Plasma Electrodynamics

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Magnetic Induction and Amperes Law

• Definition of magnetic induction
• Biot-Savart Law
• magnetic field of current -carrying element
• Moving point charges - Lorenz force
• Ampere's law in integral form
• Differential form of Amperes law
• Magnetic flux and Gausss law for magnetism
• Vector Potential

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Force between Two Complete Circuits Amperes
Force (lt--gt Coulomb Force)

• Total force on C,
• As expected from Newtons third law
• macroscopically and microscopically

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Definition of the Magnetic Induction (or
Magnetic Flux Density)

• Magnetic induction or magnetic flux density or
  B-field

• Biot-Savart law

• Units teslaweber/(meter)2 and
  weberjoule/amperevolt-second
• Vector sum of the individual contribution from
  the current elements

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Examples of Direct Magnetic Calculations

• Straight current of finite length

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• Axial induction of a circular current

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• Axial induction of an ideal solenoid

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• Infinite plane uniform current sheet

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Moving Point Charges

• Considering moving point charges as volume
  current density distributions

• For a single point charge with charge q

• Lorentz force

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Amperes Law in Integral Form ( lt--gt Gausss Law )

• Biot-Savart law

• Amperes circuit law (the integral form of
  Amperes law)

Amperian circuit ( lt--gt Gaussian surface )
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Analogy between Gausss Law and Amperes Law

• Amperes Law

Amperian circuit
For a straight wire,

• Gausss Law

Gaussian surface
For a point charge,
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Infinitely long straight current
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Infinitely long ideal solenoid
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Toroidal coil
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Amperes Law in Vector Form

• Amperes circuit law (the integral form of
  Amperes law)

• Using Stokes theorem

• Vector form of Amperes law

• Boundary conditions

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Magnetic Flux and Gausss Law for Magnetism

• Define magnetic flux as for electric flux

• Gausss Law for magnetism

Gausss law for magnetism says The NET
magnetic flux through any closed surface is
ALWAYS zero.

• Boundary conditions

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Vector Potential

• Introduce vector potential

• From Biot-Savart law

Using
Then,
With various current elements
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Gauge Transformation and Uniform Induction

• Gauge transformation

Coulomb gauge

• Uniform induction

(B constant )
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Straight Currents
for a very long straight current
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Two Anti-parallel Long Currents