A long, thick, hollow cylindrical metal pipe has an inner radius of R and an out

Question

A long, thick, hollow cylindrical metal pipe has an inner radius of R and an outer radius of 3R.Its length L is very long compared to its radius (you could treat it as being infinitely long). The shaded area in this cross-section view represents the part of the pipe that carries a current inthe direction that points out of the page. The total current carried by the pipe is I, and it isdistributed uniformly through the shaded area. Use Ampere's Law to find the magnitude of themagnetic field for each of these three cases:

(a) inside the hollow interior of the pipe,

(b) outside the pipe, at a distance r away from the center,

(c) at a location inside the shaded part of the pipe, at a distance r away from the center.

Please show all your work and relevant formulae. Thank you.

Explanation / Answer

amperes law

B*ds = uo*Ienclosed

(a)

consider a circle of radius r inside the hollow

Ienclosed = 0

B*2*pi*r = uo*0

B = 0

(b)

consider a circle at a distance r away from the center

Ienclosed = I

B*2*pi*r = uo*I

B = Uo*I/(2*pi*r)

(c)

consider a circle at a distance r inside the shaded part of the pipe

Ienclosed = I*pi*r^2/pi((3R)^2-R^2)

Ienclosed = I*r^2/(9R^2-R^2)

B*2*pi*r = uo*I*r^2/(9R^2-R^2)

B = uo*I*r/(2*pi*(9R^2-R^2))

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