I-4 Simple Electrostatic Fields

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I-4 Simple Electrostatic 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 lines are radial
  and have a spherical symmetry

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A Spherically Symmetric Field II

• The magnitude of E depends only on r
• Lets move a test charge q equal to unity from
  some point A to another point B. We study
  directly the potential! Its change actually
  depends only on changes of the radius. 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
• If we move a non-unity charge q we have again to
  deal with its potential energy

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The General Formula of

• The general formula is very simple
• Gradient of a scalar function 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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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

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in Centrosymmetric Fields

• When the field has a spherical symmetry the
  general formulas simplify to
• and
• 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 field lines.
• In uniform fields equipotentials are planes
  perpendicular to the field lines.
• 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
• In non-relativistic case

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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 a 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
• Accelerations of elementary particles can be
  enormous!

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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
  energy and takes the advantage of the existence
  of the potential energy.

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Motion IV Energetic Approach

• If in the electrostatic field a free charged
  particle is at a certain time in a point A and
  after some time we find it in a point B and work
  has not been done on it by an external agent,
  then 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
  and vice versa
• In high energy physics 1eV is used as a unit of
  energy 1eV 1.6 10-19 J.

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Motion of Charged Particles in Electrostatic
Fields II

• It is simple to calculate the gain in kinetic
  energy of accelerated particles from
• When accelerating electrons by few tens of volts
  we can neglect the original speed.
• But relativistic speeds can be reached at easily
  reached voltages!

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Homework

• The homework from yesterday is due Monday!

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Things to read

• This lecture covers
• Chapter 21-10, 23-5, 23-8
• Advance reading
• The rest of chapters 21, 22, 23

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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 .

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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 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 e and p I

• What is the acceleration of an electron and a
  proton in the electric field E 2 104 V/m ?
• ae E q/m 2 104 1.76 1011 3.5 1015 ms-2
• ap 2 104 9.58 107 1.92 1012 ms-2
• J/Cm C/kg N/kg m/s2

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The Acceleration of an Electron II

• What would be the speed of an electron, if
  accelerated from zero speed by a voltage
  (potential difference) of 200 V?
• Thermal motion speed 103 m/s can be neglected
  even in the case of protons (vp 1.97 105 m/s)!

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Relativistic Effects When Accelerating an Electron

• Relativistic effects start to be important when
  the speed reaches about 10 of the speed of light
  c/10 3 107 ms-1.
• What is the accelerating voltage to reach this
  speed?
• Conservation of energy mv2/2 q V
• V mv2/2e 9 1014/4 1011 2.5 kV !
• A proton would need V 4.7 MV!

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Relativistic Approach I

• 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

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Relativistic Approach II

• The speed is usually expressed in multiples of
  the c by means of ? v/c. Since ? is very close
  to 1 a trick has to be done not to overload the
  calculator.

So for ? we have

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Example of Relativistic Approach

• Electrons in the X-ray ring of the NSLS have
  kinetic energy Ek 2.8 GeV. What is their speed.
  What would be their delay in arriving to
  ?-Centauri after light?

E0 0.51 MeV for electrons. So ? 5491 and v
0.999 999 983 c. The delay to make 4 ly is dt
2.1 s ! Not bad and the particle would find the
time even shorter!!