Physicists Jean-Baptiste Biot and Félix Savart discovered a fundamental quantitative link between an electric current and its magnetic field, based on their experiments in 1820. A magnetic field, an area in space surrounding a conductor where sensors can detect magnetic forces, is created when an electric current or moving electric charge passes through the conductor. Let’s consider all the contributions from each little element, or segment, of a current-carrying conductor. The value of the magnetic field at a point in the surrounding space can be thought of as the sum of all of these contributions.

Biot-Savart Law Definition

This law describes how the value of the magnetic field produced by a short length of current-carrying conductor at a certain place in space is dependent on each factor that impacts the field.

Elaboration of Biot-Savart Law 

• As a starting point, it should be noted that the value of the magnetic field at a given point is proportional to both the value of the current flowing through a conductor and the length of the current-carrying segment under discussion.
• It is important in determining the value of the field with respect to a particular spot when a segment of current is flowing through it.
• The field is largest when the line connecting the point and the short segment of current forms a 90-degree angle with the current segment.
• As the angle gets smaller, the field of the current segment shrinks until it reaches zero, when the point is located on a line that itself includes a segment of the current element.  
• The magnetic field at a point is also dependent on how far the point is from the current element. 
• For example, if you move twice as far away from the source of the magnetic field and the current element that makes it, you lose four times as much strength as you did before.

Biot-Savart Law Derivation

In this example, we will consider a wire carrying an electric current I and an infinitely short length of a wire dl located at a distance x from point A.

Biot Savart  Law indicates the magnetic intensity dH at a given point A owing to current I transmitting through a small element dl is –

The current (I) is directly proportional.

(a) The length of the element (dl) is directly proportional

(b) There is direct proportionality to the sine of the angle between the current direction and the line connecting element dl from point A

(c) There is inverse proportionality to the square of the distance (x) of point A from the element dl – 

dH ∝ I dl Sinθ/x2

where k is a constant that is determined by the magnetic characteristics of the medium being studied

K = μ0μr/4π

µ0 = absolute permeability of air or vacuum and its value is 4 x 10-7 Wb/A-m

µr = relative permeability of the medium

Applications of Biot-Savart Law 

Biot-Savart law is applied in a specific example by adding up the contributions concerning a magnetic field to a particular position from the entire series of small current segments that comprise a definite conductor of the distinct shape being considered. 

An extremely long straight wire carrying current, for example, will produce a magnetic field at a location nearby that is directly proportional to the value of the current and inversely proportional to the distance between the wire and the supplied point.

Differences Between Biot-Savart Law and Ampere’s Law

Before proceeding with the differences, let us take a look at Ampere’s law. Electromagnetism relies heavily on Ampere’s Law. For each closed-loop path, determining the expression is the challenge. Permeability times electric current equals the sum of length elements x magnetic field x permeability, as stated in this equation. This law enables us to maintain a proper bridge between electricity and magnetism, allowing us to bridge the distance between them. It also explains the mathematical relationship between magnetic fields and electrical currents.

•  It is possible to calculate the magnetic field produced by any shape of the loop using the Biot-Savart Law, but Ampère’s Law is a simplified and convenient version of the law for symmetrical wire and magnetic field configurations.
• These two principles are extremely helpful in determining the magnetic field generated at any given site by various current distributions.
•  From this, one can consider Ampere’s and Biot-Savart laws to be comparable.
•  Any closed path has a magnetic field line integral that is equal to the permeability of open space multiplied by the current that runs through the surface that is inside the path. This is Ampere’s law.
• The magnetic field generated by a current segment can be expressed mathematically using Biot-Savart Law.

Benefits and Limitations of Biot-Savart Law 

• The advantage of the Biot-Savart law is that it works with any magnetic field produced by a current loop, regardless of its strength. 
•  A limitation is that Biot–law Savart’s can only be used in open space. This means that magnetic fields near the ferromagnetic material in the vicinity of the end winding have big differences between the measured force and the force that was calculated.

Conclusion 

The Biot-Savart Law is a mathematical equation that describes the magnetic field generated by a constant electric current flow. A relationship exists between the amplitude, direction, length, and proximity of an electric current and the magnetic field. Neither Ampere’s circuital law nor Gauss’s theorem contradicts the Biot–Savart law. When it comes to magnetostatics, the Biot-Savart law is crucial, playing a role similar to Coulomb’s law in electrostatics.