4' Amperes Law and Applications

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4). Amperes Law and Applications

• As far as possible, by analogy with
  Electrostatics
• B is magnetic flux density or magnetic
  induction
• Units weber per square metre (Wb?m-2) or tesla
  (T)
• Magnetostatics in vacuum, then magnetic media
• based on magnetic dipole moment

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Biot-Savart Law

• The analogue of Coulombs Law is
• the Biot-Savart Law
• Consider a current loop (I)
• For element dl there is an
• associated element field dB
• dB perpendicular to both dl and r-r
• same 1/(4pr2) dependence
• ?o is permeability of free space
• defined as 4p x 10-7 Wb A-1 m-1
• Integrate to get B-S Law

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B-S Law examples
I
q

• (1) Infinitely long straight conductor
• dl and r, r in the page
• dB is out of the page
• B forms circles
• centred on the conductor
• Apply B-S Law to get

dl
r
z
r - r
dB
a
O
r
q p/2 a sin q cos a
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B-S Law examples

• (2) on-axis field of circular loop
• Loop perpendicular to page, radius a
• dl out of page and r, r in the page
• On-axis element dB is in the page,
• perpendicular to r - r, at q to axis.
• Magnitude of element dB
• Integrating around loop, only z-components of dB
  survive
• The on-axis field is axial

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On-axis field of circular loop
Introduce axial distance z, where r-r2 a2
z2 2 limiting cases
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Magnetic dipole moment
The off-axis field of circular loop is much more
complex. For z gtgt a it is identical to that of
the electric dipole m current times
area vs p charge times distance
m
q
r
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B field of large current loop

• Electrostatics began with sheet of electric
  monopoles
• Magnetostatics begin sheet of magnetic dipoles
• Sheet of magnetic dipoles equivalent to current
  loop
• Magnetic moment for one dipole m I a area a
• for loop M I A area A
• Magnetic dipoles one current loop
• Evaluate B field along axis passing through loop

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B field of large current loop

• Consider line integral B.dl from loop
• Contour C is closed by large semi-circle which
  contributes zero to line integral

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Electrostatic potential of dipole sheet

• Now consider line integral E.dl from sheet of
  electric dipoles
• m I a I m/a (density of magnetic moments)
• Replace I by Np (dipole moment density) and mo by
  1/eo
• Contour C is again closed by large semi-circle
  which contributes zero to line integral

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Differential form of Amperes Law
Obtain enclosed current as integral of current
density Apply Stokes theorem Integration
surface is arbitrary Must be true point wise
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Amperes Law examples

• Infinitely long, thin conductor
• B is azimuthal, constant on circle of radius r
• Exercise find radial profile of B inside and
  outside conductor of radius R

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Solenoid
Distributed-coiled conductor Key parameter n
loops/metre If finite length, sum individual
loops via B-S Law If infinite length, apply
Amperes Law B constant and axial inside, zero
outside Rectangular path, axial length L (use
label Bvac to distinguish from core-filled
solenoids) solenoid is to magnetostatics what
capacitor is to electrostatics
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Relative permeability
Recall how field in vacuum capacitor is reduced
when dielectric medium is inserted always
reduction, whether medium is polar or
non-polar is the analogous expression
when magnetic medium is inserted in the vacuum
solenoid. Complication the B field can be
reduced or increased, depending on the type of
magnetic medium
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Magnetic vector potential
For an electrostatic field We cannot therefore
represent B by e.g. the gradient of a
scalar since Magnetostatic field, try B is
unchanged by