Biot–savart law – wikipedia gas pain relief


The Biot–Savart law is used for computing the resultant magnetic field B at position r in 3D-space generated by a steady 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 is [3]

where d ℓ {\displaystyle d{\boldsymbol {\ell }}} is a vector along the path C {\displaystyle C} 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} -{\boldsymbol {\ell }}} is the full displacement vector from the wire element ( d ℓ {\displaystyle d{\boldsymbol {\ell }}} ) to the point at which the field is being computed ( r {\displaystyle \mathbf {r} } ), and μ 0 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{\boldsymbol {\ell }}\times \mathbf {{\hat {r}}’} }{|\mathbf {r’} |^{2}}}}

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 (as used in the definition of the SI unit of electric current – the Ampere).

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 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. [4]

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} } ( current density). 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} \,d\ell )\times \mathbf {r} ‘}{|\mathbf {r} ‘|}}={\frac {\mu _{0}}{2\pi }}\int _{C}\ (\mathbf {J} \,d\ell )\times \mathbf {{\hat {r}}’} } Electric current density (throughout conductor volume) [ edit ]

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

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: [5] 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}}}} H = v × D {\displaystyle \mathbf {H} =\mathbf {v} \times \mathbf {D} } or 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 v 2 ≪ c 2, the electric field and magnetic field can be approximated as [5] 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" [6] due to its closely analogous form to the "standard" Biot–Savart law given previously. These equations were first derived by Oliver Heaviside in 1888. Magnetic responses applications [ edit ]

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 [ edit ]

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 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 B in electromagnetism.

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 B lines in isolation, we see exactly the aerodynamic scenario in so much as that B is the vortex axis and H is the circumferential velocity as in Maxwell’s 1861 paper.

B ( r ) = μ 0 4 π ∭ V d 3 l J ( l ) × r − l | r − l | 3 {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}d^{3}l\mathbf {J} (\mathbf {l} )\times {\frac {\mathbf {r} -\mathbf {l} }{|\mathbf {r} -\mathbf {l} |^{3}}}}

and using the product rule for curls, as well as the fact that J does not depend on r {\displaystyle \mathbf {r} } , this equation can be rewritten as [8] B ( r ) = μ 0 4 π ∇ × ∭ V d 3 l J ( l ) | r − l | {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\nabla \times \iiint _{V}d^{3}l{\frac {\mathbf {J} (\mathbf {l} )}{|\mathbf {r} -\mathbf {l} |}}}

Since the divergence of a curl is always zero, this establishes Gauss’s law for magnetism. Next, taking the curl of both sides, using the formula for the curl of a curl, and again using the fact that J does not depend on r {\displaystyle \mathbf {r} } , we eventually get the result [8] ∇ × B = μ 0 4 π ∇ ∭ V d 3 l J ( l ) ⋅ ∇ ( 1 | r − l | ) − μ 0 4 π ∭ V d 3 l J ( l ) ∇ 2 ( 1 | r − l | ) {\displaystyle \nabla \times \mathbf {B} ={\frac {\mu _{0}}{4\pi }}\nabla \iiint _{V}d^{3}l\mathbf {J} (\mathbf {l} )\cdot \nabla \left({\frac {1}{|\mathbf {r} -\mathbf {l} |}}\right)-{\frac {\mu _{0}}{4\pi }}\iiint _{V}d^{3}l\mathbf {J} (\mathbf {l} )\nabla ^{2}\left({\frac {1}{|\mathbf {r} -\mathbf {l} |}}\right)}

Finally, plugging in the relations [8] ∇ ( 1 | r − l | ) = − ∇ l ( 1 | r − l | ) , {\displaystyle \nabla \left({\frac {1}{|\mathbf {r} -\mathbf {l} |}}\right)=-\nabla _{l}\left({\frac {1}{|\mathbf {r} -\mathbf {l} |}}\right),} ∇ 2 ( 1 | r − l | ) = − 4 π δ ( r − l ) {\displaystyle \nabla ^{2}\left({\frac {1}{|\mathbf {r} -\mathbf {l} |}}\right)=-4\pi \delta (\mathbf {r} -\mathbf {l} )}

(where δ is the Dirac delta function), using the fact that the divergence of J is zero (due to the assumption of magnetostatics), and performing an integration by parts, the result turns out to be [8] ∇ × B = μ 0 J {\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }