0.2 Biot - savart law  (Page 2/4)

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

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

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.

Magnetic field due to small thin current element

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.

Magnetic field due to small thin current element

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 :

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

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 :

Magnetic field due to small thin current element

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 ${\mathbf{B}}_1$ , ${\mathbf{B}}_2$ etc. as their directions are different.