Let’s work with the solenoid used to measure the earth’s magnetic ﬁeld. The solenoid was wound with 79 turns over a distance of 79cm. The diameter was d = 11cm. A TA mistakenly runs the wire feeding the solenoid back across the end of the solenoid and along the top. This problem asks you to analyze the error created in the experiment by this foolish setup. The experiment measures the magnetic ﬁeld at point P at the center of the solenoid. The distance from the wire segment to the point P is ℓ = 0.5m. Since the percent error will be the same for any current, let’s analyze the system at the maximum output of out power supplies I = 17A. Note the coordinate system for the problem has the plane of the paper as the x − z plane.
(a) On the ﬁgure below draw the direction of the magnetic ﬁeld of the solenoid, ~ Bsole, the wire segment from A to B, ~ BAB, and the inﬁnite straight wire from B to C, ~ BBC, at point P.
(b) Calculate the ﬁeld of the solenoid at point P in the inﬁnite solenoid approximation. Report the ﬁeld as a vector.
(c) Calculate the ﬁeld of the wire segment AB in the ﬁnite current element approximation at point P. Report the ﬁeld as a vector.
(d) Calculate the ﬁeld of the wire segment BC in the approximation that it is an inﬁnite straight wire at point P. Report the ﬁeld as a vector.
(e) Calculate the magnetic ﬁeld of the segment BC exactly at point P. Report the ﬁeld as a vector. Report both a symbolic and numeric answer. Use the point P as the origin for this calculation.
(f) Does the TA bungle introduce more than 10% relative error in the experiment?
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