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Magnetic field due to small thin current element

Magnetic field acts perpendicular to the plane formed by current element and displacement vectors

The current in the arm AB and CD do not produce magnetic field at point P as the point lies on the extended line of the current length element. Recall that θ=0, sinθ=0, hence B=0. On the other hand the wire segment BC is designed to be far off from point P in comparison to small wire segment AD. Since magnetic field due to individual current element of segment AD is inversely proportional to the square of linear distance, the magnetic field at P due to AC is relatively negligible with respect to magnetic field due to small wire element AD. Clearly, magnetic field at P is nearly equal to magnetic field due to small current element AD. The measurement of magnetic field at P with this arrangement allows us to determine magnetic field due to small current element AD and thus, allows us to verify the law.

Electromagnetism

We study magnetism under the nomenclature “electromagnetism” to emphasize that magnetism is actually a specific facet of electrical phenomenon. This is not farther from the reality as well. Let us see what happens when charge flows through the wire. Every particle carrying charge is capable of producing electrical field. In this case of a wire carrying steady current, however, charge is moving with certain velocity through the wire (conductor). Though, there is net velocity associated with the charge, the net electric charge in any infinitesimal volume element is zero. This means that the "charge density" at any point is zero but the "current density" at that point is non-zero for a conductor carrying current.

Since there is no charge density, there is no electric field. Recall that a net charge stationary or moving produces electric field. On the other hand, since there is net motion of charge, there is magnetic field.

Subsequently, we shall learn that a varying or changing magnetic field sets up an electric field. This aspect is brought out by Faraday's induction law. The electromagnetic induction sets up the basis of interlinking of electrical and magnetic phenomena. The production of electric field (and hence current in a conductor) due to varying magnetic field suggests that its inverse should also be true. As a matter of fact, this is so. Maxwell discovered that a varying electric field sets up a magnetic field. Thus, two phenomena are reciprocal of each other and prove the strong connection between electricity and magnetism.

In general, we consider electrical property to be the precursor of magnetic property. One of the most important arguments that advances this thinking is the existence of electrical monople i.e. a charge of specific polarity. There is no such magnetic monopole as yet. Magnetic polarities exist in pair (recall a magnet has a pair of north and south pole).

The connection between electric and magnetic field is futher verified by the fact that a stationary charge in one frame of reference sets up only electric field in that reference. But the same stationary charge in one frame of reference sets up both electric and magnetic fields in a frame of reference, which is moving at certain relative velcoity with respect to first reference. Similarly, a moving charge in one frame of reference sets up both electric and magnetic fields in that frame of reference, but it sets up only electric field in a reference in which the moving charge is stationary (we can always imagine one such frame to exist).

The above discussion also draws an important distinction between “current in wire” and “moving charge”, which have been said to be equivalent in earlier text. Current in wire sets up only magnetic field. Moving charge, on the other hand, sets up magnetic field in addition to electric field as there is net charge – unlike the case of current in wire in which there is no net charge. Clearly, equivalence of "current in small element of wire" and "moving charge" is limited to production of magnetic field only.

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Source:  OpenStax, Electricity and magnetism. OpenStax CNX. Oct 20, 2009 Download for free at http://cnx.org/content/col10909/1.13
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