III

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III2 Magnetic Fields Due to Currents
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Main Topics

• Forces on Moving Electric Charges
• Biot-Savart Law
• Amperes Law.
• Calculation of Some Magnetic Fields.

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Forces on Electric Currents III

• From the formula describing force on electric
  currents the units can be derived.
• The SI unit for the magnetic intensity B is 1
  Tesla, abbreviated as T, 1T 1 N/Am
• Some older are units still commonly used for
  instance 1 Gauss 1G 10-4 T

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Forces on Electric Currents IV

• Now, we can qualitatively show that two parallel
  currents will attract and the force will be in
  the straight line which connects these currents.
  This seems to be similar to a force between two
  point charges but now the force is the result of
  a double vector product as we shall see very soon.

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Forces on Moving Electric Charges I

• Since currents are in reality moving charges it
  can be expected that all what is valid for
  interaction of magnetic fields with currents will
  be valid also for moving charges.
• The force of a magnetic field B acting on a
  charge q moving by a velocity v is given by the
  Lorentz formula
• F q(v x B)

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Forces on Moving Electric Charges II

• Lorentz force is in fact part of a more general
  formula which includes both electric and magnetic
  forces
• F qE (v x B)
• This relation can be taken as a definition of
  electric and magnetic forces and can serve as a
  starting point to study them.

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Forces on Moving Electric Charges III

• Lorentz force is a central issue in whole
  electromagnetism. We shall return to it by
  showing several examples. Moreover we shall find
  out that it can be used as a basis of explanation
  of almost all magnetic and electromagnetic
  effects.
• But at this point we need to know how are
  magnetic fields created quantitatively.

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Biot-Savart Law I

• There are many analogies between electrostatic
  and magnetic fields and of course a question
  arises whether some analog of the Coulombs law
  exists, which would describe how two short pieces
  of wires with current would affect themselves. It
  exists but it is too complicated to use. For this
  reason the generation and influence of magnetic
  fields are separated.

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Biot-Savart Law II

• All what is necessary to find the mutual forces
  of two macroscopic wires of various sizes and
  shapes with currents is to employ the principle
  of superposition, which is valid in magnetic
  fields as well and integrate.
• It is a good exercise to try to make a few
  calculations but we do something better!

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Ampères Law

• As in electrostatics also in magnetism a law
  exists which can considerably simplify
  calculations in cases of a special symmetry and
  can be used to clarify physical ideas in many
  important situations.
• It is the Ampères law which relates the line
  integral of B over a closed path with currents
  which are surrounded by the path.

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Magnetic Field Due to a Straight Wire I

• As it is the case with using the Gauss law, we
  have to find a path which is tangential to B
  everywhere and on which the magnitude of B would
  be constant. So it must be a special field line.
  Then we can move B out of the integral, which is
  then simply the length of the path.

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Magnetic Field Due to a Straight Wire II

• Let us have a straight wire with current I.
• We expect B(r) and axial symmetry where the wire
  is naturally its axis. The field lines are
  circles and our path will be a circle with the
  radius r equal to the distance where we want to
  find the field. Then simply
• 2?r B(r) ?0I ?
• B(r) ?0I/2?r

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Magnetic Field Due to a Straight Wire III

• So the vectors of the magnetic induction B are
  tangents to circles centered on the wire, which
  thereby are the field lines, and the magnitude of
  B decreases with the first power of the distance.
  It is similar as with the electrostatic field of
  an straight, infinite and uniformly charged wire.

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Force Between Two Straight Wires I

• Let us have two straight parallel wires in which
  currents I1 and I2 flow in the same direction
  separated by a distance d.
• First, we can find the directions and then simply
  deal only with the magnitudes. It is convenient
  to calculate a force per unit length.
• F/l ?0/2? I1I2/d

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Force Between Two Straight Wires II

• This is used for the definition of 1 ampere
• 1 ampere is a constant current which, if
  maintained in two straight parallel conductors of
  infinite length, of negligible cross section, and
  placed 1 meter apart in vacuum, would produce
  between these conductors a force equal to 2 10-7
  N per meter of length.

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Homework

• No homework!

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Things to read

• Chapter 28 1, 2, 3, 4,6

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Magnetic interaction of two currents I
Let us have two currents I1 and I2 flowing in two
pieces of wire dl1(r1) and dl2(r2). Then the
force acting on the second piece due to the first
piece, can be described by
This very general formula covers almost all the
magnetism physics but would be hard to use in
practice.
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Magnetic interaction of two currents II
That is the reason why it is divided into the
formula using the field (we already know)
and the formula to calculate the field, which is
the Biot-Savart law
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Magnetic interaction of two currents III
If we realize that
is a unit vector pointing in the direction from
the first current r1 to the second one r2, we se
that magnetic forces decrease also with the
second power of the distance.
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Magnetic interaction of two currents IV
The scaling constant ?0 4? 10-7 Tm/A is
called the permeability of vacuum or of free
space. Some authors dont use it since it is not
an independent parameter of the Nature. It is
related to the perimitivity of vacuum ?0 and the
speed of light c by

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Ampères Law
Let us have none, one, two ore more wires with
currents I1, I2 then

• All the current must be added but their polarity
  must be taken into account !