Biot–Savart law

Shown are the directions of ${\displaystyle Id{\boldsymbol {\ell }}}$, ${\displaystyle \mathbf {{\hat {r}}'} }$, and the value of ${\displaystyle |\mathbf {r'} |}$

The Biot–Savart law[4]^{: Sec 5-2-1} is used for computing the resultant magnetic flux density B at position r in 3D-space generated by a filamentary current I (for example due to a wire). A steady (or stationary) current is a continual flow of charges which does not change with time and the charge neither accumulates nor depletes at any point. The law is a physical example of a line integral, being evaluated over the path C in which the electric currents flow (e.g. the wire). The equation in SI units teslas (T) is[5]

where ${\displaystyle d{\boldsymbol {\ell }}}$ is a vector along the path ${\displaystyle C}$ whose magnitude is the length of the differential element of the wire in the direction of conventional current. ${\displaystyle {\boldsymbol {\ell }}}$ is a point on path ${\displaystyle C}$. ${\displaystyle \mathbf {r'} =\mathbf {r} -{\boldsymbol {\ell }}}$ is the full displacement vector from the wire element (${\displaystyle d{\boldsymbol {\ell }}}$) at point ${\displaystyle {\boldsymbol {\ell }}}$ to the point at which the field is being computed (${\displaystyle \mathbf {r} }$), and μ₀ is the magnetic constant. Alternatively:

$${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {{\hat {r}}'} }{|\mathbf {r'} |^{2}}}}$$

where ${\displaystyle \mathbf {{\hat {r}}'} }$ is the unit vector of ${\displaystyle \mathbf {r'} }$. The symbols in boldface denote vector quantities.

The integral is usually around a closed curve, since stationary electric currents can only flow around closed paths when they are bounded. However, the law also applies to infinitely long wires (this concept was used in the definition of the SI unit of electric current—the Ampere—until 20 May 2019).

To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen (${\displaystyle \mathbf {r} }$). Holding that point fixed, the line integral over the path of the electric current is calculated to find the total magnetic field at that point. The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually.[6]

For example, consider the magnetic field of a loop of radius ${\displaystyle R}$ carrying a current ${\displaystyle I.}$ For a point a distance ${\displaystyle x}$ along the center line of the loop, the magnetic field vector at that point is:

$${\displaystyle \mathbf {B} ({\hat {\mathbf {x} }})={\frac {\mu _{0}IR^{2}}{2(x^{2}+R^{2})^{3/2}}}{\hat {\mathbf {x} }},}$$

where ${\displaystyle {\hat {\mathbf {x} }}}$ is the unit vector of along the center-line of the loop (and the loop is taken to be centered at the origin).[4]^{: Sec 5-2, Eqn (25)} Loops such as the one described appear in devices like the Helmholtz coil, the solenoid, and the Magsail spacecraft propulsion system. Calculation of the magnetic field at points off the center line requires more complex mathematics involving elliptic integrals that require numerical solution or approximations.[7]

The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire. If the conductor has some thickness, the proper formulation of the Biot–Savart law (again in SI units) is:

where ${\displaystyle \mathbf {r'} }$ is the vector from dV to the observation point ${\displaystyle \mathbf {r} }$, ${\displaystyle dV}$ is the volume element, and ${\displaystyle \mathbf {J} }$ is the current density vector in that volume (in SI in units of A/m²).

In terms of unit vector ${\displaystyle \mathbf {{\hat {r}}'} }$

$${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}\ dV{\frac {\mathbf {J} \times \mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}$$

In the special case of a uniform constant current I, the magnetic field ${\displaystyle \mathbf {B} }$ is

$${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}I\int _{C}{\frac {d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}$$

i.e., the current can be taken out of the integral.

In the case of a point charged particle q moving at a constant velocity v, Maxwell's equations give the following expression for the electric field and magnetic field:[8]

$${\displaystyle {\begin{aligned}\mathbf {E} &={\frac {q}{4\pi \epsilon _{0}}}{\frac {1-{\frac {v^{2}}{c^{2}}}}{\left(1-{\frac {v^{2}}{c^{2}}}\sin ^{2}\theta \right)^{\frac {3}{2}}}}{\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}\\\mathbf {H} &=\mathbf {v} \times \mathbf {D} \\\mathbf {B} &={\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {E} \end{aligned}}}$$

where ${\displaystyle \mathbf {\hat {r}} '}$ is the unit vector pointing from the current (non-retarded) position of the particle to the point at which the field is being measured, and θ is the angle between ${\displaystyle \mathbf {v} }$ and ${\displaystyle \mathbf {r} '}$.

When v² ≪ c², the electric field and magnetic field can be approximated as[8]

$${\displaystyle \mathbf {E} ={\frac {q}{4\pi \varepsilon _{0}}}\ {\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}$$

$${\displaystyle \mathbf {B} ={\frac {\mu _{0}q}{4\pi }}\mathbf {v} \times {\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}$$

These equations were first derived by Oliver Heaviside in 1888. Some authors[9][10] call the above equation for ${\displaystyle \mathbf {B} }$ the "Biot–Savart law for a point charge" due to its close resemblance to the standard Biot–Savart law. However, this language is misleading as the Biot–Savart law applies only to steady currents and a point charge moving in space does not constitute a steady current.[11]

• André-Marie Ampère
• James Clerk Maxwell
• Pierre-Simon Laplace

• Darwin Lagrangian