Curl of magnetic field is free current density and electric displacement derivative

The magnetic field circulation theorem states that an electric current and a change in electric displacement generate a rotational magnetic field. Also known as the Ampère’s circuital law.

Notes:

1. The \(\text{curl}\) operator is only defined for a 3D space.

Links:

1. Wikipedia, second line in table.

2. Physics LibreTexts, formula 15.5.3.

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

position_vector

Position vector of a point in space. See distance_to_origin.

Symbol:

r

Latex:

\({\vec r}\)

Dimension:

length

magnetic_field

Vector of the magnetic field as a function of position_vector and time. See :symbols:~magnetic_field_strength`.

Symbol:

H(r, t)

Latex:

\({\vec H} \left( {\vec r}, t \right)\)

Dimension:

current/length

electric_displacement

Vector of the electric_displacement field as a function of position_vector and time.

Symbol:

D(r, t)

Latex:

\({\vec D} \left( {\vec r}, t \right)\)

Dimension:

charge/area

free_current_density

Vector of the free (i.e. unbound) current_density field as a function of position_vector and time.

Symbol:

J_f(r, t)

Latex:

\({\vec J}_\text{f} \left( {\vec r}, t \right)\)

Dimension:

current/area

law

curl(H(r, t)) = J_f(r, t) + Derivative(D(r, t), t)

Latex:

\[\text{curl} \, {\vec H} \left( {\vec r}, t \right) = {\vec J}_\text{f} \left( {\vec r}, t \right) + \frac{\partial}{\partial t} {\vec D} \left( {\vec r}, t \right)\]