BIOT-SAVART'S LAW OR AMPERE'S LAW FOR CURRENT ELEMENT
- Biot Savart Law Explained
- Biot Savart Law Wire
- Application Of Biot Savart Law
- Biot Savart Law In Terms Of Current Density Definition
The Biot Savart Law.doc 1/4 Jim Stiles The Univ. Of EECS The Biot-Savart Law So, we now know that given some current density, we can find the resulting magnetic vector potential A(r): 0 (r) r 4rr V dv µ π ′ = ′ ∫∫∫ − ′ J A and then determine the resulting magnetic flux density B(r) by taking the curl. Below is the Biot Savart law expression before the substitution of j. Substituting for j (current density) where I is the vector current and dl is element length. After substituting for j, we obtain Biot Savart expression. Biot Savart law for point charges. Following is the Biot Savart law derivation for point charge.
Biot-Savart's law deals with the magnetic field induction at a point due to a small current element.
Current element
A current element is a conductor carrying current.It is the product of current,I and length of very small segment of current carrying wire ,dL.
Let us consider a small element AB of length dl of the conductor RS carrying a current I.
Let r be the position vector of the point P from the current element I dL.and θ be the angle dl and r.
According to Biot-Savart's law,the magnetic field induction dB or magnetic flux density at a point P due to current element depends upon the following factors.
(i) dB∞I
(ii) dB∞dl
(iii) dB∞sinθ
(iv) dB∞1/r2
Combining these factors,we get
dB∞IdLsinθ/r2
or dB=K Idl sinθ/r2
where K is a constant of perportionality.
In S.I units, K=μ0/4π
thus , dB=μ0/4π I dl sinθ/r2
where μ0 is absolute premeability of free space and
μ0=4π*10-7 Wb A-1m-1
= 4π*10-7*TA-1m [ 1T=1 Wb m-2]
In C.G.S units,K=1 (In free space)
Thus dB=Idl sinθ/r2
In vector form,
dB=μ0/4π I(dl*r)/r3
magnetic field induction at point P due to current through entire wire is
B=∫μ0/4πIdl*r/r3
Or B=∫μ0/4π Idl sin θ/r2
BiotSavart's law in terms of magnetising force or magnetic intensity (H) of the magnetic field:
In S.I System,
dH=dB/μ0=1/4π Idl*r/r3=1/4π Idl*αr/r2
H=∫1/4π Idl*αr/r2
H= ∫1/4π Idl sinθ/r2
Importance OF BIOT SAVART'S LAW:-
- This law is analogous to Coulomb's law in electrostatics.
- Biot Savart's law is valid for a symmetrical current distribution.
- This law cannot be easily verified experimentally as the current carrying conductor of very small length cannot be obtained pratically.
- Biot Sarvart's law is applicable only to very small length conductor carrying current.
- The direction of dB is perpendicular to both Idl and r.
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Biot Savart Law Explained
In physics, specifically electromagnetism, the Biot–Savart law (/ˈbiːoʊsəˈvɑːr/ or /ˈbjoʊsəˈvɑːr/)[1] is an equation describing the magnetic field generated by a constant electric current. 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 that of 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 consistent with both Ampère's circuital law and Gauss's law for magnetism.[2] It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.
- 1Equation
- 5See also
Equation
Electric currents (along a closed curve/wire)
The Biot–Savart law is used for computing the resultant magnetic fieldB at position r in 3D-space generated by a flexible currentI (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]
[math] mathbf{B}(mathbf{r}) = frac{mu_0}{4pi} int_C frac{I , dboldsymbol elltimesmathbf{r'}}{|mathbf{r'}|^3}[/math]
where [math]dboldsymbol ell[/math] is a vector along the path [math]C[/math] whose magnitude is the length of the differential element of the wire in the direction of conventional current. [math]boldsymbol ell[/math] is a point on path [math]C[/math]. [math]mathbf{r'} = mathbf{r} - boldsymbol ell[/math] is the full displacement vector from the wire element ([math]dboldsymbol ell[/math]) at point [math]boldsymbol ell[/math] to the point at which the field is being computed ([math]mathbf{r}[/math]), and μ0 is the magnetic constant. Alternatively:
- [math] mathbf{B}(mathbf{r}) = frac{mu_0}{4pi}int_C frac{I , dboldsymbol elltimesmathbf{hat r'}}{|mathbf{r'}|^2}[/math]
where [math]mathbf{hat r'}[/math] is the unit vector of [math]mathbf{r'}[/math]. The symbols in boldface denote vector quantities.
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 (this concept was used in the definition of the SI unit of electric current—the Ampere—until 20 May 2019).
To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen ([math]mathbf r[/math]). 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 [math]mathbf{J}[/math] (current density). The resulting formula is:
- [math] mathbf{B}(mathbf{r}) = frac{mu_0}{2pi}int_C frac{(mathbf J, dell)timesmathbf r'}{|mathbf r'|} = frac{mu_0}{2pi}int_C (mathbf J, dell)timesmathbf{hat r'} [/math]
Electric current density (throughout conductor volume)
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:
[math] mathbf B (mathbf r) = frac{mu_0}{4pi}iiint_V frac{(mathbf{J},dV)timesmathbf r'}{|mathbf r'|^3}[/math]
where [math]mathbf{r'}[/math] is the vector from dV to the observation point [math]mathbf{r}[/math], [math]dV[/math] is the volume element, and [math]mathbf{J}[/math] is the current density vector in that volume (in SI in units of A/m2).
In terms of unit vector [math]mathbf{hat r'}[/math]
- [math] mathbf B (mathbf r) = frac{mu_0}{4pi}iiint_V dV frac{mathbf Jtimesmathbf{hat r'}}{|mathbf r'|^2} [/math]
Constant uniform current
In the special case of a uniform constant current I, the magnetic field [math]mathbf{B}[/math] is
- [math] mathbf{B}(mathbf{r}) = frac{mu_0}{4pi} I int_C frac{dboldsymbol ell times mathbf{r'}}{|mathbf{r'}|^3}[/math]
i.e. the current can be taken out of the integral.
Point charge at constant velocity
In the case of a point charged particleq moving at a constant velocityv, Maxwell's equations give the following expression for the electric field and magnetic field:[5]
- [math]begin{align} mathbf{E} &= frac{q}{4pi epsilon_0} frac{1 - frac{v^2}{c^2}}{left(1 - frac{v^2}{c^2}sin^2thetaright)^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{align}[/math]
where [math]mathbf hat r'[/math] 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 [math]mathbf v[/math] and [math]mathbf r'[/math].
When v2 ≪ c2, the electric field and magnetic field can be approximated as[5]
- [math] mathbf{E} =frac{q}{4piepsilon_0} frac{mathbf{hat r'}}{|mathbf r'|^2} [/math]
- [math] mathbf{B} =frac{mu_0 q}{4pi} mathbf{v} times frac{mathbf{hat r'}}{|mathbf r'|^2} [/math]
These equations were first derived by Oliver Heaviside in 1888. Some authors[6][7] call the above equation for [math]mathbf{B}[/math] 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.[8]
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 in aerodynamic theory to calculate the velocity induced by vortex lines.
Biot Savart Law Wire
In the aerodynamic 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',[9] magnetic field strength H was directly equated with pure vorticity (spin), whereas 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,
- Magnetic induction current
- [math]mathbf{B} = mu mathbf{H}[/math]
- Electric convection current
- [math]mathbf{J} = rho mathbf{v}[/math]
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 form solenoidal rings around a vortex axis. 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) into the equivalent role of the magnetic induction vector B in electromagnetism.
In electromagnetism the B 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 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.
In two dimensions, for a vortex line of infinite length, the induced velocity at a point is given by
- [math]v = frac{Gamma}{2pi r}[/math]
where Γ is the strength of the vortex and r 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):
- [math]v = frac{Gamma}{4 pi r} left[cos A - cos B right][/math]
where A and B are the (signed) angles between the line and the two ends of the segment.
The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism
- See also: Curl (mathematics) and Vector calculus identities
In a magnetostatic situation, the magnetic field B as calculated from the Biot–Savart law will always satisfy Gauss's law for magnetism and Ampère's law:[10]
The Biot–Savart law is used for computing the resultant magnetic fieldB at position r in 3D-space generated by a flexible currentI (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]
[math] mathbf{B}(mathbf{r}) = frac{mu_0}{4pi} int_C frac{I , dboldsymbol elltimesmathbf{r'}}{|mathbf{r'}|^3}[/math]
where [math]dboldsymbol ell[/math] is a vector along the path [math]C[/math] whose magnitude is the length of the differential element of the wire in the direction of conventional current. [math]boldsymbol ell[/math] is a point on path [math]C[/math]. [math]mathbf{r'} = mathbf{r} - boldsymbol ell[/math] is the full displacement vector from the wire element ([math]dboldsymbol ell[/math]) at point [math]boldsymbol ell[/math] to the point at which the field is being computed ([math]mathbf{r}[/math]), and μ0 is the magnetic constant. Alternatively:
- [math] mathbf{B}(mathbf{r}) = frac{mu_0}{4pi}int_C frac{I , dboldsymbol elltimesmathbf{hat r'}}{|mathbf{r'}|^2}[/math]
where [math]mathbf{hat r'}[/math] is the unit vector of [math]mathbf{r'}[/math]. The symbols in boldface denote vector quantities.
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 (this concept was used in the definition of the SI unit of electric current—the Ampere—until 20 May 2019).
To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen ([math]mathbf r[/math]). 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 [math]mathbf{J}[/math] (current density). The resulting formula is:
- [math] mathbf{B}(mathbf{r}) = frac{mu_0}{2pi}int_C frac{(mathbf J, dell)timesmathbf r'}{|mathbf r'|} = frac{mu_0}{2pi}int_C (mathbf J, dell)timesmathbf{hat r'} [/math]
Electric current density (throughout conductor volume)
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:
[math] mathbf B (mathbf r) = frac{mu_0}{4pi}iiint_V frac{(mathbf{J},dV)timesmathbf r'}{|mathbf r'|^3}[/math]
where [math]mathbf{r'}[/math] is the vector from dV to the observation point [math]mathbf{r}[/math], [math]dV[/math] is the volume element, and [math]mathbf{J}[/math] is the current density vector in that volume (in SI in units of A/m2).
In terms of unit vector [math]mathbf{hat r'}[/math]
- [math] mathbf B (mathbf r) = frac{mu_0}{4pi}iiint_V dV frac{mathbf Jtimesmathbf{hat r'}}{|mathbf r'|^2} [/math]
Constant uniform current
In the special case of a uniform constant current I, the magnetic field [math]mathbf{B}[/math] is
- [math] mathbf{B}(mathbf{r}) = frac{mu_0}{4pi} I int_C frac{dboldsymbol ell times mathbf{r'}}{|mathbf{r'}|^3}[/math]
i.e. the current can be taken out of the integral.
Point charge at constant velocity
In the case of a point charged particleq moving at a constant velocityv, Maxwell's equations give the following expression for the electric field and magnetic field:[5]
- [math]begin{align} mathbf{E} &= frac{q}{4pi epsilon_0} frac{1 - frac{v^2}{c^2}}{left(1 - frac{v^2}{c^2}sin^2thetaright)^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{align}[/math]
where [math]mathbf hat r'[/math] 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 [math]mathbf v[/math] and [math]mathbf r'[/math].
When v2 ≪ c2, the electric field and magnetic field can be approximated as[5]
- [math] mathbf{E} =frac{q}{4piepsilon_0} frac{mathbf{hat r'}}{|mathbf r'|^2} [/math]
- [math] mathbf{B} =frac{mu_0 q}{4pi} mathbf{v} times frac{mathbf{hat r'}}{|mathbf r'|^2} [/math]
These equations were first derived by Oliver Heaviside in 1888. Some authors[6][7] call the above equation for [math]mathbf{B}[/math] 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.[8]
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 in aerodynamic theory to calculate the velocity induced by vortex lines.
Biot Savart Law Wire
In the aerodynamic 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',[9] magnetic field strength H was directly equated with pure vorticity (spin), whereas 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,
- Magnetic induction current
- [math]mathbf{B} = mu mathbf{H}[/math]
- Electric convection current
- [math]mathbf{J} = rho mathbf{v}[/math]
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 form solenoidal rings around a vortex axis. 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) into the equivalent role of the magnetic induction vector B in electromagnetism.
In electromagnetism the B 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 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.
In two dimensions, for a vortex line of infinite length, the induced velocity at a point is given by
- [math]v = frac{Gamma}{2pi r}[/math]
where Γ is the strength of the vortex and r 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):
- [math]v = frac{Gamma}{4 pi r} left[cos A - cos B right][/math]
where A and B are the (signed) angles between the line and the two ends of the segment.
The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism
- See also: Curl (mathematics) and Vector calculus identities
In a magnetostatic situation, the magnetic field B as calculated from the Biot–Savart law will always satisfy Gauss's law for magnetism and Ampère's law:[10]
Outline of proof[10] (Click 'show' on the right.) |
---|
Starting with the Biot–Savart law:
Substituting the relation
and using the product rule for curls, as well as the fact that J does not depend on [math]mathbf r[/math], this equation can be rewritten as[10]
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 [math]mathbf r[/math], we eventually get the result[10]
Finally, plugging in the relations[10]
(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[10]
i.e. Ampère's law. (Due to the assumption of magnetostatics, [math]partial mathbf E / partial t = mathbf 0[/math], so there is no extra displacement current term in Ampère's law.) |
In a non-magnetostatic situation, the Biot–Savart law ceases to be true (it is superseded by Jefimenko's equations), while Gauss's law for magnetism and the Maxwell–Ampère law are still true.
See also
People
Electromagnetism
- Ampère's law
- Darwin Lagrangian
Notes
- ↑'Biot-Savart law'. Random House Webster's Unabridged Dictionary.
- ↑Jackson, John David (1999). Classical Electrodynamics (3rd ed.). New York: Wiley. Chapter 5. ISBN0-471-30932-X.
- ↑Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN:978-0-471-92712-9
- ↑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'.
- ↑ 5.05.1Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 222–224, 435–440. ISBN0-13-805326-X. https://archive.org/details/introductiontoel00grif_0/page/222.
- ↑Knight, Randall (2017). Physics for Scientists and Engineers (4th ed.). Pearson Higher Ed.. p. 800.
- ↑'Archived copy'. Archived from the original on 2009-06-19. https://web.archive.org/web/20090619185915/http://maxwell.ucdavis.edu/~electro/magnetic_field/pointcharge.html. Retrieved 2009-09-30.
- ↑See the cautionary footnote in Griffiths p. 219 or the discussion in Jackson p. 175–176.
- ↑Maxwell, J. C.. 'On Physical Lines of Force'. Wikimedia commons. http://commons.wikimedia.org/wiki/File:On_Physical_Lines_of_Force.pdf. Retrieved 25 December 2011.
- ↑ 10.010.110.210.310.410.5See Jackson, page 178–79 or Griffiths p. 222–24. The presentation in Griffiths is particularly thorough, with all the details spelled out.
References
- Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN0-13-805326-X. https://archive.org/details/introductiontoel00grif_0.
- Feynman, Richard (2005). The Feynman Lectures on Physics (2nd ed.). Addison-Wesley. ISBN978-0-8053-9045-2.
Further reading
- 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
- Electromagnetism, B. Crowell, Fullerton College
- MISN-0-125The Ampère–Laplace–Biot–Savart Law by Orilla McHarris and Peter Signell for Project PHYSNET.
- Magnetic field of a circular loop with electric current, Illustration of Biot-Savart law
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