Biot-Savart Law/Proof
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Theorem
Let $s$ be a wire carrying a steady current $I$.
Let $\map {\mathbf B} {\mathbf r}$ be the total magnetic field due to $I$ flowing through $s$ at a point $P$ whose position vector is $\mathbf r$.
Then:
- $\ds \map {\mathbf B} {\mathbf r} = \dfrac {\mu_0 I} {4 \pi} \oint_s \dfrac {\d \mathbf l \times \paren {\mathbf r - \mathbf r'} } {\size {\mathbf r - \mathbf r'}^3}$
where:
- $\d \mathbf l$ is an infinitesimal vector associated with $s$
- $\mathbf r$ is the position vector of $P$
- $\mathbf r'$ is the position vector of $\d \mathbf l$
- $\mu_0$ denotes the vacuum permeability.
Proof
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Source of Name
This entry was named for Jean-Baptiste Biot and Félix Savart.
Sources
- 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Appendix $\text C$: The derivation of the Biot-Savart law