Solutions 285
librium position determined by the condition
a 4 a,z
L zo = — mg, 2 0 = a4a2 •The frequency of these oscillations will be
a 2 cc
=
LmThe coordinate of the loop in a certain time t
after the beginning of motion will be
z
a4cc2[ 1+ cos (
a2ct3.55. The cross-sectional areas of the coils are
S 1 = :01/4 and S2 = aD, 21 /4. We shall use the
well-known formula for the magnetic flux (I) =
LI = BSN, which gives B = LI/(SN). Therefore,
B2 == L2 S iNi /2
B1 L1 S2N2 Ii•
But ./. 1 = I since the wire and the current source
remain unchanged. The ratio of the numbers ofturns can be found from the formula N (^1) /N 2 =
D 2 /Di. This gives
B2 L2S1D2 L2D1
LiSaDi L1D2 •
Therefore, the magnetic induction in the new coil is
B 1 L 2 D 1
B 2 =
Llp2
3.56*. Let N 1 be the number of turns of the coil of
inductance L 1 , and N2 be the number of turns of
the coil of inductance L2. It should be noted that
the required composite coil of inductance L can be
treated as a coil with N = N 1 + N2 turns. If the
relation between the inductance and the number of
turns is known, L can be expressed in terms of Li
and L2. For a given geometrical configuration of
the coil, such a relation must actually exist because
mgL