# Biot-Savart Law statement, formula, derivation

Last updated on August 23rd, 2023 at 09:16 am

In this post, we will cover the Biot-Savart Law statement, formula, and derivation.

Biot and Savart conducted many experiments to determine the factors on which the magnetic field due to the current in a conductor depends. The results of the experiments are summarized by the **Biot-Savart law**.

**Biot-Savart law**

Let us consider a conductor XY carrying a current I (Fig 1).

AB = dl is a small element of the conductor. P is a point at a distance r from the midpoint O of AB.

**Biot-Savart Law statement**

According to Biot and Savart, the magnetic induction dB at P due to the element of length dl is

(i) directly proportional to the current (I)

(ii) directly proportional to the length of the element (dl )

(iii) directly proportional to the sine of the angle between dl and the line joining element dl and the point P (sin θ)

(iv) inversely proportional to the square of the distance of the point from the element (1/r^{2})

**Biot-Savart Law formula & derivation**

∴ dB α (I dl sin θ) /r^{2}

=> dB = K (I dl sin θ) /r^{2}

Here, K is the constant of proportionality

The constant K =μ/4π, where μ is the permeability of the medium.

dB = (μ/4π) (I dl sin θ) /r^{2}

dB = (μ Idl sin θ) /(4πr^{2})

μ = μ_{r} μ_{0} where μ_{r} is the relative permeability of the medium and μ_{0} is the permeability of free space. μ_{0} = 4π × 10^{–7} henry/metre.

In air,dB = (μ_{0}Idl sin θ) /(4πr^{2})

**scalar and vector forms of Biot-Savart law**

**Here are the scalar and vector forms of the Biot-Savart law.**

The direction of dB is perpendicular to the plane containing the current element Idl and r (i.e. plane of the paper) and acts inwards. The unit of magnetic induction is tesla (or) Weber m^{-2}.

**[ Also read Ampere’s Circuital Law – formula, derivation]**