19'7 Magnetic Fields Long Straight Wire

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19.7 Magnetic Fields Long Straight Wire

• A current-carrying wire produces a magnetic field
• The compass needle deflects in directions tangent
  to the circle
• The compass needle points in the direction of the
  magnetic field produced by the current

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Direction of the Field of a Long Straight Wire

• Right Hand Rule 2
• Grasp the wire in your right hand
• Point your thumb in the direction of the current
• Your fingers will curl in the direction of the
  field

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Magnitude of the Field of a Long Straight Wire
Magnetic field
I
For a long straight wire ? Ampères law

• The magnitude of the field at a distance r from a
  wire carrying a current of I is given by the
  formula above, where µo 4 ? x 10-7 T m / A
• µo is called the permeability of free space

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Ampères Law

• André-Marie Ampère found a procedure for deriving
  the relationship between the current in a
  arbitrarily shaped wire and the magnetic field
  produced by the wire
• Ampères Circuital Law
• ?B ?l µo I ?
• Integral (sum) over the closed path

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Ampères Law, cont

• Choose an arbitrary closed path around the
  current
• Sum all the products of B ?l around the closed
  path B is the component of B parallel to ?l.

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Ampères Law to Find B for a Long Straight Wire

• ? B ?l B ? ?l B (2?r)µo I

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19.8 Magnetic Force Between Two Parallel
Conductors

• F1B2I1l
• B2m0I2/(2?d)
• F1m0I1I2 l /(2?d)

The field B2 at wire 1 due to the current I2 in
wire 2 causes the force F1 on wire 1.
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Force Between Two Conductors, cont

• Parallel conductors carrying currents in the same
  direction attract each other
• Parallel conductors carrying currents in the
  opposite directions repel each other

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Defining Ampere and Coulomb

• The force between parallel conductors can be used
  to define the Ampere (A)
• If two long, parallel wires 1 m apart carry the
  same current, and the magnitude of the magnetic
  force per unit length is 2 x 10-7 N/m, then the
  current is defined to be 1 A
• The SI unit of charge, the Coulomb (C), can be
  defined in terms of the Ampere (A)
• If a conductor carries a steady current of 1 A,
  then the quantity of charge that flows through
  any cross section in 1 second is 1 C

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If I1 2 A and I2 6 A in the figure below,
which of the following is true(a) F1 3F2, (b)
F1 F2, or (c) F1 F2/3?
QUICK QUIZ 19.5
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(b). The two forces are an action-reaction pair.
They act on different wires, and have equal
magnitudes but opposite directions.
QUICK QUIZ 19.5 ANSWER
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19.9 Magnetic Field of a Current Loop

• The strength of a magnetic field produced by a
  wire can be enhanced by forming the wire into a
  loop
• All the segments, ?x, contribute to the field,
  increasing its strength

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Magnetic Field of a Current Loop Total Field
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19.10 Magnetic Field of a Solenoid
Length L

• If a long straight wire is bent into a coil of
  several closely spaced loops, the resulting
  device is called a solenoid
• It is also known as an electromagnet since it
  acts like a magnet only when it carries a current

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Magnetic Field of a Solenoid, cont.

• Magnetic field at the center of a
  current-carrying solenoid (N is the number of
  turns)
• Bm0NI/L, where L is the length of the solenoid
  and with nN/L (number of turns per unit lengths)
  we get
• Bm0nI (? Ampères law)

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Magnetic Field of a Solenoid, cont.

• The longer the solenoid, the more uniform is the
  magnetic field across the cross-sectional area
  with in the coil.
• The exterior field is nonuniform, much weaker,
  and in the opposite direction to the field inside
  the solenoid

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Magnetic Field in a Solenoid, final

• The field lines of the solenoid resemble those of
  a bar magnet

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Magnetic Field in a Solenoid from Ampères Law

• A cross-sectional view of a tightly wound
  solenoid
• If the solenoid is long compared to its radius,
  we assume the field inside is uniform and outside
  is zero
• Apply Ampères Law to the red dashed rectangle

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Magnetic Field in a Solenoid from Ampères Law,
cont.

• ? B ?l BL, since contributions from side 2, 3
  , and 4 are zero
• BLm0NI, where N is the number of turns
• Bm0(N/L)Im0nI, where nN/L is the number of
  turns per unit length

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19.11 Magnetic Effects of Electrons -- Orbits

• An individual atom should act like a magnet
  because of the motion of the electrons about the
  nucleus
• Each electron circles the atom once in about
  every 10-16 seconds
• This would produce a current of 1.6 mA and a
  magnetic field of about 20 T at the center of the
  circular path
• However, the magnetic field produced by one
  electron in an atom is often canceled by an
  oppositely revolving electron in the same atom

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Magnetic Effects of Electrons Orbits, cont.

• The net result is that the magnetic effect
  produced by electrons orbiting the nucleus is
  either zero or very small for most materials

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Magnetic Effects of Electrons -- Spins

• Electrons also have spin
• The classical model is to consider the electrons
  to spin like a top
• It is actually a quantum effect

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Magnetic Effects of Electrons Spins, cont

• The field due to the spinning is generally
  stronger than the field due to the orbital motion
• Electrons usually pair up with their spins
  opposite each other, so their fields cancel each
  other
• That is why most materials are not naturally
  magnetic

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Magnetic Effects of Electrons -- Domains

• Permanent magnetism is an atomic effect due to
  electron spin. In atoms with two or more
  electrons, the  electrons are usually arranged in
  pairs with their spins oppositely aligned ? NOT
  MAGNETIC
• If the spin does not pair ? ferromagnetic
  materials ? magnetic domains produce a net
  magnetic field.

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Magnetic Effects of Electrons -- Domains

• Large groups of atoms in which the spins are
  aligned are called domains
• When an external field is applied, the domains
  that are aligned with the field tend to grow at
  the expense of the others
• This causes the material to become magnetized

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Domains, cont

• (a) Random alignment shows an unmagnetized
  material
• (b) When an external magnetic field is applied,
  the domains aligned parallel to B grow

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Domains and Permanent Magnets

• Two possibilities
• a) Soft magnetic materials
• If the external field is removed, magnetism
  disappears
• b) Hard magnetic materials
• Permanent magnets