Question

Derive an expression for Biot-Savart law along with the diagram. (5 marks)

Answer 1

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/r²

Combining these factors,we get

dB∞IdLsinθ/r²

or dB=K Idl sinθ/r²

where K is a constant of perportionality.

In S.I units, K=μ₀/4π

thus , dB=μ₀/4π I dl sinθ/r²

where μ₀ is absolute premeability of free space and

μ₀=4π*10^-7 Wb A^-1m^-1

= 4π*10^-7*TA^-1m [ 1T=1 Wb m^-2]

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

Thus dB=Idl sinθ/r²

In vector form,

dB=μ₀/4π I(dl*r)/r³

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

B=∫μ₀/4πIdl*r/r³

Or B=∫μ₀/4π Idl sin θ/r²

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

In S.I System,

dH=dB/μ₀=1/4π Idl*r/r³=1/4π Idl*α_r/r²

H=∫1/4π Idl*α_r/r²

H= ∫1/4π Idl sinθ/r²

Importance OF BIOT SAVART’S LAW:-

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