A conductor is the shape of a square frame with side `a` suspended by an elastice thread is located in a uniform horizontal magnetic field with induction `B`. In equuilibrium the plane of the fame is parallel to the vector `B` (figure). Having been displaced from the equilibrium position, the frame performs small oscillation about a vertical axis passing through its centre. The moment of inertia of the grame relative to that axis is equal to `I`, it electric resistance is `R`. Neglecting the inductance of the frame, find the time interval after which the amplitude of the frame's deviation angle decreases `e-` fold.

Explanation

If `varphi-=` angle of devition of the frame from its n ormal position, then an `e.m.f.`
`epsilon=B a^(2) dot(varphi)`
is induced in the frame in the displaced position and a current `( epsilon)/( R)=(Ba^(2)dot(varphi))/( R)` flows in it. A couple
`(B a^(2) dot (varphi))/(R). B. a.a= (B^(2)a^(4))/( R) dot ( varphi)`
then acts on the frame in addition to any elastice restoring coupel `c varphi`. We write the equation of the frame as
`Iddot(varphi)+(B^(2)a^(4))/(R)dot(varphi) c varphi=0`
Thus `beta=(B^(2)a^(4))/(2 I R)` where` beta` is defined in the book.
Amplitude of oscillation die out according to `e^(- beta t)` so time required for the oscillations to decrease to `(1)/(e)` of its value is
`(1)/(beta)=( 2 IR)/( B^(2)a^(4))`