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Biot-Savart Law

In these Study material notes on Biot, we will discuss formulas, definitions, laws from the Biot - Savart Law. Learn the concepts of the magnetic field due to current carrying elements.


Electromagnetic fields are composed of electric fields and magnetic fields, and together they produce what is known as electromagnetic fields. The connection between both fields is known as Biot- Savart Law.

Jean-Baptiste Biot and Felix Savart were the first scientists to develop a theory that there is a relationship between electric fields and magnetic fields. Together, in 1820, they came up with an equation that proved that a magnetic field is generated in the presence of electric current. The Biot-Savart Law forms the basis of magnetostatics and is Coulomb’s law equivalent in magnetostatics. Let’s look at and understand more about the Biot-Savart Law and its applications. 

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What is the Biot-Savart Law? 

An electric current flowing in a conductor or any moving electric charge is set to produce a magnetic field. The amount of magnetic field built at any point in the nearby space is considered the sum of all the influences from each small current-carrying conductor nearby. The law states that the magnetic field at any point in space is directly from a current-carrying conductor, depending on several factors that can influence the field. The factors are: 

  1. Firstly, the value of the magnetic field at any point is directly proportional to two things – the value of current (I) in the conductor and the length of the current-carrying segment. 
  2. Secondly, the field’s value also depends on the position of the point concerning the segment of the current. The field is the greatest if the line from the point to the shortest segment of current is at a right angle to the current segment. The magnetic field starts to reduce if the angle () gets smaller and eventually becomes zero when the point lies on the line of which the current element is a part. 
  3. Lastly, the magnetic field is impacted by the distance between the point and the current element. It has been established that if the distance is r between them, the field is r2 smaller. In simple words, the value of a magnetic field is inversely proportional to the square of the distance from the magnetic field source.


How to derive the Biot-Savart Law:

There are several ways to check the Biot-Savart Law and its application to derive the equation. For example, there is a long wire with current I, and there is a P point in the space. There is also a tiny length of the wire dl at a distance of r from the point P. In this case, r is the vector at an angle θ with the direction of the current. 

Due to the insignificant length dl of the wire, there is a magnetic density at the point P, which is directly proportional to the current carried by this portion of the wire. The current in this small length of the wire is the same as the current carried by the whole wire itself. We can say, 

 dB∝ I

dB∝ 1r2

If the angle formed by distance vector r and the direction of the current is θ, then the part of wire dl facing the perpendicular of point P is dlsinθ

dB∝ dl sin

Now combining all the three above-mentioned points, 

dB∝ I dl sinr2

If we introduce the constant K, which in the SI system of unit is

k=μo 4π

Here μo is the permeability of vacuum.

Then the equation becomes

dB=k I dl sinr2


dB=μo4π I dl sinr2

Application of the Biot- Savart Law

There are several applications of the Biot-Savart Law, some of them being: 

  1. The force between two parallel and long current-carrying conductors: 

Force per unit length of wire B is

F2l=I2 B1=μo I1I22πd

Force per unit length of A due to current in B is

F1l=I1 B2=μo I1I22πd

And is directed opposite to the force on B due to A. 

Thus, it is safe to conclude that the conductors attract each other if the current flows in the same direction and repel if it flows in the opposite direction.

2. Magnetic induction at the centre of a circular coil carrying current

B=μo4πI (2r)r2=μoI2πr

For a coil of n turns,

B=μon I2πr

Importance of the Biot-Savart Law

  • The law is very relevant when it comes to small conductors carrying current
  • It applies to the symmetrical current distribution
  • The law is very similar to Coulomb’s law in electrostatics


So what we have discussed today is the Biot study material and why the law is significant in electromagnetism. After reviewing the application of the Biot-Savart Law, one can easily understand the basic concept of the rule. The key finding is that current flowing through a conductor or any moving particle electrically charged will produce the magnetic field. The amount of magnetic field generated depends on several factors, which form the formula of the Biot-Savart Law.