Biot Savart Law - Statement, Derivation and Applications

The magnetic field created by a constant electric field is described by the Biot Savart Law equation. According to this law, the magnitude, length, direction, and proximity of the electric current is related to the magnetic field. Also, this law is consistent with both Gauss's theorem and Ampere's circuital law. As Coulomb's law is fundamental for electrostatics, similarly, Biot Savart Law is also fundamental for magnetostatics. In 1820 two French physicists Jean Baptiste Biot and Felix Savart created the Biot Savart Law statement, which derived the mathematical formula for magnetic flux density at a point due to a nearby current-carrying conductor. Witnessing the deflection of the magnetic compass needle, these scientists concluded that the current particle projects a magnetic field into the space surrounding it.

The mathematical expression was derived through observation and calculations, which showed the magnetic flux density. This flux density dB is proportional to the length of the current I, element dl, the sine of angle θ between the direction of the current and the vector joining a given point of a magnetic field and the current element. It is inversely proportional to the square of the distance of the given point from the current element r.

Biot Savart Law Derivation

The biot savart law formula can be given as

\[dB\ \alpha\frac{Idl\sin\theta}{r^{2}}\]

Or

\[dB=k\frac{Idl\sin\theta}{r^{2}}\]

Where, k is constant, depending upon the magnetic properties of the medium and system of the units employed. In the SI system of the unit,

\[k = \frac {\mu_0 \mu_r}{4\pi}\]

The final Biot-Savart law derivation is expressed as,

\[db = \frac {\mu_0 \mu_r}{4\pi}\times \frac{Idl\sin\theta}{r^{2}}\]

Consider a long wire carrying current I and at a point p in space. The wire is represented in the picture below by red color. Also, consider an infinitely small length of the wire dl at a distance r from the point P as shown in the diagram. Here, r is the distance vector, which makes the angle θ with the direction of the current in the minute portion of the wire.

Imagining the condition, you can understand that the magnetic field density at point P due to the minute length dl of the wire is proportional to the current carried by this portion of the wire. As the current through the minute length of the wire is similar to the current carried by the whole wire itself, we can express it as:

dB ∝ I

It is also natural to think that the magnetic field density at point P due to the minute length dl of wire is inversely proportional to the square of the straight distance from Point P to the center of dl. Mathematically it can be expressed as:

dB ∝ \[\frac{1}{r^{2}}\]

Also, the magnetic field density at the point P due to the minute portion of the wire is proportional to the actual length of the minute length dl of wire. θ is the angle between distance vector r and direction of current through this minute portion of the wire, the component dl of the wire facing perpendicular to the point P is dlsinθ. 

Hence, dB ∝ dl sinθ

Now, merging these three statements, we can write,

\[dB\ \alpha k\frac{I.dl.sin\theta}{r^{2}}\]

The formula for Biot-savart law can be stated here as follows:

Now, introducing the value of constant k (which we have already mentioned at the starting of this derivation) in the above formula, we get,

\[dB = \frac{kl.dl.\sin\theta}{r^{2}}\]

\[ K = \frac{\mu_{0}\mu_{r}}{4\pi}\]

Here, μ0 is the absolute permeability of air or vacuum used in the expression of constant k, and its value is 4π10-7 Wb/ A-m in the SI system of units. μr in the expression of constant k is the relative permeability of the medium. 

Now, flux density (B) at point P due to the total length of the current-carrying conductor or wire can be expressed as,

\[B = \int dB = dB= \int\frac{\mu_{0}\mu_{r}}{4\pi}\times\frac{kl.dl.\sin\theta}{r^{2}}\]

\[= \frac{\mu_{0}\mu_{r}}{4\pi}I \int\frac{\sin\theta. dl}{r^{2}}\]

If D is the perpendicular distance of point P from the wire, then

r sinθ = D or r =\[\frac{D}{\sin\theta}\]

Now, the formula of flux density B at point P can be rewritten as,

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

Again, \[\frac{l}{D}\] = cotθ

 ⇒l = D cotθ

Therefore, dl = -Dcosec2θdθ

Finally, the expression of B comes as,

\[B = \frac {I\mu_0 \mu_r}{4\pi}\int \frac{\sin^{3}\theta}{D^{2}} [-Dcosec^{2}\theta d\theta\]

=\[-\frac {I\mu_0 \mu_r}{4\pi D}\int \sin^{3}cosec^{2}\theta d\theta\]

=\[-\frac {I\mu_0 \mu_r}{4\pi D}\int \sin\theta d\theta\]

This angle θ is dependent upon the length of the wire and position of point P. Say for a certain partial length of the wire, angle θ, as indicated in the figure above, varies from  θ1 to θ2. Hence, magnetic flux density at the point P due to the total length of the conductor is,

\[B = -\frac {I\mu_0 \mu_r}{4\pi D}\int_{\theta 1}^{\theta 2}\sin\theta d\theta\]

= \[-\frac {I\mu_0 \mu_r}{4\pi D}[-\cos\theta]_\theta1^\theta2\]

= \[\frac {I\mu_0 \mu_r}{4\pi D} [\cos\theta_{1}- \cos\theta_{2}]\]

Let's consider the wire is infinitely long, then θ will change from 0 to π that is  θ1 = 0 to  θ2 = π.

Placing these two values in the above final expression of Biot Savart law, we get,

\[B = \frac {I\mu_0 \mu_r}{4\pi D}[\cos0 - \cos\pi =  \frac {I\mu_0 \mu_r}{4\pi D} 1-(-1)\]

\[B = \frac{\mu_{0}\mu_{r}I}{2\pi D}\]

Applications of Biot Savart Law 

The applications of Biot Savart Law are mentioned below. 

• The Biot Savart Law is used to calculate magnetic responses at the molecular and atomic levels. 

• The Biot Savart Law is used in aerodynamic theory while calculating velocity generated by vortex lines. 

Importance of Biot Savart Law 

The Biot Savart Law is important for the following reasons: 

• The Biot Savart law can be applied to small conductors that carry current. 

• This law can also be applied to symmetrical current distribution.