Question

Apply ampere's circuital law to determine the magnetic field at a point due to long straight current carrying conductor

Answer 1

Answer:

Ampere's circuital law states that the integral lines of the magnetic field B around any closed circuit is equal to μ₀ (permeability constant) times the total current 'I' passing through this closed circuit.

Mathematically;  

∲B.dl= μ₀*I  

Proof for a straight current  carrying conductor:

Consider a long straight current carrying conductor 'I'. According to Biot-Savart law, the magnitude of the magnetic field B due to the current carrying conductor at any point at a distant 'r' from it is mathematically given  by;

B= μ₀*I*2π*r  

The magnetic field B is directed along the circumference of the circle of radius 'r' with the wire as center. The magnitude of the field B is same all points on the circle. To evaluate the line integral of the magnetic field B along the circle, we consider a small current element dI along the circle. At every point on the circle, both B and dl are tangential to the circle so that the angle between them is zero.  

B.dI= B*dl cos(0°)= B*dl (1) = B*dl

Hence the line integral of the magnetic field along the circular path is  

∲B.dl= ∲B*dl= B ∲(dl)= μ₀*I*2π*r*I  = μ₀*I*2πr*2πr  

   ∴∲B.dl=μ₀*I  

This proves Ampere's law. This law is valid for any assembly of current and for any arbitrary closed loop.