# BIOT-SAVART’S LAW DERIVATION

**BIOT-SAVART’S LAW OR AMPERE’S LAW FOR CURRENT ELEMENT**

Biot-Savart’s law deals with the magnetic field induction at a point due to a small current element.

Current element

A current element is a conductor carrying current.It is the product of current,I and length of very small segment of current carrying wire ,dL.

Let us consider a small element AB of length dl of the conductor RS carrying a current I.

Let r be the position vector of the point P from the current element I dL.and θ be the angle dl and r.

According to Biot-Savart’s law,the magnetic field induction dB or magnetic flux density at a point P due to current element depends upon the following factors.

(i) dB∞I

(ii) dB∞dl

(iii) dB∞sinθ

(iv) dB∞1/r^{2}

Combining these factors,we get

dB∞IdLsinθ/r^{2}

or dB=K Idl sinθ/r^{2}

where K is a constant of perportionality.

**In S.I units, K=**μ_{0}/4π

thus , dB=μ_{0}/4π I dl sinθ/r^{2}

where μ_{0} is absolute premeability of free space and

μ_{0}=4π*10^{-7} Wb A^{-1}m^{-1}

= 4π*10^{-7}*TA^{-1}m [ 1T=1 Wb m^{-2}]

**In C.G.S units**,K=1 (In free space)** **

**Thus **dB=Idl sinθ/r^{2}

**In vector form,**

dB=**μ _{0}**/4π I(dl*r)/r

^{3}

magnetic field induction at point P due to current through entire wire is

B=∫μ_{0}/4πIdl*r/r^{3}

Or B=∫μ_{0}/4π Idl sin θ/r^{2}

**BiotSavart’s law in terms of magnetising force or magnetic intensity (H) of the magnetic field:**

**In S.I System,**

** dH=dB/**μ_{0}=1/4π Idl*r/r^{3}=1/4π Idl*α_{r/r}^{2}

** H=∫1/4π Idl*α _{r}/r^{2}**

** H= ∫1/4π Idl sinθ/r ^{2}**

**Importance OF BIOT SAVART’S LAW:-**

**This**law is analogous to Coulomb’s law in electrostatics.**Biot**Savart’s law is valid for a symmetrical current distribution.- This law cannot be easily verified experimentally as the current carrying conductor of very small length cannot be obtained pratically.
- Biot Sarvart’s law is applicable only to very small length conductor carrying current.
- The direction of dB is perpendicular to both Idl and r.

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