A thin wire carrying a steady current lies in the -plane

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Magnetostatics

Electricity, Magnetism and Electromagnetism, Ch The Biot-Savart law and the fields of steady currents

Problem

A thin wire carrying a steady current I=10 AI = 10\,\mathrm{A} lies in the xyxy-plane, with +x+x to the right, +y+y up the page and +z+z out of the page. The current arrives from x=−∞x = -\infty along the straight line y=−Ry = -R, travelling in the +x+x direction, reaches the point (0,−R)(0,-R), follows a semicircle of radius R=5.0 cmR = 5.0\,\mathrm{cm} centred on the origin through the point (R,0)(R,0) to the point (0,+R)(0,+R), and then returns along the straight line y=+Ry = +R to x=−∞x = -\infty. Both straight arms may be treated as semi-infinite, the wire as a line, and the surroundings as vacuum. ((a)) Starting from the Biot-Savart law, show that the semicircle alone gives a field of magnitude μ0I/4R\mu_0 I/4R at the origin, and state its direction. Evaluate it. ((b)) Using the field of a straight segment, show that each semi-infinite straight arm gives μ0I/4πR\mu_0 I/4\pi R at the origin, and state the direction of each. Evaluate one of them. ((c)) Find the total field at the origin, magnitude and direction.

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