The Biot – Savart law is formulated in a restricted context. This law is true for (i) a small element "dl " of a thin wire carrying current and (ii) steady current i.e. flow of charge per unit time through the wire is constant. Biot – Savart’s law for free space is given by :

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The ratio

Note the vector cross product in the numerator. Direction of magnetic field produced is given by the direction of vector cross product dlXr . Further, it is also clear that as far as magnitude of magnetic field is concerned it is inversely proportional to the square of the linear distance i.e.

Now, the unit vector in the direction of line joining current element and point is given by :

Substituting in the Biot-Savart expression for r, we have :

Some important deductions arising from Biot-Savart law are given in the following subsections.

** Direction of magnetic field and superposition principle
**

The direction of magnetic field is the direction of vector cross product dl X r. In the figure shown below, the wire and displacement vector are considered to be in the plane of drawing (xy plane). Clearly, direction of magnetic field is perpendicular to the plane of drawing. In order to know the orientation, we align or curl the fingers of right hand as we travel from vector dl to vector r as shown in the figure. The extended thumb indicates that magnetic field is into the plane of drawing (-z direction), which is shown by a cross (X) symbol at point P.

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This was a simplified situation. What if wire lies in three dimensional space (not in xy plane of reference shown in figure) such that different parts of the wire form different planes with displacement vectors. In such situations, magnetic fields due to different current elements of the current carrying wire are in different directions as shown here.

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It is clear that directions of magnetic field due to different elements of the wire may not be along the same line. On the other hand, a single mathematical expression such as that of Biot-Savart can not denote multiple directions. For this reason, Biot-Savart’s law is stated for a small element of wire carrying current – not for the extended wire carrying current. However, we can find magnetic field due to extended wire carrying current by using superposition principle i.e. by using vector additions of the individual magnetic fields due to various current elements. We shall see subsequently that as a matter of fact we can integrate Biot-Savart’s vector expression for certain situations like straight wire or circular coil etc as :

For better appreciation of directional property of magnetic field, yet another visualization of three dimensional representation of magnetic field due to a small element of current is shown here :

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The circles have been drawn such that their centers lie on the tangent YY’ drawn along the current length element dl and the planes of circles are perpendicular to it as shown in the figure. Note that magnetic field being perpendicular to the plane formed by vectors dl and r are tangential to the circles drawn. Also, each point on the circle is equidistant from the current element. As such, magnitudes of magnetic field along the circumference are having same value. Note, however, that they have shown as different vectors

For the time being we shall use the right hand rule for the vector cross product to determine the direction of magnetic field for each current element. There are, however, few elegant direction finding rules for cases of extended wires carrying current like straight wire or circular coil. These rules will be described in separate modules on the respective topics.

** Magnitude of magnetic field
**

The magnitude of magnetic field is given by :

The magnitude depends on angle (θ) between two vector elements "dl" and "r". For a point on the wire element or on the tangent drawn to it, the angle θ = 0° or 180° and the trigonometric sine ratio of the angle is zero i.e. sinθ = 0. Thus, magnetic field at a point on the extended line passing through vector "dl" is zero.

Further magnetic field is very small due to small value of proportionality constant, which is equal to

** Other form of Biot – Savart’s law
**

We have stated earlier that source of magnetic field is a small element of current or a moving charge. After all, current is nothing but passage of charge. Clearly, there needs to be an alternative expression for the Biot-Savart’s law in terms of charge and its velocity. Now, for steady current :

This equivalence for current with moving charge with respect to production of magnetic field helps us to formulate Biot – Savart’ law for a charge q, which is moving with constant speed v as :

The equivalence noted for current and moving charge is quite interesting for sub-atomic situations. An electron moving around nucleus can be considered to be equivalent to current. In Bohr’s atom,

where T is time period of revolution. Now,

where v is the speed of electron moving around. Combining above two equations, we have :

Thus, an electron moving in circular path is equivalent to a steady current I. Negative sign here indicates that the equivalent current is opposite to the direction of motion of electron around nucleus.

** The source (cause) of magnetic field
**

The basic source (cause) of electric field is a scalar point charge. What is the correspondence here? Is current (I) the corresponding basic source for the magnetic field? An examination of the Biot – Savart’s law reveals that it is not “I” alone which is basic source (cause) – rather it is the vector Idl, referred as "current element". This means that the source responsible for magnetic field is identified by current (I) and length of element (dl) together. Equivalently, the basic source of magnetism is a moving charge represented by the vector qv.