The Biot–Savart Law (Magnetic Field of a Current Carrying Conductor)

A Danish scientist Hans Christian Oersted first discovered in 1820 that a compass needle is deflected near a wire carrying current. After that Jean-Baptiste Biot (1774–1862) and Félix Savart conducted experiments and determined an expression for the magnetic field caused by the wire. And the summarized version of their experiments is what we call Biot-Savart law.

From the experiments Biot and Savart summarized the following observations for the magnetic field $d\vec B$ due to the current element of length $dl$ carrying current $I$ at a field point $p$ (see Figure).

  1. The magnetic field $d\vec B$ is directly proportional to the current $I$.
  2. The magnetic field $d\vec B$ is directly proportional to the magnitude of the length of current element $dl$.
  3. The magnetic field $d\vec B$ is directly proportional to the sine of the angle between displacement $d\vec l$ and position vector of field point $\vec r$
  4. The magnetic field $d\vec B$ is inversely proportional to the distance from the source point to the field point $r$.

The current element of length $dl$ represents the source point of magnetic field. The mathematical expression of the magnetic field at the field point at $p$ is

\[B = \frac{\mu_0}{4\pi}\frac{I\,dl \,\sin \theta}{r^2} \tag{1} \label{1}\]

The magnetic field in vector form can be expressed in terms of the cross product between $d\vec l$ and unit vector of $\vec r$, that is

\[\vec B = \frac{\mu_0}{4\pi}\frac{I \, d\vec l \times \hat r}{r^2} \tag{2} \label{2}\]

From the above expression of magnetic field in vector form you can easily guess the direction of magnetic filed which is perpendicular to the plane containing $d\vec l$ and $\vec r$. More specifically the direction of magnetic field is determined by the right hand rule, that is curl the fingers of your right hand in the sense of $d\vec l$ moving into $\hat r$ keeping thumb straight and the thumb gives the direction of magnetic field.

Figure 1 The inward (into the screen) magnetic field is represented by a collection of crosses.
Figure 2 The outward (out of the screen) magnetic field is represented by a collection of dots.

In the Figure 1, at the field point the direction of magnetic field is outward from the screen (towards you). Or point your thumb in the direction of current and your fingers of right hand curl in the direction of magnetic field (see Figure 2).This is one of the interesting differences between electric and magnetic fields. In electric fields the direction is radial, that is it is along the line from the source point to field point.

The part $\mu_0/4\pi$ is the proportionality constant in Equations \eqref{1} and \eqref{2} where $\mu_0$ is called permeability of free space and it's value is $4\pi \times 10^{-7} T \cdot m/A$. You may find it interesting that this constant includes exactly $4\pi$.

You are not done yet until you know one important thing. The Equations \eqref{1} and \eqref{2} can not be verified experimentally because they are based on an isolated current element. An isolated current element or moving charge is not possible as already noted in magnetic field of moving charge. There must be a complete circuit for an electric current.

We can only conduct experiments for an entire current distribution. You can find the magnetic field by integrating Equation \eqref{1} for the entire length of the current carrying conductor.

\[B = \frac{\mu_0}{4\pi} \int \, \frac{I \, dl \, \sin \theta}{r^2} \tag{3} \label{3}\]

The real experiments conducted to determine the magnetic field is represented by the above equation. The Equation \eqref{1} or \eqref{2} is only the initial step of the integration process. So Equation \eqref{3} determines the magnetic field of the field point (the difference is it is integrated to include the entire current distribution).

Electromagnetism
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