A long, cylindrical wire having radius b carries a uniformly distributed current i across its cross section. Determine the magnitude of the magnetic field at a point situated inside the wire, at a distance a from its axis.

Step-by-step solution

Using Ampere's circuital law: B d l = ₀ I_ enclosed For a point inside the wire at a distance a from the axis, consider a circular Amperian loop of radius a . Since the current is uniformly distributed over the cross-section of radius b , the current density is: J = i b^2 The current enclosed by the Amperian loop is: I_ enclosed = J a^2 = i b^2 a^2 = i a^2 b^2 Applying Ampere's law: B(2 a) = ₀ ( i a^2 b^2 ) B = ₀ i a 2 b^2