Biot Savart Law

In physics , especially electricity , the Biot Savart law An equation describing the electric current generated by a constant magnetic field . It relates the magnetic field to the magnitude, direction, length and proximity of the electric current. The Biot Savart law is fundamental to magnetostatics. Playing a role similar to Coulomb’s law in electrostatics . When magnetostatics does not apply, the Biot –Savart law should be replaced by Jefimenko ‘s equations . The law is valid in the magnetostatic approximation , and is consistent with both Ampere’s circuit law and Gauss’s law for magnetism . [2] It is named after Jean-Baptiste Biot and Felix Savart , who discovered this relationship in 1820.

The equation

Electric currents (along a closed curve/wire)

Biot Savart Law
Biot Savart Law

The Biot–Savart rule is used to calculate the resultant magnetic field B at position r in 3D-space generated by a flexible current I (e.g. due to a wire) . A steady (or steady) current is a continuous flow of charges that does not change with time and charge neither accumulates nor dissipates at any point. The law is a physical example of a line integral , being evaluated on a path C in which electric currents flow (such as a wire). Equation [3] in SI units is

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

where is is a vector along the path whose magnitude is the length of the differential element of the wire in the direction of conventional current .

{\displaystyle d{\boldsymbol {\ell }}}C{\displaystyle {\boldsymbol {\ell }}}

There is a point on the path . The wire is the absolute displacement vector from the element at the point to the point at which the field is being calculated, and μ 0 is the magnetic constant . as an alternative:

C{\displaystyle \mathbf {r'} =\mathbf {r} -{\boldsymbol {\ell }}}{\displaystyle d{\boldsymbol {\ell }}}{\displaystyle {\boldsymbol {\ell }}}\mathbf {r}
{\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 is the unit vector of . Symbols in boldface represent vector quantities.

\mathbf {{\hat {r}}'}\mathbf {r'}

The integral is usually around a closed curve , because static electric currents can only flow around closed paths when they are bound. However, the law also applies to infinitely long wires (the concept was used in the definition of the SI unit of electric current – the ampere – as of 20 May 2019).

To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen ( ) , keeping that point constant, calculate the integral line on the path of the electric current as the total magnetic field at that point. is done to find it. The application of this law depends indirectly on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the fields created by each microscopic section of wire individually .R

There is also a 2D version of the Biot–Savart equation, which is used when the sources are irreversible in one direction. In general, current does not need to flow only in a plane that is normal in the inductive direction and is given by ( current density ). The resulting formula is:

{\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 flux density (in conductor volume)

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

\mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}\ {\frac {(\mathbf {J} \,dV)\times \mathbf {r} '}{|\mathbf {r} '|^{3}}}

where dV is the vector from the observation point , is the volume element, and is the current density vector in that quantity (in SI in units of A / m 2 ).

\mathbf {r'}\mathbf {r}dV\mathbf {J}~~
In~~ terms~~ of~~ unit~~ vector~\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}}}}

Constant uniform current

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

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

That is, the current can be drawn out of the integral.

Point charge at constant velocity

In the case of a point charged particle q moving with a constant velocity v , Maxwell’s equations give the following expressions for the electric field and the magnetic field:

{\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 is is the unit vector indicating from the particle’s current (non-dim) position to the point at which the field is being measured, and is the angle between and.

{\mathbf {\hat r}}'\mathbf {v}{\mathbf r}'

When 2 « 2 , the electric field and the magnetic field can be estimated as

{\mathbf {E}}={\frac {q}{4\pi \epsilon _{0}}}\ {\frac {{\mathbf {{\hat r}'}}}{|{\mathbf r}'|^{2}}}
{\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 [6] [7] refer to the above equation as the “Biot-Savart law for a point charge” because of its close resemblance to the standard Biot Savart law. However, this language is misleading because the Biot Savart law only applies to stationary currents and a point charge moving in space does not constitute a steady current. B

Magnetic feedback applications

The Biot Savart law can also be used at the atomic or molecular level in the calculation of magnetic reactions, such as chemical shielding or magnetic susceptibility, provided that the current density can be obtained from quantum mechanical calculations or theory.

Aerodynamics applications

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

In aerodynamic applications, the roles of vorticity and current are reversed compared to magnetic applications.

In Maxwell’s 1861 paper ‘On Physical Lines of Force’, [9] the magnetic field strength H was directly correlated with the net vortex (spin), while B was a weighted vortex weighted for the density of the vortex ocean. had gone. Maxwell considered the magnetic permeability μ to be a measure of the density of the vortex ocean. Hence the relationship

  1. magnetic induction current
{\mathbf {B}}=\mu {\mathbf {H}}

was essentially a rotational analogy for the linear electric current relation,

  • electric conduction current
{\mathbf {J}}=\rho {\mathbf {v}}
  1. where is the electric charge density. B was seen as a kind of magnetic flux of vortices aligned in their axial planes, with the circumferential velocity of H vortices.

The electric current equation can be thought of as a convection current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current that involves spin. In the inductive current there is no linear motion in the direction of the vector B. The magnetic inductive current represents the lines of force. In particular, this inverse square law represents lines of force.

In aerodynamics, induced air currents form solenoid rings around a vortex axis. An analogy can be made that the vortex axis is playing the role that electric current plays in magnetism. This puts the air currents of aerodynamics (fluid velocity field) in the equivalent role of the magnetic induction vector B in electromagnetism .

In electromagnetism, B lines form solenoid rings around the source current, while in aerodynamics, air currents (velocities) form solenoid 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 B lines in isolation, we see exactly the aerodynamic scenario because B is the vortex axis and H is the circumferential velocity in Maxwell’s 1861 paper.

In two dimensions, for a vortex line of infinite length, the induced velocity at a point is given by

v={\frac {\Gamma }{2\pi r}}

where is the strength of the vortex and r is the perpendicular distance between the point and the vortex line. It is analogous to the magnetic field produced on a plane by an infinitely long straight thin wire.

This is a finite case of the formula for vortex segments of finite length (similar to a finite wire):

v={\frac {\Gamma }{4\pi r}}\left[\cos A-\cos B\right]

where a and b are the (signed) angles between the two ends of the line and the segment.

Biot-Savart’s Law, Ampere’s Circuit Law and Gauss’s Law for Magnetism

In a magnetostatic state, the magnetic field B will always satisfy Gauss’s law for magnetism and Ampere’s law as calculated from the BiotSavart law: 

Starting with the Biot-Savart law:

{\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}}}

replace relationship

{\frac {{\mathbf r}-{\mathbf l}}{|{\mathbf r}-{\mathbf l}|^{3}}}=-\nabla \left({\frac {1}{|{\mathbf r}-{\mathbf l}|}}\right)

and using the product rule for curl, as well as the fact that J does not depend , this equation can be rewritten as r

{\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 curl deviation is always zero, it establishes Gauss’s law for magnetism. Next, using the formula to take the curl of both sides of the curl of a curl, and again using the fact that does not depend on J , we finally get the result r

\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 ties

\nabla \left({\frac {1}{|{\mathbf r}-{\mathbf l}|}}\right)=-\nabla _{l}\left({\frac {1}{|{\mathbf r}-{\mathbf l}|}}\right),
\nabla ^{2}\left({\frac {1}{|{\mathbf r}-{\mathbf l}|}}\right)=-4\pi \delta ({\mathbf r}-{\mathbf l})

(where is the Dirac delta function), using the fact that the deviation of J is zero (due to the assumption of magnetostatics), and integrating by parts, the result is

\nabla \times {\mathbf B}=\mu _{0}{\mathbf J}

That is, Ampere’s law. (Due to the notion of magnetism, , so there is no additional displacement current term in Ampere’s law.)

{\displaystyle \partial \mathbf {E} /\partial t=\mathbf {0} }

In a non- magnetic state, the Biot Savart law ceases to be true (it is replaced by Jefimenko’s equations), while for magnetism Gauss’s law and Maxwell–Ampre law are still true.