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In physics, specifically electromagnetism, the Biot–Savart law ( /
Physics is the natural science that studies matter and its motion and behavior through space and time and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as electric fields, magnetic fields, and light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.
A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field varies with location. As such, it is an example of a vector field.
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
An electric current is a flow of electric charge. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges; positive and negative. Like charges repel and unlike attract. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane.
where is a vector along the path whose magnitude is the length of the differential element of the wire in the direction of conventional current . is the full displacement vector from the wire element () to the point at which the field is being computed (), and μ0 is the magnetic constant. Alternatively:
In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.
where is the unit vector of . The symbols in boldface denote vector quantities.
In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.
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).
The ampere, often shortened to "amp", is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics.
To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen (). 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.
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input produces response.
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 (current density). The resulting formula is:
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:
where is the volume element and is the current density vector in that volume (in SI in units of A/m2).
In the special case of a steady constant current I, the magnetic field is
i.e. the current can be taken out of the integral.
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:
where 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 and .
When v2 ≪ c2, the electric field and magnetic field can be approximated as
These equations are called the "Biot–Savart law for a point charge"due to its closely analogous form to the "standard" Biot–Savart law given previously. These equations were first derived by Oliver Heaviside in 1888.
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.
The Biot–Savart law is also used in aerodynamic theory to calculate the velocity induced by vortex lines.
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',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,
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
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):
where A and B are the (signed) angles between the line and the two ends of the segment.
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:
|Outline of proof (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 , this equation can be rewritten as
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 , we eventually get the result
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.
The centimetre–gram–second system of units is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism.
An electromagnetic field is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature.
In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.
An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. Mathematically the electric field is a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strengh. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.
In classical electromagnetism, Ampère's circuital law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.
The magnetic moment is a quantity that represents the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.
In electromagnetism, displacement current density is the quantity ∂D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potentialψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.
Magnetostatics is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory. Magnetostatic focussing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis.
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector M.
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has an exactly defined value that can be approximated as
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
In electromagnetism, Jefimenko's equations give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay of the fields due to the finite speed of light and relativistic effects. Therefore they can be used for moving charges and currents. They are the general solutions to Maxwell's equations for any arbitrary distribution of charges and currents.
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.
In magnetostatics, the force of attraction or repulsion between two current-carrying wires is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, following the Biot–Savart law, and the other wire experiences a magnetic force as a consequence, following the Lorentz force law.
Magnets exert forces and torques on each other due to the rules of electromagnetism. The forces of attraction field of magnets are due to microscopic currents of electrically charged electrons orbiting nuclei and the intrinsic magnetism of fundamental particles that make up the material. Both of these are modeled quite well as tiny loops of current called magnetic dipoles that produce their own magnetic field and are affected by external magnetic fields. The most elementary force between magnets, therefore, is the magnetic dipole–dipole interaction. If all of the magnetic dipoles that make up two magnets are known then the net force on both magnets can be determined by summing up all these interactions between the dipoles of the first magnet and that of the second.
Weber electrodynamics is an alternative to Maxwell electrodynamics developed by Wilhelm Eduard Weber. In this theory, Coulomb's Law becomes velocity dependent. In mainstream contemporary physics, Maxwell electrodynamics is treated as the uncontroversial foundation of classical electromagnetism, while Weber electrodynamics is generally unknown.