0.2 Biot - savart law  (Page 3/4)

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 :

$$đB=\frac{{\mu }_0}{4\pi }\frac{Iđl\sin \theta }{r^2}$$

The magnitude depends on angle (θ) between two vector elements "d l " 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 "d l " is zero.

Further magnetic field is very small due to small value of proportionality constant, which is equal to ${10}^{-7}$ SI unit. The relative weakness of magnetic field is evident from the fact that proportionality constant for Coulomb’s law has the value $9X{10}^9$ in SI unit.

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 :

$$Iđl=\frac{q}{t}đl=q\frac{đl}{đt}=qv$$

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 :

$$đ\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{q\mathbf{v}X\underset{}{\overset{‸}{\mathbf{r}}}}{r^2}$$

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,

$$I=\frac{-e}{T}$$

where T is time period of revolution. Now,

$$T=\frac{2\pi r}{v}$$

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

$$\Rightarrow I=-\frac{ev}{2\pi r}$$

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 Id l , 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 q v .

Experimental verification of biot-savart’s law

Current flows through a closed circuit. As such, it would be difficult to determine magnetic field due to a small current element as required for verification of Biot-Savart’s law. There is, however, a cleverly designed circuit arrangement which allows us to approximate requirements of determining magnetic field due to small current element. Look at the circuit arrangement shown in the figure. The parts of the wire along AB and CD when extended meet at point P. We arrange the layout in such a manner that the segment AD represents a small current element. The direction of magnetic field produced at P due to this small current element is out of the plane of drawing (shown by a filled circle i.e. dot) as current flows upward in the arm AD (apply right hand vector product rule).