Derivation of Biot Savart Law

Jean Baptiste Biot and Felix Savart, two French physicists, came up with the Biot-Savart law in 1820. They came up with the mathematical expression for magnetic flux density at a point due to a nearby current-carrying conductor. It was because of this that these two scientists thought that any object with a current in it creates a magnetic field around itself. They looked at how a magnetic compass needle moved. The Biot Savart Law is an equation that shows how magnetic fields are made when there is a steady flow of electricity. Magnetic field strength and direction are linked to the magnitude of the electric current, its length, and how close it is to the magnetic field. Biot–Savart law is in line with both Ampere’s circuital law and Gauss’s theorem. The Biot Savart law is very important to magnetostatics, and it has a role like Coulomb’s law in electrostatics.

Biot Savart law

A basic magnetic field source is a conductor that carries current (I) and has a length (dl) equal to the current (I). The power applied to a second conductor that is connected to the primary can be easily stated in terms of the magnetic field (dB) generated by the primary. It was Biot and Savart who made the most significant contributions to our understanding of the magnetic field dependency on the ‘I’ current, dimension as well as direction of the length dl & on distance ‘r’.

There is a magnetic field or a space around a conductor where magnetic forces can be felt when an electric current moves through it. Each small piece of a current-carrying conductor adds up to make the magnetic field at a certain point in the surrounding space bigger than it would be without the parts of the conductor. The Biot-Savart law shows how the value of the magnetic field at a certain point in space from a short piece of the current-carrying conductor changes with each factor that affects the field. This is called the Biot-Savart law. This means that when you think about the magnetic field, both the amount of current in a conductor and its length will have an effect on how strong it is. The value of the field also depends on where the point is in relation to the current. As a line that goes from one point to another makes a 90° angle with the short piece of current, the field is the largest. As this angle gets smaller, the field of the current segment gets smaller. When the point is on a line of which the current element itself is a segment, the field of the current element is 0 at that point. Another thing that affects the magnetic field at a certain point is how far away the point is from the source of the electricity. In this case, the magnetic field is four times smaller at twice the distance. The value of the magnetic field is inversely proportional to the square of the distance from the source of the current that makes it.

Derivation of Biot Savart law

The formula of biot savart law

dB ∝ Idlsinθ / r²

dB = k Idlsinθ / r² …….1

here k is the constant 

value of k,

k = μ₀μ_r4π

by putting value of k in equation 1. 

dB = μ₀μ_r4πx Idlsinθr²

Biot Savart law real-life examples 

A raft of compasses with a 200-amp cable flows through the center of the raft. Alternatively, three 100-amp cables with three-phase alternating current powering them, where the resulting field rotates and can spin a conductive item through induction. There is a simpler solution than attempting to create a 2-volt, 200-amp power source from scratch.

The Biot–Savart law can be applied in the calculation of magnetic responses even at the atomic or molecular level, e.g. chemical shieldings or magnetic susceptibilities, provided that the current density can be determined from a quantum mechanical calculation or theory.

Biot Savart law expression 

B= μ₀NI/2R

B= magnetic field intensity 

μ= permeability of free space

N= number of turns

I = current intensity 

R = radius

Application of Biot savart law

The Biot-Savart law is used to analyze magnetic response at the molecular or atomic level.

In aerodynamic theory, it is used to determine the velocity induced by the vortex line and is used to calculate the speed.

The significance of the Biot-Savart Law

Biot-Savart Law is a law of electrostatics that is nearly identical to Coulomb’s law.

Biot-Savart Law is applicable when very small conductors are used to convey current. Biot-Savart Law is also applicable when there is a symmetrical current distribution.

Conclusion 

Jean Baptiste Biot and Felix Savart, two French physicists, came up with the Biot-Savart law in 1820. The Biot Savart Law is an equation that shows how magnetic fields are made when there is a steady flow of electricity. Magnetic field strength and direction are linked to the magnitude of the electric current, its length, and how close it is to the magnetic field. Biot-Savart law states how the value of the magnetic field at a specific point in space from one short segment of the current-carrying conductor depends on each factor that influences the field. The magnetic field is inversely proportional to the square of the distance from the current element that produces it.