Introduction :
In previous chapter we have discussed about electric fields due to some simple charge distributions , here we are going to discuss about the magnetic fields produced by some simple current distributions (like straight wire and circular loops).
Here we are going to introduce two methods for calculating magnetic fields . One of the methods (Biot-Savart law) is based on a direct technique like Coulomb’s Law, and the other (Ampere’s law) is based on arguments of symmetry, analogous to Gauss’s law.
Biot-Savart’s Law :
In order to understand Biot-Savart law, we need to understand the term current-element. Current element is the product of current and length of an infinitesimal segment of a current carrying wire. The current element is taken as a vector quantity. Its direction is the same as the direction of current. The magnetic field (any such field) is defined at a point.
In the figure shown, there is a segment of a current carrying wire and P is the point where the magnetic field is to be calculated. $I\vec{dl}$ is a current element and $\vec{r}$ is the position vector of the point ‘ P ‘ with respect to the current element $I\vec{dl}$.
According to Biot-Savart’s Law, magnetic field at point P due to the current element $I\vec{dl}$ is
(i)is directly proportional to the current (I) flowing through the element .
(ii)is directly proportional to the length of the element .
(iii)is directly proportional to the sine of the angle (θ) between length of the element and the line joining the element to the point P.
(iv)is inversely proportional to the square of the distance (r) of the point from the element .
$ \displaystyle dB = \frac{\mu_0}{4 \pi} \frac{I dl sin\theta }{r^2}$
$ \displaystyle \vec{dB} = \frac{\mu_0}{4 \pi} \frac{I (\vec{dl} \times \vec{r})}{r^3}$
Since the entire segment is made-up of infinite such current elements and direction of magnetic field due to each element of P is same, the magnetic field due to the entire wire segment can be found by integrating the magnetic field due to the current elements over entire length of the wire.
Hence ,
$ \displaystyle \vec{B} = \int\frac{\mu_0}{4 \pi} \frac{I (\vec{dl} \times \vec{r})}{r^3}$
Here μo is called permeability of free space.
In SI units the value of ( $ \large \; \frac{\mu_0}{4 \pi}= 10^{-7} \;tesla-meter/ampere $ ) .
The SI units of magnetic fields are Tesla, (weber/m2)
In the above expression limits of the integral depend on the shape and size of the current carrying wire.