Magnetostatics

1
Magnetostatics

• Magnetostatics is the branch of electromagnetics
  dealing with the effects of electric charges in
  steady motion (i.e, steady current or DC).
• The fundamental law of magnetostatics is Amperes
  law of force.
• Amperes law of force is analogous to Coulombs
  law in electrostatics.

2
Magnetostatics (Contd)

• In magnetostatics, the magnetic field is produced
  by steady currents. The magnetostatic field does
  not allow for
• inductive coupling between circuits
• coupling between electric and magnetic fields

3
Amperes Law of Force

• Amperes law of force is the law of action
  between current carrying circuits.
• Amperes law of force gives the magnetic force
  between two current carrying circuits in an
  otherwise empty universe.
• Amperes law of force involves complete circuits
  since current must flow in closed loops.

4
Amperes Law of Force (Contd)

• Experimental facts
• Two parallel wires carrying current in the same
  direction attract.
• Two parallel wires carrying current in the
  opposite directions repel.

5
Amperes Law of Force (Contd)

• Experimental facts
• A short current-carrying wire oriented
  perpendicular to a long current-carrying wire
  experiences no force.

F12 0
?
I2
I1
6
Amperes Law of Force (Contd)

• Experimental facts
• The magnitude of the force is inversely
  proportional to the distance squared.
• The magnitude of the force is proportional to the
  product of the currents carried by the two wires.

7
Amperes Law of Force (Contd)

• The force acting on a current element I2 dl2 by a
  current element I1 dl1 is given by

Permeability of free space m0 4p ? 10-7 F/m
8
Amperes Law of Force (Contd)

• The total force acting on a circuit C2 having a
  current I2 by a circuit C1 having current I1 is
  given by

9
Amperes Law of Force (Contd)

• The force on C1 due to C2 is equal in magnitude
  but opposite in direction to the force on C2 due
  to C1.

10
Magnetic Flux Density

• Amperes force law describes an action at a
  distance analogous to Coulombs law.
• In Coulombs law, it was useful to introduce the
  concept of an electric field to describe the
  interaction between the charges.
• In Amperes law, we can define an appropriate
  field that may be regarded as the means by which
  currents exert force on each other.

11
Magnetic Flux Density (Contd)

• The magnetic flux density can be introduced by
  writing

12
Magnetic Flux Density (Contd)

• where

the magnetic flux density at the location of dl2
due to the current I1 in C1
13
Magnetic Flux Density (Contd)

• Suppose that an infinitesimal current element Idl
  is immersed in a region of magnetic flux density
  B. The current element experiences a force dF
  given by

14
Magnetic Flux Density (Contd)

• The total force exerted on a circuit C carrying
  current I that is immersed in a magnetic flux
  density B is given by

15
Force on a Moving Charge

• A moving point charge placed in a magnetic field
  experiences a force given by

The force experienced by the point charge is in
the direction into the paper.
16
Lorentz Force

• If a point charge is moving in a region where
  both electric and magnetic fields exist, then it
  experiences a total force given by
• The Lorentz force equation is useful for
  determining the equation of motion for electrons
  in electromagnetic deflection systems such as
  CRTs.

17
The Biot-Savart Law

• The Biot-Savart law gives us the B-field arising
  at a specified point P from a given current
  distribution.
• It is a fundamental law of magnetostatics.

18
The Biot-Savart Law (Contd)

• The contribution to the B-field at a point P from
  a differential current element Idl is given by

19
The Biot-Savart Law (Contd)
20
The Biot-Savart Law (Contd)

• The total magnetic flux at the point P due to the
  entire circuit C is given by

21
Types of Current Distributions

• Line current density (current) - occurs for
  infinitesimally thin filamentary bodies (i.e.,
  wires of negligible diameter).
• Surface current density (current per unit width)
  - occurs when body is perfectly conducting.
• Volume current density (current per unit cross
  sectional area) - most general.

22
The Biot-Savart Law (Contd)

• For a surface distribution of current, the B-S
  law becomes
• For a volume distribution of current, the B-S law
  becomes

23
Amperes Circuital Law in Integral Form

• Amperes Circuital Law in integral form states
  that the circulation of the magnetic flux
  density in free space is proportional to the
  total current through the surface bounding the
  path over which the circulation is computed.

24
Amperes Circuital Law in Integral Form (Contd)
By convention, dS is taken to be in the direction
defined by the right-hand rule applied to dl.
Since volume current density is the most
general, we can write Iencl in this way.
25
Amperes Law and Gausss Law

• Just as Gausss law follows from Coulombs law,
  so Amperes circuital law follows from Amperes
  force law.
• Just as Gausss law can be used to derive the
  electrostatic field from symmetric charge
  distributions, so Amperes law can be used to
  derive the magnetostatic field from symmetric
  current distributions.

26
Applications of Amperes Law

• Amperes law in integral form is an integral
  equation for the unknown magnetic flux density
  resulting from a given current distribution.

known
unknown
27
Applications of Amperes Law (Contd)

• In general, solutions to integral equations must
  be obtained using numerical techniques.
• However, for certain symmetric current
  distributions closed form solutions to Amperes
  law can be obtained.

28
Applications of Amperes Law (Contd)

• Closed form solution to Amperes law relies on
  our ability to construct a suitable family of
  Amperian paths.
• An Amperian path is a closed contour to which the
  magnetic flux density is tangential and over
  which equal to a constant value.

29
Amperes Law in Differential Form

• Amperes law in differential form implies that
  the B-field is conservative outside of regions
  where current is flowing.

30
Fundamental Postulates of Magnetostatics

• Amperes law in differential form
• No isolated magnetic charges

B is solenoidal
31
Vector Magnetic Potential

• Vector identity the divergence of the curl of
  any vector field is identically zero.
• Corollary If the divergence of a vector field
  is identically zero, then that vector field can
  be written as the curl of some vector potential
  field.

32
Vector Magnetic Potential (Contd)

• Since the magnetic flux density is solenoidal, it
  can be written as the curl of a vector field
  called the vector magnetic potential.

33
Vector Magnetic Potential (Contd)

• The general form of the B-S law is
• Note that

34
Vector Magnetic Potential (Contd)

• Furthermore, note that the del operator operates
  only on the unprimed coordinates so that

35
Vector Magnetic Potential (Contd)

• Hence, we have

36
Vector Magnetic Potential (Contd)

• For a surface distribution of current, the vector
  magnetic potential is given by
• For a line current, the vector magnetic potential
  is given by

37
Vector Magnetic Potential (Contd)

• In some cases, it is easier to evaluate the
  vector magnetic potential and then use B
  ?? A, rather than to use the B-S law to directly
  find B.
• In some ways, the vector magnetic potential A is
  analogous to the scalar electric potential V.

38
Vector Magnetic Potential (Contd)

• In classical physics, the vector magnetic
  potential is viewed as an auxiliary function with
  no physical meaning.
• However, there are phenomena in quantum mechanics
  that suggest that the vector magnetic potential
  is a real (i.e., measurable) field.

39
Divergence of B-Field

• The B-field is solenoidal, i.e. the divergence of
  the B-field is identically equal to zero
• Physically, this means that magnetic charges
  (monopoles) do not exist.
• A magnetic charge can be viewed as an isolated
  magnetic pole.

40
Divergence of B-Field (Contd)

• No matter how small the magnetic is divided, it
  always has a north pole and a south pole.
• The elementary source of magnetic field is a
  magnetic dipole.

41
Magnetic Flux

• The magnetic flux crossing an open surface S is
  given by

Wb
Wb/m2
42
Magnetic Flux (Contd)

• From the divergence theorem, we have
• Hence, the net magnetic flux leaving any closed
  surface is zero. This is another manifestation
  of the fact that there are no magnetic charges.

43
Magnetic Flux and Vector Magnetic Potential

• The magnetic flux across an open surface may be
  evaluated in terms of the vector magnetic
  potential using Stokess theorem

44
Fundamental Laws of Magnetostatics in Integral
Form
Amperes law
Gausss law for magnetic field
Constitutive relation
45
Fundamental Laws of Magnetostatics in
Differential Form
Amperes law
Gausss law for magnetic field
Constitutive relation
46
Fundamental Laws of Magnetostatics

• The integral forms of the fundamental laws are
  more general because they apply over regions of
  space. The differential forms are only valid at
  a point.
• From the integral forms of the fundamental laws
  both the differential equations governing the
  field within a medium and the boundary conditions
  at the interface between two media can be derived.

47
Boundary Conditions

• Within a homogeneous medium, there are no abrupt
  changes in H or B. However, at the interface
  between two different media (having two different
  values of m), it is obvious that one or both of
  these must change abruptly.

48
Boundary Conditions (Contd)

• The normal component of a solenoidal vector field
  is continuous across a material interface
• The tangential component of a conservative vector
  field is continuous across a material interface

49
Boundary Conditions (Contd)

• The tangential component of H is continuous
  across a material interface, unless a surface
  current exists at the interface.
• When a surface current exists at the interface,
  the BC becomes

50
Boundary Conditions (Contd)

• In a perfect conductor, both the electric and
  magnetic fields must vanish in its interior.
  Thus,

• a surface current must exist
• the magnetic field just outside the perfect
  conductor must be tangential to it.