The fundamental equations magnetostatics are **linear equations**,

**∇∙B** = 0, **∇**×**B** = μ_{0}**j** = **j**/(ε_{0}c^{2})
(SI units).

The **principle of superposition **holds.

The **magnetostatic force** on a particle with charge q is **F** = q**v
**×** B**.

Definitions:**Drift velocity:** <**v**> =
(1/N)∑_{i}**v**_{i}, N
= number of charge carriers**.
Current density:**

Current:
I = ∫**j**∙d**A** or I = dQ/dt.

Resistance:
R = ΔV/I.

Resistance of a straight wire:
R = ρl/A.

Power:
P = IΔV = I^{2}R = (ΔV)^{2}/R.

Resistors in series:
R = R_{1 }+ R_{2 }+ R_{3} + ... ._{
}Parallel Resistors:
1/R = (1/R_{1}) + (1/R_{2}) + (1/R_{3}) + ... .

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.)

∮_{Γ}**B**∙d**r** =
μ_{0}I_{through Γ}.

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.
∫_{closed surface} **B·**d**A **= 0.

**B**(**r**) = (μ_{0}/(4π))∫_{ }dV' **j(r'**)×(**r**-**r**')/|**r**-**r**'|^{3}.

For filamentary currents we have **B**(**r**) = (μ_{0}/(4π))∫_{
}I d**l'**×(**r**-**r**')/|**r**-**r**'|^{3}.

**∇∙B** = 0 --> **B** = **∇**×**A.
A** is not unique.

In magnetostatics we choose

Then

**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.

(**B**_{2} - **B**_{1})∙**n**_{2}
= 0,
(**B**_{2} - **B**_{1})∙**t** = μ_{0}**k**∙**n**.**
A** is continuous across the boundary.

**F** = ∫_{V }**j**(**r**)** **×** B**(**r**)
dV.

For filamentary currents we have **F** = ∫_{V }I d**l **
×** B**(**r**).

**m** = IA**n** = ½∫_{V }**r**×**j**(**r**) dV.

The **vector potential of a magnetic dipole** at the origin is **A**(**r**)
= (μ_{0}/4π)**m**×**r**/r^{3}.

The **magnetic field of a magnetic dipole** at the origin is **B**(**r**) = (μ_{0}/4π)(3(**m∙r**)**r**/r^{5}
- **m**/r^{3}).

The **energy of a magnetic dipole in an external magnetic field** is
U_{mech} = -**m∙B**.

This is the mechanical work done to bring the dipole from infinity to its
present position.

The **force on a dipole **is **F** =
**∇**(**m∙B**).

The **torque on a dipole** is ** τ** =
**m**
× **B**.

The **magnetization** **M** = d**m**/dV is defined as the magnetic
dipole moment per unit volume.

The total current density is due to free and to magnetization current
densities.

**j** = **j**_{f }+ **j**_{m}, **k _{m}** =

H

For

Χ

(**H**_{2} - **H**_{1})∙**t**_{2}
= **k**_{f}∙**n**,
** ∇∙H** ≠ 0 in general.

The magnetostatic energy stored in a current distribution is given by

U = (2μ_{0})^{-1}∫_{all
space}**B∙B ** dV.

In the presence of a magnetic material, the total work done in establishing a
free current distribution is

W = ½∫_{all
space}**B∙H ** dV,

or, in the presence of a lih magnetic material

U = (2μ)^{-1}∫_{all
space}**B∙B ** dV.