Biot-Savart Law

The Biot–Savart law is named after Jean-Baptiste Biot and Félix Savart is an equation describing the magnetic field generated by an electric current who discovered this relationship in 1820. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.

## Electric Current

[edit | edit source]

### Electric currents (along closed curve)

[edit | edit source]

The Biot–Savart law is used for computing the resultant magnetic field B at position r generated by a steady current I (for example due to a wire): a continual flow of charges which is constant in 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. The equation in SI units is[1]

Template:Equation box 1

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

B ( r ) = μ 0 4 π ∫ C I d l × r ^ ′ | r ′ | 2 {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {Id\mathbf {l} \times \mathbf {{\hat {r}}'} }{|\mathbf {r'} |^{2}}}}

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

The integral is usually around a closed curve, since electric currents can only flow around closed paths. An infinitely long wire (as used in the definition of the SI unit of electric current - the Ampere) is a counter-example.

To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen ( r {\displaystyle \mathbf {r} } ). Holding that point fixed, the line integral over the path of the electric currents 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.[2]

There is also a 2D version of the Biot-Savart equation, used when the sources are invariant in one direction. In general, the current need not flow only in a plane normal to the invariant direction and it is given by  J {\displaystyle \mathbf {J} } . The resulting formula is:

B ( r ) = μ 0 2 π ∫ C ( J d l ) × r ′ | r ′ | = μ 0 2 π ∫ C ( J d l ) × r ^ ′ {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{2\pi }}\int _{C}\ {\frac {(\mathbf {J} \,dl)\times \mathbf {r} '}{|\mathbf {r} '|}}={\frac {\mu _{0}}{2\pi }}\int _{C}\ (\mathbf {J} \,dl)\times \mathbf {{\hat {r}}'} }

### Electric currents (throughout conductor volume)

[edit | edit source]

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:

Template:Equation box 1

or, alternatively:

B ( r ) = μ 0 4 π ∭ V ( J d V ) × r ^ ′ | r ′ | 2 {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}\ {\frac {(\mathbf {J} \,dV)\times \mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}

where  d V {\displaystyle dV} is the volume element and  J {\displaystyle \mathbf {J} } is the current density vector in that volume (in SI in units of A/m2).

The Biot–Savart law is fundamental to magnetostatics, playing a similar role to Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations.

### Constant uniform current

[edit | edit source]

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

B ( r ) = μ 0 4 π I ∫ C d l × r ^ ′ | r ′ | 2 {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}I\int _{C}{\frac {d\mathbf {l} \times \mathbf {{\hat {r}}'} }{|\mathbf {r'} |^{2}}}}

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

### Point charge at constant velocity

[edit | edit source]

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:[3]

E = q 4 π ϵ 0 1 − v 2 / c 2 ( 1 − v 2 sin 2 ⁡ θ / c 2 ) 3 / 2 r ^ ′ | r ′ | 2 {\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}}}{\frac {1-v^{2}/c^{2}}{(1-v^{2}\sin ^{2}\theta /c^{2})^{3/2}}}{\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}
B = 1 c 2 v × E {\displaystyle \mathbf {B} ={\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {E} }

where  r ^ ′ {\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  v {\displaystyle \mathbf {v} } and  r ′ {\displaystyle \mathbf {r} '} .

When v2 ≪ c2, the electric field and magnetic field can be approximated as[3]

E = q 4 π ϵ 0 r ^ ′ | r ′ | 2 {\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}}}\ {\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}
B = μ 0 q 4 π v × r ^ ′ | r ′ | 2 {\displaystyle \mathbf {B} ={\frac {\mu _{0}q}{4\pi }}\mathbf {v} \times {\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}

These equations are called the "Biot–Savart law for a point charge"[4] due to its closely analogous form to the "standard" Biot–Savart law given previously. These equations were first derived by Oliver Heaviside in 1888.

## See also

[edit | edit source]

w:Biot-Savart_law

1. ↑ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
2. ↑ The superposition principle holds for the electric and magnetic fields because they are the solution to a set of linear differential equations, namely Maxwell's equations, where the current is one of the "source terms".
3. ↑ 3.0 3.1 Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 222–224, 435–440. ISBN 0-13-805326-X.
4. ↑ "Magnetic Field From a Moving Point Charge". Archived from the original on 2009-06-19. Retrieved 2009-09-30.