Classical Electrodynamics · Problem 5.10

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · circular-loop-vector-potential-integral-representations

Problem

A circular current loop of radius

a

carrying a current

I

lies in the

x

y

plane with its center at the origin. (a) Show that the only nonvanishing component of the vector potential is

A_\phi(\rho, z) = \frac{\mu_0 Ia}{\pi}\int_0^\infty dk\,\cos kz\, I_1(k\rho_<) K_1(k\rho_>)

where

\rho_< (\rho_>)

is the smaller (larger) of

a

and

\rho

. (b) Show that an alternative expression for

A_\phi

A_\phi(\rho, z) = \frac{\mu_0 Ia}{2}\int_0^\infty dk\, e^{-k|z|} J_1(ka) J_1(k\rho)

(c) Write down integral expressions for the components of magnetic induction, using the expressions of parts a and b. Evaluate explicitly the components of

\mathbf{B}

on the

z

axis by performing the necessary integrations.

Related theory

Magnetic Field On Axis Of Circular Loop

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