PHYS-222 Worksheet 8 for Section 25 & 36
TA: Yang Li, leeyoung@iastate.edu
October 7, 2012
Problem 8-1
A Helmholtz coil consists of two identical circular magnetic coils that are placed symmetrically one on each
side of the experimental area along a common axis, and separated by a distance d (See Fig. (a)). Each coil
carries an equal electrical current I flowing in the same direction. Suppose the radius of each coil is a.
(a) Is the force between the two coils attractive or repulsive? attractive
µ0 Ia2
µ0 Ia2
~ along the axis?
(b) What is the magnetic field B
+ 2((d/2−x)
2 +a2 )3/2
2((d/2+x)2 +a2 )3/2
2~
B(x) (c) The distance d is chosen such that ∂ ∂x
= 0, so that the magnetic field B is as uniform as
2
x=0
possible. What is the best value of d, in terms of a? Then what is the magnetic field at the center
8 µ0 I
(x = 0) of the coils ? d = a, B = 5√
5 a
4
~
(d*) The magnetic field at the center of coils still has non-vanishing 4-th order derivative, i.e. ∂ B(x=0)
6= 0.
∂x4
To fix this, we can add another larger coil between the two. That makes a Maxwell coil. A Maxwell
coil generates more uniform magnetic field. Work out the spacial configuration for Maxwell coil.
(a) A Helmholtz coil
(b) forces
charges
bewteen
moving
Useful formula: (ref. textbook Eq. 18.15) the magnetic field generated by a loop ( with radius a,
carrying current I ) along the axis is
µ0 Ia2
B=
2
2(x + a2 )3/2
Problem 8-2
Two point charges q and q 0 move with velocity ~v and v~0 respectively. What is the magnetic force between q
and q 0 when their relative displacement is d (See Fig. (b))? Compare with the electric force.
0
µ0 qq 0 vv 0
1 qq
vv 0
fm = 4π
d2 , fe = 4πε0 d2 , fm /fe = c2 1
1