0.2 Biot - savart law  (Page 7/4)

Biot – Savart law is the basic law providing a relation between cause and effect in electromagnetism. In electrostatics, Coulomb’s law tells us the relation between point charge (cause) and electric field (effect) that the charge produces in its surrounding. Similarly, Biot-Savart’s law tells us the relation between current element or moving charge (cause) and magnetic field (effect) that the current element or moving charge produces in its surrounding. Besides, Biot – Savart law is an empirical law (a result of experimental observations) just like Coulomb’s law.

Clearly, there is a strong evidence of parallelism in the study of electrostatics and electromagnetism. There is, however, one important distinction between them. The electric field is along the straight line joining charge and the position in space i.e. along displacement vector. The relationship here is linear. The magnetic field, on the other hand, is along the perpendicular direction of the plane constituted by the small current element and displacement vector. This feature of magnetic field introduces a new dimension to the formulation of Biot-Savart law. We have to compulsorily rely on vector notations and operations. In a nutshell, we are required to be a bit conscious of the direction of magnetic field, which often requires visualization in three dimensional space.

Magnetic field due to small thin current element

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 :

$$đ\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Iđ\mathbf{l}X\mathbf{r}}{r^3}$$

Magnetic field due to small thin current element

The ratio ${\mu }_0/4\pi$ is the proportionality constant and has the value of ${10}^{-7}$ Tm/A. The constant ${\mu }_0$ is known as permeability of free space. The SI unit of magnetic field is Tesla (T), which is defined in the context of magnetic force on a moving charge in magnetic field ( See module Lorentz force ). It is expressed as 1 Newton per Ampere - meter. Further, the vector representation of small length element of wire "d l " in the expression is referred as "current length element" and the vector "Id l " is referred simply as "current element". The direction of current length vector "d l " is the direction of tangent drawn to it in the direction of current in the wire.

Note the vector cross product in the numerator. Direction of magnetic field produced is given by the direction of vector cross product d lXr . 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. $1/r^2$ (one of r in the numerator cancels with that in the denominator). This means that Biot-Savart Law is also inverse square law like Coulomb's law.

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

$$\underset{}{\overset{‸}{\mathbf{r}}}=\frac{\mathbf{r}}{r}$$ $$\Rightarrow \mathbf{r}=r\underset{}{\overset{‸}{\mathbf{r}}}$$