Homework #2 (due 02/06)
Boas Chapter 1
1.15; 10.24; 13.14; 13.29 (please give first 4 terms of the series); 15.3; 15.14; 15.28;
16.11; 16.17 (please give first 4 terms of the series); 16.22;
Extra-credit problem – Helmholtz coils
Consider a pair of two identical circular
magnetic coils of radius R that are placed
symmetrically one on each side of the
experimental area along a common axis x, and
separated by the distance d. Each coil carries
an equal electrical current I flowing in the
same direction. (Note: for all the calculations
in this problem consider points only along x-
axis.)
1) Start with the formula for the on-axis magnetic field due to a single wire loop (which
is itself derived from the Biot-Savart law):
Where:
= the permeability constant =
= total coil current
= coil radius
= distance from the plane of the coil on axis.
Calculate the value of the magnetic field exactly between two coils.
2) Find the ratio between the radius of the coils R and the distance between the coils d
which produces nearly spatially uniform magnetic field in the central region of the coils
(Helmholtz configuration). To find this ratio consider a small on-axis displacement ∆x
from the central point, present the magnetic field as a power series of ∆x. Then find the
ratio between R and d which makes the first non-trivial term of the expansion zero.
(Hint: your answer should look very simple!)
3) Use computer to plot on-axis magnetic field in the whole region between two coils,
and estimate the length (in terms of coils’ radius) of the central region with magnetic
field changing (a) less that 1% and (b) less than 5%.