Inductance- II

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Inductance- II

• Plan
• Energy Storage in a Magnetic field
• Energy density and the magnetic field

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Recap
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• Units volt-second/ampere (Henry)

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Toroid

• Solenoid

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Filling the magnetic materials can increase the
inductance
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Inductors in series

• Inductors in parallel

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Energy storage in magnetic field

• Energy is stored in the electric field due to the
  charges.
• Similar manner, there is energy stored around a
  current carrying wire, where magnetic field
  exists.

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Example

• Work done in separating the current carrying
  wires is stored in the energy of the magnetic
  field of the wire.
• This energy can be recovered by allowing the
  wires to move.

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Energy in Magnetic fields

• When the current is turned on, it has to work
  against the back emf.
• This energy is recovered if the current is turned
  off.
• This energy can be regarded as the energy stored
  in the magnetic field.

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• Work done on a charge dq against the back emf in
  one trip around the circuit

• Work done per unit time

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The energy stored in the magnetic field.

• Work done, from zero to build a current I

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Energy density and magnetic field
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• For a solenoid

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• A solenoid plays a role for magnetic field
  similar to that of the parallel plate capacitor
  for the electric fields.

Solenoid
Capacitor
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A length of cupper wire carries a current of 10
A, uniformly distributed. Calculate (a) magnetic
energy density just outside the surface of the
wire. The wire diameter is 2.5 mm.
uB 1J/m3
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Find the magnetic energy density of a circulating
electron in the hydrogen atom.Electron circulates
about the nucleus in a circular path of radius
5.29 x 10-11m at a frequency f of 6.60 x 1015 Hz
(rev/s).
B 12.6 T uB 6.32 x 107 J/m3
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A long coaxial cable carries a current I (the
current flows down the surface of the inner
cylinder, radius a, and back along the outer
cylinder, radius b. Find the magnetic energy
stored in a section of length l.
I
a
b
I
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• Energy in a cylindrical shell of length l, radius
  r and thickness ds is

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Area element is dr dz
b
r
a
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Calculate the energy stored in a section of
length l of the solenoid.
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Calculate the energy stored in the toroidal coil.
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A long wire carries a current I uniformly
distributed over a cross section of the wire.
(a) Find the magnetic energy of a length l
stored within the wire.
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Find the inductance of the length l of the wire
associated with the flux inside the wire
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Two long parallel wires, each radius a, whose
centers are at a distance d apart carry equal
current in opposite directions. Neglecting the
flux within the wire themselves, find the
inductance of a length l of such a pair of wires.
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a
I
d-y
d
y
a
I
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A uniform magnetic field B fills a cylindrical
volume of radius R. A metal rod of length L is
placed as shown. If B is changing at the rate
dB/dt, find the emf that is produced by the
changing magnetic field and that acts between the
ends of the rod.
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Facts

• For electrostatics

Time varying magnetic fields
Path dependent
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Electric field at a distance r from center
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Along the triangle via path AOBA
O
B
A
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• A is the area of the triangle AOB

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If you choose other path
Via Path ACBA
B
A
C
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Important laws