Two French physicists, Jean Baptiste Biot and Felix Savart derived the mathematical expression for magnetic flux density at a point due a nearby current carrying conductor, in 1820. Viewing the deflection of a magantic compass needle, two scientists concluded that any current element projects a magnetic field in the space.
After, through observations and calculations they had derived a mathematical expression, which shows, the magnetic flux density of which dB, is directly proportional to the length of the element dl, the current I, the sine of the angle and θ between direction of the current and the vector joining a given point of the field and the current element and is inversely proportional to the square of the distance of the given point from the current element, r. This is Biot Savart law statement.
Where, k is a constant, depends upon the magnetic properties of the medium and system of the units employed. In SI system of unit,
Therefore, 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.
If you try to visualize the condition, you can easily understand the magnetic field density at the point P due to that infinitesimal length dl of the wire is directly proportional to current carried by this portion of the wire.
As the current through that infinitesimal length of wire is same as the current carried by the whole wire itself, we can write,
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 center of dl. Mathematically we can write this as,
Lastly, magnetic field density at that point P due to that infinitesimal portion of wire is also directly proportional to the actual length of the infinitesimal length dl of wire. 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 it’s value is 4π10-7 Wb/ A-m in SI system of units. μr of the expression of constant k is relative permeability of the medium.
Now, flux density(B) at the point P due to total length of the current carrying conductor or wire can be represented as,
If D is the perpendicular distance of the point P form 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 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 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.