Magnetic Fields Due to Currents

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CHAPTER-29

• Magnetic Fields Due to Currents

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Ch 29-2 Calculating Magnetic Field Due to a
Current

• Biot Savart Law
• Magnetic field dB due to a differential
  current-length element ids-element of a length
  vector ds in the direction of current i at a
  point P at a distance r from the current-length
  element ds is given by
• dB? (i ds sin?)/r2
• dB(?0/4?) (i ds sin?)/r2
• dB(?0/4?) (i ds x r)/r3 (Biot-Savart)

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Ch 29-2 Calculating Magnetic Field Due to a
Current

• Magnetic Field lines due to a Current in a Long
  Straight Wire encircles the wires clockwise (
  current into the page) or counter clockwise (
  current pout of the page).
• Direction of B is tangent to those circles in the
  direction of field lines

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Ch 29-2 Calculating Magnetic Field Due to a
Current

• Magnetic Field Due to a Current in a long
  straight wire
• Right hand rule
• Grasp the element in your right hand with your
  extended thumb pointing in the direction of the
  current. Your finger will then naturally curl
  around in the direction of magnetic field line
  due to that element.

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Ch 29-2 Calculating Magnetic Field Due to a
Current

• Magnetic Field Due to a Current in a long
  straight wire
• B ?0i/2?R
• Magnetic Field Due to a Current in a semi-long
  straight wire
• B ?0i/4?R
• Magnetic Field Due to a Current in a Circular Arc
  of Wire
• B (?0i/4?R) ?(rad)

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Ch 29-3 Forces Between Two Parallel Currents

• Force on Wire b of length L with current ib due
  to magnetic field of very long wire a at the
  location of b
• Fbaib Lx Ba
• and Ba ?0ia/2?d
• Fbaib Lx Ba?0iaib (Lx Ba )/2?d
• Fba ?0iaib (LBa sin?)/2?d
• Fba ?0iaib LBa /2?d
• Fba/L ?0iaib LBa /2?d
• Parallel currents attracts each other
• Antiparallel currents repels eaxh other

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Ch 29-4 Amperes Law

• Amperes Law magnetic analogue of Gausss law
• Gauss law ?surf E.dA qenc/ ?0
• Amperes Law ?loop B.ds ?0 ienc
• integral to be evaluated along Ampereian loop,
  ienc is net current enclosed by the loop
• Curled-straight right-hand rule to determine sign
  for current direction
• Curl your right hand rule around the Amperian
  loop, with the fingers pointing in the direction
  of integration. A current through the loop in the
  direction of outstretched thumb is ve and
  opposite is ve.
• ?loop B.ds ?loop B ds cos??0 ienc?0 (i1-i2)
• ? is angle between B and ds
• ?loop Bds cos??loop Bds B?loop ds?0 (i1-i2)
• B2?r?0 ienc

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Ch 29-4 Amperes Law

• Magnetic Field outside a long straight wire with
  a current
• B ?loop Bds cos?
• B?loop dsB2?r? ienc (?0)
• B2?r?0 ienc (outside a very long wire)
• Magnetic Field inside a long straight wire with a
  current
• B ?0 ienc/2?r
• ienc (ienc/?R2)?r2 iencr2 /R2
• B ?0 ienc/2?r (?0 iencr)/2?R2

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Ch 29-5 Solenoids and Toroids

• Solenoids Helical coil of wire producing uniform
  magnetic field at the center.
• Length of the coil is much greater than the coil
  radius
• B-field of the solenoid
• ?loop B.ds ?0 ienc
• ?loop B.ds
• ?ab B.ds ?bc B.ds ?cd B.ds ?da B.ds
• ?ab B.ds Bh
• ienciNinh
• Then Bh ?0 inh
• B?0 in
• n is number of turns per unit length of the
  solenoid.

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Ch 29-5 Solenoids and Toroids

• Toroid Hollow solenoid curved to meet its two
  ends.
• B field inside a torus is given by
• ?loop B.ds ?0 ienc
• ?loop B.ds B 2?r?0 Ni
• B?0 Ni/2?r

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Ch 29-6 A Current Carrying Coil as a Magnetic
Dipole

• Z component of magnetic field of a current
  carrying coil Bz given by
• B?Bz (?0 /2?) (NiA) /z3
• (?0 /2?) ?/z3
• where z-axis is from south to north axis