Magnetostatics

Magnetostatics

The fundamental equations magnetostatics are linear equations,

 
(SI units)		  

 
(Gaussian units)

The principle of superposition holds.

The magnetostaticstatic force on a particle with charge q is

    (SI units),              (Gaussian units).

Definitions:

Drift velocity:   N = number of charge carriers
Current density:  
Current:  

The continuity equation is     In statics    .

Currents and Circuits

  
Current:
 I = ňj×dA  or  I = dQ/dt
Resistance:
 R = DV/I
Resistance of a straight wire:
 R = rl/A
Power:
 P = IDV = I2R = (DV)2/R,
Resistors in series:
 R = R1 + R2 + R3
Parallel Resistors:
 1/R = (1/R1) + (1/R2) + (1/R3)
Kirchhoff's first rule :  (Junction rule)
At any junction point in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction.  (This is a consequence of charge conservation.)
Kirchhoff's second rule :  (Loop rule)
When any closed circuit loop is traversed, the algebraic sum of the changes in the potential must equal zero.  (This is a consequence of conservation of energy.)

Ampere’s law


(SI units)		  
(Gaussian units)

In situations with enough symmetry Ampere’s law alone can be used to find the magnitude of B.  The flux of B through any closed surface is zero.  

The Biot-Savart law

(SI units)   (Gaussian units)
 
For filamentary currents we have    
  Image196.gif (1271 bytes)

The magnetic vector potential

.

A is not unique.  ,  with   an arbitrary scalar field and C an arbitrary constant vector is also a vector potential for the same field.

In magnetostatics we choose Then

(SI units)   Gaussian units)
 
 

The uniqueness theorem:

If if the current density j is specified throughout a volume V and A or its normal derivatives are specified at the boundaries of a volume V, then a unique solution exists for A inside V.

Or, if the current density j is specified throughout a volume V and and either A or B are specified at the boundaries of a volume V, then a unique solution exists for B inside V.

Boundary conditions in magnetostatics

(SI units)   (Gaussian units)
 
 
A is continuous across the boundary.   A is continuous across the boundary.

  

The force on a current distribution

(SI units)   (Gaussian units)
 
For filamentary currents we have    
 

 

The magnetic dipole moment of a charge distribution

(SI units)   (Gaussian units)
  Image215.gif (1450 bytes)
The vector potential of a magnetic dipole at the origin is    
 
The magnetic field of a magnetic dipole at the origin is    
 

The energy of a magnetic dipole in an external magnetic field is  This is the mechanical work done to bring the dipole from infinity to its present position.

The force on a dipole is  .

The torque on a dipole is  .