Magnetostatics

MagnetostaticsPresented byMoitreya AdhikaryBESU, Shibpur

Topics to be coveredOverview of magnetostaticsLaws governing magnetostaticsContinuity of current equationMaxwells Correction of Amperes law, Concept of Displacement currentMaxwells equations for static field

What is magnetostatics? Magnetostaticsis the study ofmagnetic fieldsin systems where thecurrentsare steady (not changing with time) i.e. where charges are moving at constant velocity . It is the magnetic analogue of Electrostatics where thechargesare stationary.

A magnetostatic field is produced by a constant current flow (or direct current). This current flow may be due to magnetization currents* as in permanent magnets, electron-beam currents as in vacuum tubes, or conduction currents as in current-carrying wires. *For magnetic materials the relationship between B & H is The magnetizationMmakes a contribution to thecurrent densityJ, known as themagnetization currentorbound current:

However, the magnetization need not essentially be static; the equations of magnetostatics can be used to predict fast magnetic switchingevents 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.

Applications of magnetostatics Magnetostatics is widely used in applications ofmicromagneticssuch as models ofmagnetic recordingdevices. Also the development of the motors, transformers, microphones, compasses, telephone bell ringers, television focusing controls, advertising displays, magnetically levitated high-speed vehicles, memory stores, magnetic separators, and so on, involve magnetic phenomena and play an important role in our everyday life.

Discovery of the magnetic effect of electricity In 1820, Hans Christian Oersted (1777-1851), a Danish professor of physics, proved electric current can affect a compass needle. However, Oersted could not explain why.

Hans Christian Oersted's Experiment What really happened... In Oersted's experiment, a wire carrying a current has a magnetic field, which was proved later, surrounding it as shown in the figure. When this field is explored with a pivoted magnet or compass needle, the magnetic field produces an aligning force or torque on the needle such that the needle always orients itself normal to a radial line originating at the centre of the wire. This orientation is parallel to the magnetic field.

Meanwhile...The experiments and analyses of the effect of a current element were being carried out by Ampere and by Jean Baptiste Biot and Felix Savart, around 1820. Andre Marie Ampere (1775-1836), a French physicist, developed Oersted's discovery and introduced the concept of current element and the force between current elements.

Laws governing magnetostatic fields There are two major laws governing magnetostatic fields:

(1) Biot-Savart's law and (2) Ampere's circuit law.

Like Coulomb's law, Biot-Savart's law is the general law of magnetostatics. Just as Gauss's law is a special case of Coulomb's law, Ampere's law is a special case of Biot-Savart's law and is easily applied in problems involving symmetrical current distribution.

(1) Biot-Savart's Law Biot-Savart's law states that the magnetic field intensity dH produced at a point P, as shown in the figure1, by the differential current clement Idl is proportional to the product Idl and the sine of the angle between the clement and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element.

where R = |R| and aR = R/R. Thus the direction of dH can be determined by the right-hand rule with the right-hand thumb pointing in the direction of the current, the right-hand fingers encircling the wire in the direction of dH as shown in figure2 next. Alternatively, we can use the right-handed screw rule to determine the direction of dH with the screw placed along the wire and pointed in the direction of current flow, the direction of advance of the screw is the direction of dH as in figure2.

It is customary to represent the direction of the magnetic field intensity H (or current I) by a small circle with a dot or cross sign depending on whether H (or I) is out of, or into, the page as illustrated in figure3.

Just as we can have different charge configurations in electrostatics, we can have different current distributions: line current, surface current, and volume current as shown in figure4. If we define K as the surface current density (in amperes/meter) and J as the volume current density (in amperes/meter square), the source elements are related as

Thus in terms of the distributed current sources, the Biot-Savart's law becomes

Applications of Biot-Savart's lawBiot-Savart's law can be applied for computing the magnetic field due to a line, surface or volume current distributions for geometries like current loop, finite straight wire etc. The Biot-Savart's law can be used in the calculation of magnetic responses even at the atomic or molecular level, e.g. chemical shielding or magnetic susceptibilities, provided that the current density can be obtained from a quantum mechanical calculation or theory. The Biot-Savart's law is also used in aerodynamic theory to calculate the velocity induced by vortex lines.

Example of the application of Biot-Savart's law for calculating magnetic field due a circular current loop

Field at Centre of Current Loop

The form of the magnetic field from a current element IdL in the Biot-Savart's law becomes

which in this case simplifies greatly because the angle =90 for all points along the path and the distance to the field point is constant. The integral becomes B) Field on axis of Current Loop

The application of the Biot-Savart's law on the centre line of a current loop involves integrating the z-component.

The symmetry is such that all the terms in this element are constant except the distance element dL , which when integrated just gives the circumference of the circle. The magnetic field is then

Limitation of Biot-Savart's law The fundamental limitation of Biot-Savart's law is that it is valid in magnetostatic approximation i.e. in the systems involving steady currents (not changing with time). (2) Ampere's circuit law Ampere's circuit law states that the line integral of the tangential component of H around a closed path is the same as the net current Ienc enclosed by the path.

In other words, the circulation of H equals Ienc; that is, (Integral form of Ampere's law)

By applying Stoke's theorem to the left-hand side of the former equation we obtain

But

Thus comparing two equations (differential form of Ampere's law)

This equation is also known as Maxwell's fourth equation for static field.Applications of Ampere's law Gauss's law in electrostatics is used to determine electric intensity for the surface of uniform or symmetric charge distribution. Similarly, Ampere's circuital law is used to determine magnetic induction for current distribution with sufficient symmetry. Ampere's circuital law is used for example to determine magnetic field due to a straight conductor carrying current, magnetic field due to a solenoid carrying current, magnetic field due to a current in a toroid etc.

In order to use this law, it is necessary to choose a closed path for which it is possible to determine the line integral of H. For this reason, this law has a limited use. In some applications where this law is not applicable then we use Biot-Savart's law. Example of the application of Ampere's law for calculating magnetic field due an infinite filamentary line current

Where to use Ampere's law, where to use Biot-Savart's law? It depends on whether a path integral or a normal (ordinary) integral is easier to determine. When path integral is easier to calculate we prefer Ampere's law and when ordinary integral is more readily determinable, we should go for Biot-Savarts law. However, it is true that Biot-Savart's law is more general than Ampere's law which is a derivative of the former under special condition. Hence for arbitrary current distribution Biot-Savart's law is more generally used. Limitations of Ampere's law There are two important issues regarding Ampere's law that require closer scrutiny. First, there is an issue regarding the continuity equation for electrical charge. There is a theorem in vector calculus that states the divergence of a curl must always be zero. Hence where magnetic flux density

and so the original Ampere's law implies that

But in general

which is non-zero for a time-varying charge density, for instance, in a capacitor circuit where time-varying charge densities exist on the plates.

Second, there is an issue regarding the propagation of electromagnetic waves. For example, in free space, where

Ampere's law implies that

but instead

To treat these situations, the contribution of displacement current must be added to the current term in Ampere's law. James Clerk Maxwell conceived of displacement current as a polarization current in the dielectric vortex sea, which he used to model the magnetic field hydro-dynamically and mechanically. He added this displacement current to Ampere's circuital law in his 1861 paper "On Physical Lines of Force". Electric displacement field Inphysics, theelectric displacement field, denoted byD, is avector fieldthat appears inMaxwell's equations. It accounts for the effects of free and bound charge withinmaterials. "D" stands for "displacement", as in the related concept ofdisplacement currentindielectrics. Infree space, the electric displacement field is equivalent toflux density i.e. D=E.

Why is the term Displacement? In adielectricmaterial the presence of anelectric fieldEcauses the bound charges in the material (atomicnucleiand theirelectrons) to slightly separate, inducing a localelectric dipole moment. The electric displacement fieldDis defined as

where0is thevacuum permittivity(also called permittivity of free space), andPis the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the polarization density. Here we will consider P=0.Continuity equation Due to the principle of charge conservation, the time rate of decrease of charge within a given volume must be equal to the net outward current flow through the closed surface of the volume. Thus current Iout coming out of the closed surface is

where Qin is the total charge enclosed by the closed surface. Invoking divergence theorem ,

ButSubstituting last two equations into first one gives orwhich is called the continuity of current equation. It must be kept in mind that the continuity equation is derived from the principle of conservation of charge and essentially states that there can be no accumulation of charge at any point. For steady currents, v/t = 0 and hence

Correction of Amperes law Though Maxwell did not use the continuity equation as a primary motive to arrive at the correction needed for Ampere's law, we can combine the continuity equation and Gauss's law of electrostatics to arrive at the concept of displacement current. The continuity equation states:

Using Gauss's law it is trivial to show:

Adding this extra term gives Maxwell's correction to Ampere's law. Thus for time varying field, Ampere's law now states:

A practical consideration that shows the problem with Amperes lawConsider a spherically-symmetric radial distribution of currents. We get this by placing a charged conducting sphere into a poorly conducting medium. The charge leaks out (Q is time-dependent):

Hence the spherical symmetry is preserved. What would be the magnetic field generated by this current distribution?

Weve got a PROBLEM: For any loop with a non-zero enclosed current flux, Amperes Law predicts a non-zero magnetic field. However, due to the spherical symmetry, the magnetic field must be zero everywhere! We conclude that something is missing. This missing part is an additional B generated by the time-dependent E.

Maxwell suggested generalization of Amperes Law by adding the quantity to . This modification resolves all paradoxes and makes the system of equations symmetric: not only the time-dependent generates , but also the time-dependent generates ! Displacement current:

Maxwell: One of the chief peculiarities of this thesis is the doctrine which asserts that the true electric current, that upon which electromagnetic phenomena depend, is not the same thing as the current of conduction but that the time derivative of the electric displacement must be taken into account. Displacement current in a capacitor

Although current is flowing through the capacitor, no actual charge is transported through the vacuum between its plates. Nonetheless, a magnetic field exists between the plates as though a current were present there as well. This current is what we call the Displacement current.Why was not the displacement current discovered experimentally prior to Maxwell? Because in the old-fashioned experiments with highly-conducting wires and slowly-varying fields the displacement current is much smaller than the conduction current: Lets estimate how quickly the electric field should be changed in order to make the conduction current density and displacement current density of the same order of magnitude:

For the first time the magnetic field generated by the displacement current inside a capacitor was directly measured in 1985 (!!!).Importance of Displacement current The key to electromagnetic propagation is Displacement current by virtue of which electromagnetic wave can travel through materials even through vacuum. Because of this fact, one can argue that life itself on earth is possible because of displacement current.Why? Because the heat from the sun warms our planet through radiative heat transfer (rather than by conduction or convection). This radiation of heat is an electromagnetic wave. Were it not for the displacement current and other effects there would be no heat and light from the sun warming and illuminating our planet, and consequently no life on earth. So thanks to our displacement current!

Maxwell's equations for static fields The four Maxwell's equations for static or non-time-varying EM fields are listed in the below table

The first equation comes from electrostatics (modified Coulomb's law or Gauss's law).

The second law stems from the fact that magnetic poles cannot be isolated or magnetic monopole does not exist.

The third law signifies the conservativeness (since line integral of electric field is independent of the choice of path, which is zero. Conservative fields are irrotational i.e. they have vanishing curl as is the case here for E) of electrostatic field.

The fourth law is the Ampere's law (without Maxwell's correction).

Unlike electric flux lines, magnetic flux lines always close upon themselves as in the following figure.

This is due to the fact that it is not possible to have isolated magnetic poles (or magnetic charges). For example, if we desire to have an isolated magnetic pole by dividing a magnetic bar successively into two, we end up with pieces each having north and south poles as was illustrated in figure8. Thus the total flux through a closed surface in a magnetic field must be zero; that is,

This equation is referred to as the law of conservation of magnetic flux or Gauss's law for magnetostatic field. By applying the divergence theorem we obtain

or

Hence the second law of Maxwell for static field is derived. The second law remains unchanged for time varying fields.ReferencesBooks: Elements of Electromagnetics by Matthew N.O. Sadiku Electromagnetic Waves And Radiating Systems by Jordan, Balmain Electromagnetics by John D. Kraus & Keith R. Carver Time-harmonic Electromagnetic Fields by Roger F. Harrington Introduction to Electrodynamics by David J. Griffiths

Websites: http://en.wikipedia.org http://www.maxwells-equations.com/ http://www.phy.duke.edu http://hyperphysics.phy-astr.gsu.edu http://teacher.nsrl.rochester.edu Thank You