On the fields of a torus and the role of the vector potential

Journal Article · Tue Aug 01 04:00:00 EDT 1995 · American Journal of Physics

DOI:https://doi.org/10.1119/1.17842· OSTI ID:78673

Carron, N J ^[1]

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1. Mission Research Corporation, P.O. Drawer 719, Santa Barbara, California 93102 (United States)

A toroidal current distribution has nonvanishing exterior vector potential {bold A}, but zero exterior field {bold B}={del}{times}{bold A}=0. This property, together with the absence of fringing fields as in a cylindrical solenoid, makes it convenient for studies involving the vector potential in a field-free region, such as the Aharonov--Bohm effect, or the effect of {bold A} in a Josephson junction. We present an immediate general result of magnetostatics, and use it to easily compute {bold A} for a torus, to visualize the static vector potential for any current distribution, and to show how one can construct a current distribution to produce any desired {bold A}. When the torus current {ital I} varies in time, nonzero quasistatic fields {bold E}({ital t}) and {bold B}({ital t}) are produced ({bold E}{similar_to}{omega}{ital I}/{ital r}{sup 3} and {bold B}{similar_to}{omega}{sup 2}{ital I}/{ital r}{sup 2}). Radiation is also produced, with the radiation pattern of an {ital electric} {ital dipole}. The torus provides a counterexample to the common erroneous notion that if all multipole moments of a current distribution vanish then quasistatic fields and radiation must also vanish. We then formulate Maxwell`s equations in a way that obviates the role of gauge transformations. This ``gauge irrelevant`` form clarifies the relation of potentials to current sources, isolating the role of the transverse part of {bold A}. The general result from magnetostatics is extended to time-varying sources, revealing a seldom recognized symmetry of Maxwell`s equations, and showing how one can visualize {bold A} for an arbitrary time-dependent current source. {copyright} {ital 1995} {ital American} {ital Association} {ital of} {ital Physics} {ital Teachers}.

• DOE Contract Number: FG02-90ER40542
• OSTI ID: 78673
• Journal Information: American Journal of Physics, Journal Name: American Journal of Physics Journal Issue: 8 Vol. 63; ISSN 0002-9505; ISSN AJPIAS
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
AHARONOV-BOHM EFFECT
CURRENT DENSITY
GAUGE INVARIANCE
MAGNETIC FIELDS
MAXWELL EQUATIONS
POTENTIAL ENERGY
TOROIDAL CONFIGURATION