Bukhovtsev-et-al-Problems-in-Elementary-Physics

298 ANSWERS AND SOLUTIONS
407. The intensity of the field at an arbitrary point A on the axis of the
ring is .
E - Q - Q . z
- R2+ r2
cos Ct- R2sin ex cos a, (1)
(see Problem 406). . .
Obviously, E reaches its maximum at the same values of a, as the
. 2E2Rt
expression ~. But
2E2R4
--v= 2 sin- a, cos s a,=2 cos 2 a, (l-cos 2 ex) (I-cos 2 ex) (2)
is the product of three positive factors
a=2 cos! ex, (3)
b=1-cos- a, (4)
c= I-cos s ex (5)
whose sum is constant (a+b+c=2). and b=c.
The product abc=ab' will be maximum if the factors are equal, i.e.,
and. therefore,
Let us prove this. If
2
a=b=c=3 (6)
8
abc=ab 2 = 27 (7)
2
a=3+ 2d
where d is a certain number that. as follows from Eq. (3). can be within
_1. < d < ~ (8)
3 3
then, on the basis of Eq. (4)
2
b=--d
3
The product
ab l = (~ +2d) (~ -dr=2~+2tP(d-l)
is maximum, as follows from Eq. (8), when d=O. Hence,
2 V3
a=3 ana cosa=-a-
The maximum intensity of the field
will be observed c.t points at a distance
of r= ~2 R from the centre of the ring. This intensity is equal to
2V3 Q
Emax = - 9- R2