1.76 1011 3.5 1015 ms-2
• J/Cm C/kg N/kg m/s2

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Relativistic Effects When Accelerating an Electron

• Relativistic effects start to be important when
  the speed reaches c/10 3 107 ms-2. What is the
  accelerating voltage to reach this speed?
• Conservation of energy mv2/2 q V
• Vmv2/2e9 1014/4 1011 2.5 kV !

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Relativistic Approach

• If we know the speeds will be relativistic we
  have to use the famous Einsteins formula

E is the total and EK is the kinetic energy, m is
the relativistic and m0 is the rest mass