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.

(II) A circular conducting ring of radius 𝑅 is connected to two exterior straight wires at two ends of a diameter (Fig. 28–47). The current I splits into unequal portions as shown (unequal resistance) while passing through the ring. What is B (→ above B)at the center of the ring?

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Three long parallel wires are 3.5 cm from one another. (Looking along them, they are at three corners of an equilateral triangle.) The current in each wire is 9.50 A, but its direction in wire M is opposite to that in wires N and P (Fig. 28–57). Determine the magnetic force per unit length on each wire due to the other two.

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(III) A coaxial cable consists of a solid inner conductor of radius 𝑅₁ , surrounded by a concentric cylindrical tube of inner radius 𝑅₂ and outer radius 𝑅₃ (Fig. 28–45). The conductors carry equal and opposite currents I₀ distributed uniformly across their cross sections. Determine the magnetic field at a distance 𝑅 from the axis for: (a) 𝑅 < 𝑅₁ ; (b) 𝑅₁ < 𝑅 < 𝑅₂ ; (c) 𝑅₂ < 𝑅 < 𝑅₃ ; (d) 𝑅 > 𝑅₃ . (e) Let I₀ = 1.50 A, 𝑅₁ = 1.00 cm , 𝑅₂ = 2.00 cm , and 𝑅₃ = 2.50 cm Graph B from 𝑅 = 0 to 𝑅 = 3.00 cm.

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