Question

when the current[ =Ⅰ ] flowing through the wire in the direction of the z-axis, magnetic vector potential is following that

A = e_z[μⅠ*ln(1/ρ)/2π]

show that magnetic field B is same as B = e_φ[μI/2πρ]

Answer 1

We have current through th wire in the direction of

we have,

magnetic vector potential due to this current wire given as,

where,

unit vector in the direction of z-axis,

magnetic field strength is given curl of vectro potential as,

where,

,

where,

are unit vectors in cylindrical coordinate system.

Using the distributive property of the cross product we get,

We have,

,    and  

Using these values of cross-product of unit vectors in the above equation we get,

______________________________relation 1

Now we will solve partial derivatives in the above equation,

Using    in above equation we get,

______________________relation 2

_________ relation 3

Using these values of partial derivatives in relations 2 and 3 , in relation 1 we get,

hence we proved that magnetic field arround conductor is given as,

,