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Biot–Savart law

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Law of classical electromagnetism
Electromagnetism
Solenoid

Inphysics, specificallyelectromagnetism, theBiot–Savart law (/ˈbiːoʊsəˈvɑːr/ or/ˈbjoʊsəˈvɑːr/)[1] is an equation describing themagnetic field generated by a constantelectric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.

The Biot–Savart law is fundamental tomagnetostatics. It is valid in themagnetostatic approximation and consistent with bothAmpère's circuital law andGauss's law for magnetism.[2] When magnetostatics does not apply, the Biot–Savart law should be replaced byJefimenko's equations. The law is named afterJean-Baptiste Biot andFélix Savart, who discovered this relationship in 1820.

Equation

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In the following equations, it is assumed that the medium is not magnetic (e.g., vacuum). This allows for straightforward derivation of magnetic fieldB, while the fundamental vector here isH.[3]

Electric currents (along a closed curve/wire)

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

The Biot–Savart law[4]: Sec 5-2-1  is used for computing the resultantmagnetic flux densityB at positionr in 3D-space generated by a filamentarycurrentI (for example due to a wire). A steady (or stationary) current is a continual flow ofcharges which does not change with time and the charge neither accumulates nor depletes at any point. The law is a physical example of aline integral, being evaluated over the pathC in which the electric currents flow (e.g. the wire). The equation inSI unitsteslas (T) is[5]

B(r)=μ04π∫CIdℓ×r′|r′|3{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {r'} }{|\mathbf {r'} |^{3}}}}

wheredℓ{\displaystyle d{\boldsymbol {\ell }}} is a vector along the pathC{\displaystyle C} whose magnitude is the length of thedifferential element of the wire in the direction ofconventional current,ℓ{\displaystyle {\boldsymbol {\ell }}} is a point on pathC{\displaystyle C}, andr′=r−ℓ{\displaystyle \mathbf {r'} =\mathbf {r} -{\boldsymbol {\ell }}} is the fulldisplacement vector from the wire element (dℓ{\displaystyle d{\boldsymbol {\ell }}}) at pointℓ{\displaystyle {\boldsymbol {\ell }}} to the point at which the field is being computed (r{\displaystyle \mathbf {r} }), andμ0 is themagnetic constant. Alternatively:B(r)=μ04π∫CIdℓ×r^′|r′|2{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\,d{\boldsymbol {\ell }}\times \mathbf {{\hat {r}}'} }{|\mathbf {r'} |^{2}}}}wherer^′{\displaystyle \mathbf {{\hat {r}}'} } is theunit vector ofr′{\displaystyle \mathbf {r'} }. The symbols in boldface denotevector quantities.

The integral is usually around aclosed 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—theAmpere—until 20 May 2019).

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 thesuperposition principle for magnetic fields, i.e. the fact that the magnetic field is avector sum of the field created by each infinitesimal section of the wire individually.[6]

For example, consider the magnetic field of a loop of radiusR{\displaystyle R} carrying a currentI.{\displaystyle I.} For a point at a distancex{\displaystyle x} along the center line of the loop, the magnetic field vector at that point is:B=μ0IR22(x2+R2)3/2x^,{\displaystyle \mathbf {B} ={\mu _{0}IR^{2} \over 2(x^{2}+R^{2})^{3/2}}{\hat {\mathbf {x} }},}wherex^{\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 theHelmholtz coil, thesolenoid, and theMagsail spacecraft propulsion system. Calculation of the magnetic field at points off the center line requires more complex mathematics involvingelliptic integrals that require numerical solution or approximations.[7]

Electric current density (throughout conductor volume)

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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 inSI units) is:

B(r)=μ04π∭V (JdV)×r′|r′|3{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}\ {\frac {(\mathbf {J} \,dV)\times \mathbf {r} '}{|\mathbf {r} '|^{3}}}}

wherer′{\displaystyle \mathbf {r'} } is the vector from dV to the observation pointr{\displaystyle \mathbf {r} },dV{\displaystyle dV} is thevolume element, andJ{\displaystyle \mathbf {J} } is thecurrent density vector in that volume (in SI in units of A/m2).

In terms of unit vectorr^′{\displaystyle \mathbf {{\hat {r}}'} }B(r)=μ04π∭V dVJ×r^′|r′|2{\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

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In the special case of a uniform constant currentI, the magnetic fieldB{\displaystyle \mathbf {B} } isB(r)=μ04πI∫Cdℓ×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}}}}i.e., the current can be taken out of the integral.

Point charge at constant velocity

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In the case of a pointcharged particleq moving at a constantvelocityv,Maxwell's equations give the following expression for theelectric field and magnetic field:[8]E=q4πε01−β2(1−β2sin2⁡θ)3/2r^′|r′|2B=1c2v×E{\displaystyle {\begin{aligned}\mathbf {E} &={\frac {q}{4\pi \varepsilon _{0}}}{\frac {1-\beta ^{2}}{\left(1-\beta ^{2}\sin ^{2}\theta \right)^{3/2}}}{\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}\\[1ex]\mathbf {B} &={\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {E} \end{aligned}}}where

Whenv2 ≪c2, the electric field and magnetic field can be approximated as[8]E=q4πε0 r^′|r′|2{\displaystyle \mathbf {E} ={\frac {q}{4\pi \varepsilon _{0}}}\ {\frac {\mathbf {{\hat {r}}'} }{|\mathbf {r} '|^{2}}}}B=μ04πqv×r^′|r′|2{\displaystyle \mathbf {B} ={\mu _{0} \over 4\pi }q{\mathbf {v} \times {\hat {\mathbf {r} }}' \over |\mathbf {r} '|^{2}}}

These equations were first derived byOliver Heaviside in 1888. Some authors[10][11] call the above equation forB{\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.[12]

Magnetic responses applications

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

Aerodynamics applications

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The figure shows the velocity (dV{\displaystyle dV}) induced at a point P by an element of vortex filament (dl{\displaystyle dl}) of strengthΓ{\displaystyle \Gamma }.

The Biot–Savart law is also used inaerodynamic theory to calculate the velocity induced byvortex lines.

In theaerodynamic application, the roles of vorticity and current are reversed in comparison to the magnetic application.

In Maxwell's 1861 paper 'On Physical Lines of Force',[13] magnetic field strengthH was directly equated with purevorticity (spin), whereasB 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,

Magnetic induction current
B=μH{\displaystyle \mathbf {B} =\mu \mathbf {H} } was essentially a rotational analogy to the linear electric current relationship,
Electric convection current
J=ρv,{\displaystyle \mathbf {J} =\rho \mathbf {v} ,} whereρ is electric charge density.

B was seen as a kind of magnetic current of vortices aligned in their axial planes, withH 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 theB 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 form solenoidal rings around a vortex axis. Analogy can be made that the vortex axis is playing the role that electric current plays inmagnetism. This puts the air currents of aerodynamics (fluid velocity field) into the equivalent role of the magnetic induction vectorB in electromagnetism.

In electromagnetism theB lines form solenoidal rings around the source electric current, whereas in aerodynamics, the air currents (velocity) 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 theB lines in isolation, we see exactly the aerodynamic scenario insomuch asB is the vortex axis andH is the circumferential velocity as in Maxwell's 1861 paper.

In two dimensions, for a vortex line of infinite length, the induced velocity at a point is given byv=Γ2πr{\displaystyle v={\frac {\Gamma }{2\pi r}}}whereΓ is the strength of the vortex andr is the perpendicular distance between the point and the vortex line. This is similar to the magnetic field produced on a plane by an infinitely long straight thin wire normal to the plane.

This is a limiting case of the formula for vortex segments of finite length (similar to a finite wire):v=Γ4πr[cos⁡A−cos⁡B]{\displaystyle v={\frac {\Gamma }{4\pi r}}\left[\cos A-\cos B\right]}whereA andB are the (signed) angles between the point and the two ends of the segment.

The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism

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See also:Curl (mathematics) andvector calculus identities

In amagnetostatic situation, the magnetic fieldB as calculated from the Biot–Savart law will always satisfyGauss's law for magnetism andAmpère's circuital law:[14]

Proof

Starting with the Biot–Savart law:B(r)=μ04π∭Vd3ℓJ(ℓ)×r−ℓ|r−ℓ|3{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\iiint _{V}d^{3}{\boldsymbol {\ell }}\,\mathbf {J} ({\boldsymbol {\ell }})\times {\frac {\mathbf {r} -{\boldsymbol {\ell }}}{|\mathbf {r} -{\boldsymbol {\ell }}|^{3}}}}

Substituting the relationr−ℓ|r−ℓ|3=−∇(1|r−ℓ|){\displaystyle {\frac {\mathbf {r} -{\boldsymbol {\ell }}}{|\mathbf {r} -{\boldsymbol {\ell }}|^{3}}}=-\nabla \left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)}and using theproduct rule for curls, as well as the fact thatJ does not depend onr{\displaystyle \mathbf {r} }, this equation can be rewritten as[14]B(r)=μ04π∇×∭Vd3ℓJ(ℓ)|r−ℓ|{\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\nabla \times \iiint _{V}d^{3}{\boldsymbol {\ell }}\,{\frac {\mathbf {J} ({\boldsymbol {\ell }})}{|\mathbf {r} -{\boldsymbol {\ell }}|}}}

Since the divergence of a curl is always zero, this establishesGauss's law for magnetism. Next, taking the curl of both sides, using the formula for thecurl of a curl, and again using the fact thatJ does not depend onr{\displaystyle \mathbf {r} }, we eventually get the result[14]∇×B=μ04π∇∭Vd3ℓJ(ℓ)⋅∇(1|r−ℓ|)−μ04π∭Vd3ℓJ(ℓ)∇2(1|r−ℓ|){\displaystyle \nabla \times \mathbf {B} ={\frac {\mu _{0}}{4\pi }}\nabla \iiint _{V}d^{3}{\boldsymbol {\ell }}\,\mathbf {J} ({\boldsymbol {\ell }})\cdot \nabla \left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)-{\frac {\mu _{0}}{4\pi }}\iiint _{V}d^{3}{\boldsymbol {\ell }}\,\mathbf {J} ({\boldsymbol {\ell }})\nabla ^{2}\left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)}

Finally, plugging in the relationships[14]∇(1|r−ℓ|)=−∇ℓ(1|r−ℓ|),∇2(1|r−ℓ|)=−4πδ(r−ℓ){\displaystyle {\begin{aligned}\nabla \left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)&=-\nabla _{\boldsymbol {\ell }}\left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right),\\\nabla ^{2}\left({\frac {1}{|\mathbf {r} -{\boldsymbol {\ell }}|}}\right)&=-4\pi \delta (\mathbf {r} -{\boldsymbol {\ell }})\end{aligned}}}(whereδ is theDirac delta function), using the fact that the divergence ofJ is zero (due to the assumption ofmagnetostatics), and performing anintegration by parts, the result turns out to be[14]∇×B=μ0J{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} }i.e. Ampère's circuital law. (Due to the assumption ofmagnetostatics,∂E/∂t=0{\displaystyle \partial \mathbf {E} /\partial t=\mathbf {0} }, so there is no extradisplacement current term in Ampère's law.)

In anon-magnetostatic situation, the Biot–Savart law ceases to be true (it is superseded byJefimenko's equations), whileGauss's law for magnetism and theMaxwell–Ampère law are still true.

Theoretical background

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Initially, the Biot–Savart law was discovered experimentally, then this law was derived in different ways theoretically. InThe Feynman Lectures on Physics, at first, the similarity of expressions for theelectric potential outside the static distribution of charges and themagnetic vector potential outside the system of continuously distributed currents is emphasized, and then the magnetic field is calculated through the curl from the vector potential.[15] Another approach involves a general solution of the inhomogeneous wave equation for the vector potential in the case of constant currents.[16] The magnetic field can also be calculated as a consequence of theLorentz transformations for the electromagnetic force acting from one charged particle on another particle.[17] Two other ways of deriving the Biot–Savart law include: 1) Lorentz transformation of theelectromagnetic tensor components from a moving frame of reference, where there is only an electric field of some distribution of charges, into a stationary frame of reference, in which these charges move. 2) the use of the method ofretarded potentials.

See also

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People

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Electromagnetism

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Notes

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  1. ^"Biot–Savart law".Random House Webster's Unabridged Dictionary.
  2. ^Jackson, John David (1999).Classical Electrodynamics (3rd ed.). New York: Wiley. Chapter 5.ISBN 0-471-30932-X.
  3. ^Landau, L. D.; Lifshitz, E. M. (1980).The Classical Theory of Fields: Volume 2 (4th ed.). Butterworth-Heinemann.ISBN 978-0750627689.
  4. ^abZhan, Marcus (2003)."Electromagnetic Field Theory: A Problem Solving Approach".ocw.mit.edu. RetrievedJuly 3, 2022.
  5. ^Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008,ISBN 978-0-471-92712-9
  6. ^The superposition principle holds for the electric and magnetic fields because they are the solution to a set oflinear differential equations, namelyMaxwell's equations, where the current is one of the "source terms".
  7. ^Freeland, R.M. (2015)."Mathematics of Magsail".Journal of the British Interplanetary Society.68:306–323 – via bis-space.com.
  8. ^abGriffiths, David J. (1998).Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 222–224, 435–440.ISBN 0-13-805326-X.
  9. ^Rosser, W. G. V. (1968).Classical Electromagnetism via Relativity. pp. 29–42.doi:10.1007/978-1-4899-6559-2.ISBN 978-1-4899-6258-4.
  10. ^Knight, Randall (2017).Physics for Scientists and Engineers (4th ed.). Pearson Higher Ed. p. 800.
  11. ^"Magnetic Field from a Moving Point Charge". Archived fromthe original on 2009-06-19. Retrieved2009-09-30.
  12. ^See the cautionary footnote in Griffiths p. 219 or the discussion in Jackson p. 175–176.
  13. ^Maxwell, J. C."On Physical Lines of Force"(PDF).Wikimedia commons. Retrieved25 December 2011.
  14. ^abcdeSee Jackson, page 178–79 or Griffiths p. 222–24. The presentation in Griffiths is particularly thorough, with all the details spelled out.
  15. ^The Feynman Lectures on Physics Vol. II Ch. 14: The Magnetic Field in Various Situations
  16. ^ David Tong. Lectures on Electromagnetism. University of Cambridge, Part IB and Part II Mathematical Tripos (2015).http://www.damtp.cam.ac.uk/user/tong/em.html.
  17. ^ Daniel Zile and James Overdui. Derivation of the Biot-Savart Law from Coulomb’s Law and Implications for Gravity. APS April Meeting 2014, abstract id. D1.033.https://doi.org/10.1103/BAPS.2014.APRIL.D1.33.

References

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Further reading

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  • Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974,ISBN 0-7131-2459-8
  • Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray,ISBN 0-7195-3382-1
  • The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010,ISBN 978-0-521-57507-2.
  • Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008,ISBN 0-7167-8964-7
  • Encyclopaedia of Physics (2nd Edition),R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
  • McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994,ISBN 0-07-051400-3

External links

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