A thin square loop of side carries a steady current

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 square loop of side

a = 12.0\,\mathrm{cm}

carries a steady current

I = 4.0\,\mathrm{A}

. The loop is in vacuum, the wire is thin enough to be treated as a line, and no other magnetic field acts. ((a)) Starting from the Biot-Savart law, show that one side alone produces a field of magnitude

\dfrac{\sqrt{2}\,\mu_0 I}{2\pi a}

at the centre of the square, and hence that the whole loop produces

B = \dfrac{2\sqrt{2}\,\mu_0 I}{\pi a}

there. State the direction of

\mathbf{B}

relative to the plane of the loop. ((b)) Evaluate

B

at the centre. ((c)) A circular loop of the same perimeter carries the same steady current. Find the field at its centre, and the ratio of the square loop's centre field to it.

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