# GeneralWire

class plasmapy.formulary.magnetostatics.GeneralWire(parametric_eq, t1, t2, current: Quantity)[source]

Bases: Wire

General wire class described by its parametric vector equation.

Parameters:
• parametric_eq (Callable) – A vector-valued (with units of position) function of a single real parameter.

• t1 (float) – Lower bound of the parameter, smaller than t2.

• t2 (float) – Upper bound of the parameter, larger than t1.

• current (Quantity) – Electric current.

Methods Summary

 magnetic_field(p[, n]) Calculate magnetic field generated by this wire at position p.

Methods Documentation

magnetic_field(p: Quantity, n: int = 1000) [source]

Calculate magnetic field generated by this wire at position p.

Parameters:
Returns:

B – Magnetic field at the specified position.

Return type:

astropy.units.Quantity

Notes

The magnetic field generated by a wire with constant electric current is found at a point in 3D space by approximating the Biot–Savart law.

Let the point where the magnetic field will be calculated be represented by the point $$p$$ (with associated position vector $$\vec{p}$$) and the curve $$C$$ defining the wire by the parametric vector equation $$\vec{l}(t)$$ with $$t_{\min} \leq t \leq t_{\max}$$. Further, let the displacement vector from the wire to the point $$p$$ be written as $$\vec{r}(t) = \vec{p} - \vec{l}(t)$$.

The magnetic field $$\vec{B}$$ generated by the wire with constant current $$I$$ at point $$p$$ can then be expressed using the Biot–Savart law, which takes the form

$\vec B = \frac{\mu_0 I}{4\pi} \int_C \frac{d\vec{l} \times \vec{r}}{|\vec{r}|^3}$

where $$\mu_0$$ is the permeability of free space.

This line integral is approximated by segmenting the wire into $$n$$ straight pieces of equal length. The $$i\text{th}$$ wire element can be written as $$\Delta\vec{l}_i = \vec{l}(t_i) - \vec{l}(t_{i - 1})$$ where $$t_i = t_{\min} + i(t_{\max}-t_{\min})/n$$. Further, the displacement vector from the center of the $$i\text{th}$$ wire element to the position $$\vec{p}$$ is $$\vec{r}_i = \vec{p} - \frac{\vec{l}(t_i) + \vec{l}(t_{i-1})}{2}$$.

The integral is then approximated as

$\vec B \approx \frac{\mu_0 I}{4\pi} \sum_{i=1}^{n}\frac{\vec{\Delta l}_i \times \vec{r}_i}{\left| \vec{r}_i \right|^3}.$