(Solved): (7\%) Problem 13: The foil shielding on a power cable, carrying a DC current \( I=0.31 \mathrm{~A} ...

(7\%) Problem 13: The foil shielding on a power cable, carrying a DC current \( I=0.31 \mathrm{~A} \), has broken off. The unshielded part of the wire is \( x=0.026 \) meters long. The shielded parts of the wire do not contribute to the magnetic field outside the wire. Point \( P \) is above the right-hand end of the unshielded section, a distance \( x / 2 \) above the wire. The current flows to the right (in the positive direction) as shown. The small length of wire, \( d l \), a distance \( l \) from the midpoint of the unshielded section of the wire, contributes a differential magnetic field \( d B \) at point \( P \). Ignore any edge effects. \( 17 \% \) Part (a) Input an expression for the magnitude of the differential magnetic field, \( d B \), generated at point \( P \) by the current moving through the segment of wire \( d l \) in terms of given parameters and fundamental constants. \( d B=\frac{a_{0} \mid x d l}{\left.8 \pi\left(\frac{x}{2}\right)^{2}+l^{2}\right)^{3}} \quad \checkmark \) Correct! \( \$ 17 \% \) Part (b) Perform the indefinite integral from part a. \( B=\frac{\mu_{0} I I}{2 \pi x\left(\left(\frac{1}{2}\right)^{2}+l^{2}\right)^{1 / 2}} \quad \checkmark \) Correct! \( \square 17 \% \) Part (c) Select the limits of integration that will correctly calculate \begin{tabular}{|l|} \hline\( l=-x \) to \( l=0 \) \\ \( l=-x \) to \( l=x \) \\ \( l=0 \) to \( l=\infty \) \\ \( l=-x / 2 \) to \( l=x / 2 \) \\ \hline\( l=-\infty \) to \( l=0 \) \\ \( l=-\infty \) to \( l=0 \) \\ \hline \end{tabular} Hints: deduction per hint. Hints remaining: \( \underline{2} \quad \) Feedback: \( \quad \) deduction per feedback. \( \triangle 17 \% \) Part (d) Evaluate the expression derived in part (b) using the endpoints selected in part (c). \( \triangle 17 \% \) Part (e) Determine the strength of the magnetic field (in tesla) at point \( P \). \( \triangle 17 \% \) Part (f) In what direction will the magnetic field point at point \( P \) due to the current in the unshielded portion of the wire?