A long, straight wire of radius R carries a steady current I that is uniformly d

Question

A long, straight wire of radius R carries a steady current I that is uniformly distributed through the cross section of the wire (a) Calculate the magnitude of the magnetic field at a distance r from the central axis of the wire in the region rR Express your answer in terms of any of the symbols already mentioned, as well as ,, ??, and ?& as needed. We now drill a cylindrical hole through the wire. The axis of the cylindrical hole coincides with the axis of the wire. The radius of the cross-section of the cylindrical hole is a, and the total current through the wire remains the same as before the hole was drilled. (b) What is the magnitude of the current density in the wire, now that the hole is drilled out? Express your answer in terms of any of the symbols already mentioned, as well as , eo and Ho as needed. (c) Calculate the magnitude of the magnetic field at a distance r from the central axis of the wire in the region r

Explanation / Answer

given a long straight wire of radius R, steady current = I, uniformly distributed across the cross seciton of the wire

a. for r < R

magnetic field, from ampere's law

B(r)*2*pi*r = ienc*mu = mu*I*r^2/R^2 ( here mu is permeability of the material)

hence

B(r) = mu*I*r/2*pi*R^2

b. radisu of cross section of the hole = a

magnitude of current is strill the same

hence new current density J = I/pi(R^2 - a^2)

c. for r < a

ienc = 0

hence

B(r) = 0 ( from ampere's law)

d. for a < r < R

ienc = J*pi(r^2 - a^2)

hence

B(r) = I(r^2 - a^2)*mu/2*pi*r(R^2 - a^2)

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