- Biot Savart Law Definition: The Biot Savart Law is defined as a principle describing the magnetic field created by a constant electric current.
- Magnetic Flux Density: The magnetic flux density (dB) is directly proportional to the current element’s length (dl), the current (I), and the sine of the angle (θ) between the current and distance vector (r), and inversely proportional to the distance squared.
- Constant k: The constant k depends on the medium’s magnetic properties and the unit system, with μ0 representing absolute permeability and μr representing relative permeability.
- Total Magnetic Field Calculation: The total magnetic field at a point is found by integrating the effects of all infinitesimal current elements along the conductor.
- Relation to Ampere’s Law: For an infinitely long wire, the Biot Savart Law simplifies to the expression used in Ampere’s Law.
What is Biot Savart Law
The Biot Savart Law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. Biot–Savart law is consistent with both Ampere’s circuital law and Gauss’s theorem. The Biot Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb’s law in electrostatics.
Biot-Savart law was created by two French physicists, Jean Baptiste Biot and Felix Savart derived the mathematical expression for magnetic flux density at a point due to a nearby current-carrying conductor, in 1820. Viewing the deflection of a magnetic compass needle, these two scientists concluded that any current element projects a magnetic field into the space around it.
Through their observations and calculations, they derived a mathematical expression. It shows that the magnetic flux density (dB) is directly proportional to the current element length (dl), the current (I), and the sine of the angle (θ) between the current direction and the distance vector. It is inversely proportional to the square of the distance (r) from the current element.
Biot Savart Law Statement & Derivation
The Biot-Savart law can be stated as:
Where, k is a constant, depending upon the magnetic properties of the medium and system of the units employed. In the SI system of unit,
Therefore, the final Biot-Savart law derivation is,
Let us consider a long wire carrying a current I and also consider a point p in the space. The wire is presented in the picture below, by red color. Let us also consider an infinitely small length of the wire dl at a distance r from the point P as shown. Here, r is a distance-vector which makes an angle θ with the direction of current in the infinitesimal portion of the wire.
To understand this, imagine the magnetic field density at point P due to the small length (dl) of the wire. It is directly proportional to the current carried by this wire segment.
Since the current through that small wire segment is the same as the current in the entire wire, we can express this relationship as:
It is also very natural to think that the magnetic field density at that point P due to that infinitesimal length dl of wire is inversely proportional to the square of the straight distance from point P to the center of dl. Mathematically we can write this as,
Lastly, the magnetic field density at point P due to the small segment of the wire is also directly proportional to the length of this wire segment (dl).
As θ be the angle between distance vector r and direction of current through this infinitesimal portion of the wire, the component of dl directly facing perpendicular to the point P is dlsinθ,
Now, combining these three statements, we can write,
This is the basic form of Biot Savart’s Law
Now, putting the value of constant k (which we have already introduced at the beginning of this article) in the above expression, we get
Here, μ0 used in the expression of constant k is absolute permeability of air or vacuum and its value is 4π10-7 Wb/ A-m in the SI system of units. μr of the expression of constant k is the relative permeability of the medium.
Now, flux density(B) at the point P due to the total length of the current-carrying conductor or wire can be represented as,
If D is the perpendicular distance of the point P from the wire, then
Now, the expression of flux density B at point P can be rewritten as,
As per the figure above,
Finally, the expression of B comes as,
This angle θ depends upon the length of the wire and the position of the point P. Say for a certain limited length of the wire, angle θ as indicated in the figure above varies from θ1 to θ2. Hence, magnetic flux density at point P due to the total length of the conductor is,
Let’s imagine the wire is infinitely long, then θ will vary from 0 to π that is θ1 = 0 to θ2 = π. Putting these two values in the above final expression of Biot Savart law, we get,
This is nothing but the expression of Ampere’s Law.
