Question

Explain Biot-Savart’s law. Derive the magnetic field generated by a straight and finite length current carrying conductor with the help of this law. Show that the magnetic field at a perpendicular distance d from an infinite length current carrying conductor is B = \(\frac{\mu_{0} I}{2 \pi d}\)

Answer 1

Biot-Savart’s Law:

Biot-Savart law was given by French Physicist Biot and Savart on the basis of experiments done by them for the calculation of magnitude of magnetic field at any point due to a current carrying conducting wire. Here, we shall study the relation between current and the magnetic field it produces.

Figure shows a conductor wire XY carrying p. current I. Consider an infinitesimal element dl of the conductor. The magnetic field dB due to this element is to be determined at a point P which is at a distance of r from it. Let θ be the angle between dl and the position Fig Biot Savart’s Law vector r. Direction of vector \(\vec{dl}\) is in the direction of current.

According to Biot-Savart’s Law: The magnitude of the magnetic field \(d\vec B\):
i) is directly proportional to the current I.
\(|\vec{dB}|\) ∝ I ………….. (1)

(ii) is directly proportional to the element length
\(|\overrightarrow{d B |} \propto| \overrightarrow{d l} |\) ……………….. (2)

(iii) is directly proportional to the sine of the angle between vector \(\vec r\) and \(\vec{dl}\) that is sinθ
\(|\vec{dB}|\) ∝ sin θ ………………. (3)

(iv) is inversely proportional to the square of the distance r.

Here, μ is called the permeability of free space (or vacuum). If there is some other medium around the conducting wire, then the magnitude of the magnetic field will be;
\(|\overrightarrow{d B}|=\frac{\mu_{m}}{4 \pi} \frac{I|\overrightarrow{d l}| \sin \theta}{r^{2}}\) ………….. (7)
Where, μ = μ₀μ_r is the permeability of a specific medium; which is dependent on the medium.
∴ μ_r = \(\frac{\mu_{m}}{\mu_{0}}\) = Relative permeability
Vector form : From equation (7),

It is clear from the equation (8) that the direction of \(\overrightarrow{d B}\) will be perpendicular to the surface formed by \(\overrightarrow{d l}\) and \(\vec{r}\) and would be up and down depending on the Right Handed Cork Screw Rule. In the figure the direction of \(\overrightarrow{d B}\) at point P is downwards perpendicular to the surface, and is represented by (X) sign whereas at point P’ the direction of \(\overrightarrow{d B}\) is outwards perpendicular to the surface of the paper and represented by (.) sign.

Magnetic Field due to a Straight Current Carrying Wire of Infinite Length
Since, the length of the wire is infinite, hence the ends x and y are at infinite distance. Therefore, the

internal angle made by them at point P would be
ϕ₁ = ϕ₂ = \(\frac{\pi}{2}\)
Therefore, from equation (7) magnetic field due to a straight current carrying wire of infinite length,