A single-layer coil is wound uniformly with turns over a length on a former of radius

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 single-layer coil is wound uniformly with

N = 600

turns over a length

l = 30.0\,\mathrm{cm}

on a former of radius

R = 2.5\,\mathrm{cm}

, and carries a steady current

I = 3.0\,\mathrm{A}

. Treat the winding as a stack of coaxial circular loops,

n\,dz

of them in a slice

dz

, with

n = N/l

; the wire is thin, the pitch of the winding and the leads are ignored, and the coil is in vacuum. Take the coil's axis as the

z

-axis, with

+z

along the direction of the axial field. ((a)) Starting from the on-axis field of one circular loop,

B_z = \dfrac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}}

, show that the field at a point on the axis is

B = \tfrac12\mu_0 n I(\cos\theta_1-\cos\theta_2)

, where

\theta_1

and

\theta_2

are the angles measured at the field point from the

+z

direction to the rim of the

+z

end and to the rim of the

-z

end. ((b)) Evaluate

B

at the centre of the coil, and compare it with

\mu_0 n I

. ((c)) Evaluate

B

on the axis at one end of the coil. ((d)) Evaluate

B

on the axis at a point

5.0\,\mathrm{cm}

outside the coil, beyond that end.