29. Sources of Magnetic Field

A solid, cylindrical conductor carries a uniform current density, J. If the radius of the cylindrical conductor is R, what is the magnetic field at a distance ? from the center of the conductor when r < R? What about when r > R?

(I) A 2.5-mm-diameter copper wire carries a 28-A dc current (uniform across its cross section). Determine the magnetic field: (a) at the surface of the wire; (b) inside the wire, 0.50 mm below the surface; (c) outside the wire 2.5 mm from the surface.

You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that a transmission wire is about 12 m above the ground. The local power company tells you that the lines operate at 145 kV and provide a maximum of 45 MW to the local area. Estimate the maximum magnetic field you might experience walking under one such power line, and compare to the Earth’s field. [For an ac current, values are rms, and the magnetic field will be changing.]

(III) Suppose the current in the coaxial cable of Problem 34, Fig. 28–45, is not uniformly distributed, but instead the current density j varies linearly with distance from the center: j₁ = C₁𝑅 for the inner conductor and j₂ = C₂𝑅 for the outer conductor. Each conductor still carries the same total current I₀ , in opposite directions. Determine the magnetic field in terms of I₀ in the same four regions of space as in Problem 34.