ELC-28

Wire field B=μI/2πr

B = μ I /(2π r) around a long straight wire.

Reading speed

MagneticsBiot–Savart

Governing equation

B=\dfrac{\mu I}{2\pi r}

$r \in \left[1, 500\right]$

where

I: Current (A)
r: Radius (mm)
\mu_r: Relative μ (—)
B: Flux density (µT)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

B=\dfrac{\mu I}{2\pi r}

$r \in \left[1, 500\right]$

$r = 1$ $B = 4000$

Along the curve

②

Domain of the variable — blue

r_{\min} = 1,\quad r_{\max} = 500

$r \in \left[1, 500\right]$

$r = 1$ $r = 1$

Along the curve

③

Response curve — green

B=\dfrac{\mu I}{2\pi r}

$r \in \left[1, 500\right]$

$r = 1$ $B = 4000$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}B}{\mathrm{d}r}

$r \in \left[1, 500\right]$

$r = 4.138$ $B^{\prime} = -549.7$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-28 — Wire field B=μI/2πr) is the form associated with Biot–Savart. Working symbols: $I$ , $r$ , $\mu_r$ $\rightarrow$ $B$ . Ampère's law ∮ B·dl = μ I on a circle of radius r, or Biot–Savart integrated along the line.

Purpose

Purpose: compute

B

from

I

r

\mu_r

in Electrical via

B=\dfrac{\mu I}{2\pi r}

B = μ I /(2π r) around a long straight wire. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I = 20.000\,\mathrm{A}

r = 25.000\,\mathrm{mm}

\mu_r = 1.000\,\mathrm{—}

, the governing relation

B=\dfrac{\mu I}{2\pi r}

yields

B = 160.000\,\mathrm{\mu T}

. A wire, concentric B circles, a radius r. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Current I20 ARadius r25 mmRelative μ \mu_r1 —

Outputs

Flux density B160.000 µT

Reading speed

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Narration of this film

A wire, concentric B circles, a radius r.

Ampère's law ∮ B·dl = μ I on a circle of radius r, or Biot–Savart integrated along the line.

Reading speed

Watch on YouTube