Magnetic field due to constant filamentary current

Known as the Biot—Savart law, it is an equation describing the magnetic field due to a constant electric current.

Notation:

1. \(\mu_0\) (mu_0) is vacuum_permeability.

Notes:

1. This version of the law deals with a current in an infinitely thin wire. For a conductor of a finite thickness, the following relation must be used:

   \[I d \vec{\ell} = \vec{J} dV\]

2. To find the total magnetic flux density, calculate the line integral over the whole contour.

Conditions:

1. The system is in a vacuum.

Links:

1. Wikipedia — Biot—Savart law.

magnetic_flux_density_change

Infinitesimal change of the magnetic_flux_density at a given point in space.

Symbol:

dB

Latex:

\(d \vec{B}\)

Dimension:

magnetic_density

absolute_permeability

absolute_permeability of the medium.

Symbol:

mu

Latex:

\(\mu\)

Dimension:

inductance/length

current

current in the contour.

Symbol:

I

Latex:

\(I\)

Dimension:

current

position_vector

Position vector of the point at which the magnetic flux density is measured. Also see distance_to_origin.

Symbol:

r

Latex:

\({\vec r}\)

Dimension:

length

contour_element_position_vector

Position vector of a point on the integration path. Also see distance_to_origin.

Symbol:

l

Latex:

\(\vec{\ell}\)

Dimension:

length

contour_element_displacement

A vector along the integration path whose magnitude is the length of the differential element in the direction of conventional current. Also see euclidean_distance.

Symbol:

dl

Latex:

\(d \vec{\ell}\)

Dimension:

length

law

dB = mu / (4 * pi) * I * cross(dl, r - l) / norm(r - l)^3

Latex:

\[d \vec{B} = \frac{\mu}{4 \pi} \frac{I \left[ d \vec{\ell}, {\vec r} - \vec{\ell} \right]}{\left \Vert {\vec r} - \vec{\ell} \right \Vert^{3}}\]