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 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 acting on
  a charge q moving by a velocity is given by
  the Lorentz formula

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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
• 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 then try do something better!

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Magnetic Field Due to a Straight Wire I

• Lets have an infinite wire which we coincide
  with the x-axis. The current I flows in the x
  direction. We are interested in magnetic
  induction in the point P 0, a.
• The main idea is to use the principle of
  superposition. Cut the wire into pieces of the
  same length dx and add contribution of each of
  them.

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Magnetic Field Due to a Straight Wire II

• For a contribution from a single piece we use
  formula derived from the Biot-Savart law
• Since both vectors which are multiplied lie in
  the x, y plane only the z component of
  will be non-zero which leads to a great
  simplification. We see where the right hand rule
  comes from!

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Magnetic Field Due to a Straight Wire III

• So a piece of the length dx with the coordinate x
  contributes
• Here r is the distance of dx and P and ? is the
  angle between the line joining dx and P and the
  x-axis. We have to express all these quantities
  as a function of one variable e.g. the ?.

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Magnetic Field Due to a Straight Wire IV

• For r we get
• and for x and dx (- is important to get negative
  x at angles ? lt ? /2 !)

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Magnetic Field Due to a Straight Wire V

• So finally we get
• The conclusions we can derive from the symmetry
  we postpone for later!

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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 over a closed path with currents
  which are surrounded by the path.

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Magnetic Field Due to a Straight Wire VI

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

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Magnetic Field Due to a Straight Wire VII

• Let us have a long straight wire with current I.
• We expect B to depend on r and have axial
  symmetry where the wire is naturally the axis.
• The field lines, as we already know are circles
  and therefore our integration path will be a
  circle with a radius r equal to the distance
  where we want to find the field. Then

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Magnetic Field Due to a Straight Wire VIII

• The vectors of the magnetic induction 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 in the case of the electrostatic
  field of an straight, infinite and uniformly
  charged wire but there electric field lines were
  radial while here magnetic are circular, thereby
  perpendicular in every point.

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Magnetic Field in a Center of a Square Loop of
Current I

• Apparently by employing the Amperes law we have
  obtained the same information in a considerable
  easier way. But, unfortunately, this works only
  in special cases.
• Lets calculate magnetic induction in the center
  of a square loop a x a of current I. We see that
  it is a superposition of contributions of all 4
  sides of the square but to get these we have to
  use the formula for infinite wire with
  appropriate limits.

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Magnetic Field in a Center of a Square Loop of
Current II

• The contribution of one side is
• etc.

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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.

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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

• This Lecture Covers
• Chapter 28 1, 2, 3, 4, 6
• Advance reading
• Chapter 27 5 28 4, 5

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Magnetic interaction of two currents I
Let us have two currents I1 and I2 flowing in two
short straight pieces of wire and
Then the force acting on the second piece
due to the existence of the first piece is
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
particularly 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 to the second one , 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 permitivity 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
  polarities must be taken into account !