Biot-Savart law

The Biot-Savart Law is an equation in electromagnetism that describes the magnetic field vector B in terms of the magnitude and direction of the source electric current, the distance from the source electric current, and the magnetic permeability weighting factor.

The significance of the Biot-Savart Law is that it is an inverse square law solution to Ampère's Law. It is also a solution to the vorticity equation curl A = B, i.e., A can be regarded as the magnetic vector potential of B. It therefore provides the B field solution to Maxwell's equations much as the Lorentz force provides the E field solution.

Introduction
The Biot-Savart law and the Lorentz force are fundamental to electromagnetism just as Coulomb's law is fundamental to electrostatics.

In particular, if differential element of current is defined as
 * $$I d\mathbf{l}$$

then the corresponding differential element of magnetic field is


 * $$ d\mathbf{B} = K_m \frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2} $$

where


 * $$K_m = \frac{\mu_0}{4\pi} \,$$, where $$\mu_0$$ is the magnetic constant
 * $$I\mathbf{}$$ is the current, measured in amperes
 * $$d\mathbf{l}$$ is the differential length vector of the current element
 * $$\mathbf{\hat r}$$ is the unit displacement vector from the current element to the field point and
 * $$r\mathbf{}$$ is the distance from the current element to the field point.

Alternatively, differential element of current can be defined as


 * $$\mathbf{j}dV$$

and the corresponding differential element of magnetic field will be


 * $$ d\mathbf{B} = K_m \frac{\mathbf{j}dV \times \mathbf{\hat r}}{r^2} = K_m \frac{\mathbf{j} \times \mathbf{\hat r}}{r^2}dV $$

where


 * $$\mathbf{j}$$ is the current density vector and
 * $$ \ dV$$ is the differential element of volume.

General
In the magnetostatic approximation, the magnetic field can be determined if the current density j is known:


 * $$\mathbf{B}= K_m\int{\frac{\mathbf{j} \times \mathbf{\hat r}}{r^2}dV}$$

where
 * $$\mathbf{\hat{r}} = { \mathbf{r} \over r } $$ is the unit vector in the direction of r.
 * $$ \ dV$$ = is the differential element of volume.

Constant uniform current
In the special case of a constant, uniform current I, the magnetic field B is


 * $$ \mathbf B = K_m I \int \frac{d\mathbf l \times \mathbf{\hat r}}{r^2}$$

Point charge at constant velocity
In the special case of a charged point particle $$q\mathbf{}$$ moving at a constant, non-relativistic velocity $$\mathbf{v}$$, then the magnetic field is :


 * $$ \mathbf{B} = K_m \frac{ q \mathbf{v} \times \mathbf{\hat{r}}}{r^2} $$

This equation is also sometimes called the Biot-Savart law, due to its closely analogous form to the "standard" Biot-Savart law given above.

Microscopic Scale
On the microscopic scale, the Biot-Savart law becomes,


 * $$ \mathbf{H} = \epsilon \mathbf{v} \times \mathbf{E} $$

where the solution to $$\mathbf{E}$$ is the Coulomb force, and where,


 * $$\mathbf{B} = \mu \mathbf{H}$$

and hence,



\mathbf{B} = \mathbf{v}\times \frac{1}{c^2}\mathbf{E} $$

Magnetic responses applications
The Biot-Savart law can be used in the calculation of magnetic responses even at the atomic or molecular level, e.g. chemical shieldings or magnetic susceptibilities, provided that the current density can be obtained from a quantum mechanical calculation or theory.

Aerodynamics applications


The Biot-Savart law is also used to calculate the velocity induced by vortex lines in aerodynamic theory.

In the aerodynamic application, the roles of vorticity and current are reversed as when compared to the magnetic application.

In Maxwell's 1861 paper 'On Physical Lines of Force', magnetic field strength $$\mathbf{H}$$ was directly equated with pure vorticity (spin), whereas $$\mathbf{B}$$ was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability μ to be a measure of the density of the vortex sea. Hence the relationship,

(1) Magnetic Induction Current

$$\mathbf{B} = \mu \mathbf{H}$$

was essentially a rotational analogy to the linear electric current relationship,

(2) Electric Convection Current

$$\mathbf{J} = \rho \mathbf{v}$$

where ρ is electric charge density. $$\mathbf{B}$$ was seen as a kind of magnetic current of vortices aligned in their axial planes, with $$\mathbf{H}$$ being the circumferential velocity of the vortices.

The electric current equation can be viewed as a convective current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the $$\mathbf{B}$$ vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square law force.

In aerodynamics the induced air currents are forming solenoidal rings around a vortex axis that is playing the role that electric current plays in magnetism. This puts the air currents of aerodynamics into the equivalent role of the magnetic induction vector $$\mathbf{B}$$ in electromagnetism.

In electromagnetism the $$\mathbf{B}$$ lines form solenoidal rings around the source electric current, whereas in aerodynamics, the air currents form solenoidal rings around the source vortex axis.

Hence in electromagnetism, the vortex plays the role of 'effect' whereas in aerodynamics, the vortex plays the role of 'cause'. Yet when we look at the $$\mathbf{B}$$ lines in isolation, we see exactly the aerodynamic scenario in so much as that $$\mathbf{B}$$ is the vortex axis and $$\mathbf{H}$$ is the circumferential velocity as in Maxwell's 1861 paper.

For a vortex line of infinite length, the induced velocity at a point is given by


 * $$v = \frac{\Gamma}{2\pi d}$$

where


 * Γ is the strength of the vortex
 * d is the perpendicular distance between the point and the vortex line.

This is a limiting case of the formula for vortex segments of finite length:


 * $$v = \frac{\Gamma}{4 \pi d} \left[\cos A + \cos B \right]$$

where A and B are the (signed) angles between the line and the two ends of the segment.

People

 * Jean-Baptiste Biot
 * Felix Savart
 * André-Marie Ampère
 * James Clerk Maxwell

Electromagnetism

 * Maxwell's equations
 * Ampère's law
 * magnetism
 * Coulomb's law

Aerodynamics

 * vorticity
 * thin-airfoil theory