(a) A circular loop of wire of radius R
carries a current I. Find the magnetic induction B on the axis of the loop,
as a function of the distance z from the center of the loop.
(b) Use the result to find B at points on the axis of a solenoid of
radius R and length L wound with n turns per unit length.
(c) Use this result to find the self-inductance of a very long solenoid (L>>R).
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Show that the magnitude of the magnetic induction B at the center of a loop of wire carrying a current I and shaped like a regular plane polygon of 2n sides, the distance between parallel sides being 2a, is
using the Biot-Savart law. The n=3
case is shown.
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In practical magnetic structures, uniform magnetic fields are frequently necessary. The uniformity of the field produced by Helmholtz coils, or two co-axial loops which carry currents in the same direction, is one of their most important characteristics. Assume that the coils have a radius a, have axes on the x-axis, carry current I each, and are separated by a distance b.
(a) Find the magnetic field at a point P on the axis of the loops and a
distance x from the midpoint O.
(b) Expand the expression for the field in a power series, retaining terms to order x2.
(c) What relationship must exist between a and b such that the x2
terms vanish? What is the significance of this?
(d) Show that the field created by the coils to this order and under the conditions
established in part c is given by Bx=8I/(53/2ae0c2).
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A point magnetic dipole m in vacuum (region 1 in the diagram below) is pointing toward (and is normal to) the plane surface of a material with permeability m (region 2). The distance between the dipole and the surface is d.
(a) Use the method of images to find the magnetic field B in both regions, as follows: Place an image dipole m’=am a distance d into medium 2 and take the field B1 in region 1 to be due to dipoles m and m’. Take the field B2 in region 2 to be due to a single dipole m"=bm at the location of the real dipole m. Solve the boundary value problem at the interface to evaluate B1 and B2.
(b) Describe physically how each of the image dipoles m’ and m" arise and the role they play in determining the fields and the forces on the real dipole and the material of medium 2.
(c) Evaluate the force acting on the dipole m.
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A circular loop of wire with radius a=1cm and center at the origin is bent, so half lies in the y-z plane and half in the x-y plane. A current I=2A flows in the wire.
(a) What is the magnetic moment of this loop?
(b) What is the magnetic field at (x,y,z)=(3,4,0) meters from the origin?
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